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The puzzle is born in the Far East, in the vast area of the Indochina region.

The age of birth of the first puzzle is unknown because we do not have historical books that describe in detail the birth of this type of game. We have the first historical data from the journey undertaken by the Venetian Marco Polo who reached China by land and returned to Venice by sea between 1271 and 1295. His book "Il Milione" made a massive contribution to introducing Europeans to the central and eastern regions of Asia, history, culture and craftsmanship.

VelieroAnother important contribution to our knowledge of the East is attributed to Vasco da Gama, a Portuguese explorer, the first European to sail directly to India rounding the Cape of Good Hope. Starting from that time, in Europe we begin to learn about the first mind puzzles.

From the historical data we have, it is possible to understand that the puzzles were born and developed in parallel with the Buddhist religion, born in the 6th century BC in India. In the Buddhist religious tradition solitary and reflective games help to meditate, relax the mind, distract from everyday problems and enter into a relationship with God.


Lucchetto_medio_ordienteThe oldest puzzle known comes from Greece and appeared in the 3rd century b.C. The game consists of a square divided into 14 parts and the purpose is to generate different figures in these parts. This is not easy to do.

In Iran "puzzle-locks" were made before the 17th century a.D.

The next mind puzzle event known is recorded in Japan. In 1742 there is a hint in a book of a game called "Sei Shona-gon Chie No-Ita". Around 1800, Tangram puzzle became popular in China and 20 years later it also spread to Europe and America.

The Richter Company from Rudolstadt began to produce in large quantities puzzles similar to Tangram, made in different forms, also called "Anker-puzzle".

Libro_puzzleIn 1893 Professor Hoffman wrote the book "Puzzles, Old and New" which, among many other things, contained more than 40 descriptions of puzzles with secret opening mechanisms. This book became a real reference point for this kind of puzzles and was the basis for modern modifications.

The beginning of the 20th century was a period in which mind puzzles became fashionable, so the first patents were registered. The model shown in the image, made up of 12 identical parts by W. Altekruse in the 1890, is an excellent example.

After the invention of materials such as plastic, which is very easy to model, the variety of puzzles developed. For example, the probably most famous mind puzzle in the world, the Rubik's Cube, would not be possible without modern polymers.


Tangram is China's most famous puzzle. His Chinese name is Qiqiao Bang 七巧板, which means "seven ingenious pieces."
In the early XIX century, traders who arrived in Canton on sailing ships from Europe and America returned home with beautiful ivory versions of the puzzle. And very quickly that of the Tangram became the first international puzzle mania, comparable to that of the Rubik's Cube in more recent times. Among his admirers we find Lewis Carrol and Edgar Allan Poe.
Tangram is also known as "The Seven Stones of Wisdom" because the mastery of this game was said to be the key to gaining wisdom and talent.

History and legend

There is a legend about the origin of the game, it says that a monk gave his disciple a square of porcelain and a brush, telling him to travel and paint on the porcelain the beauties he would have encountered on his way. The disciple, excited, dropped the square, which broke into seven pieces. In an attempt to recompose the square, he formed interesting figures. He understood from this that he no longer needed to travel, because he could represent the beauties of the world with those seven pieces. Another legend of unknown age tells a mysterious story that happened in a Chinese monastery where a boy came in one day to learn Buddhism and learn about himself. The boy was assigned to a master who gave him a square ceramic plate. The disciple, in transporting the gift to his cell, dropped it and so the plate broke into seven pieces of perfect shape: various triangles, a square and a parallelogram.

The boy ran to his teacher crying and with great regret showed him the pieces, apologizing for the destruction of the gift.
His teacher did not scold him and told him with wise tranquility: "When you know how to assemble these pieces to form the perfect square that was, you will get the wisdom you were looking for in this monastery."
Thus, taking this legend as a reference, still today the Tangram is often called "the game of wisdom".
There are many stories that describe the origin and age of the game.
Various books speak of his birth. The book by the English researcher Sam Loyd, written in 1903, claims that there is a 4000-year-old legend about the Chinese god Tan which describes the creation of the world and the origin of the species in his seven books.

However, besides legends, there is also a research by Jerry Slocum considered official, which indicates that Tangram was invented in China between 1796 and 1801.
It was later brought to Europe by nineteenth-century English merchants, who had a strong commercial connection to the tea trade with China, the game became very popular at first in England and later in France, Italy, Germany , Holland, Switzerland etc... Around 1817 Tangram was brought to the United States and from there we now know celebrities of the century who were particularly fond of this intelligent game: Lewis Carrol and Edgar Allan Poe.

The origin of Tangram

Many Chinese scholars believe that the roots of Tangram date back to the Northern Song dynasty (960-1127), when the scholar Huang Bosi (1079-1118) invented a set of rectangular tables and a collection of illustrations that showed many possible combinations to arrange these tables, which would have welcomed banquet guests. There were seven tables in the set, and they were made in three different lengths.


Photo 1: The three dimensions of the banquet tables
Photo 2: A combination with all seven tables
Photo 3: Two other combinations for banquets

Huang Bosi's study of banquet tables led to the creation of another more versatile set of tables during the Ming dynasty (1368-1644). These were called "butterfly tables" and were described by Ge Shan in his book written in 1617. There were a total of thirteen tables in the set, and they had six shapes of triangles and trapezoids and different sizes. Ge Shan called them butterfly tables because their angular shapes resembled butterfly wings.


Photo 4: Butterfly tables form a square
Photo 5: There are thirteen tables in six different shapes
Photo 6: They look like butterfly wings

A simplified version of butterfly tables appeared around the end of the XVIII century. It is indeed the tangram puzzle that we know today. The first known tangram schemes were published in 1813 in the "Complete Book of Tangram Schemes" by Bi Wu Jushi with illustrations by Sang Xia Ke.


Cover and pages with Tangram diagrams of the book written by Bi Wu Jushi with illustrations by Sang Xia Ke, published in 1813.

The beginnings of Tangram Puzzle


Tangram carved in ivory with its original box purchased in Canton, in 1802 approx.
Courtesy of Ryerss Museum, Philadelphia

A set of Tangram ivory carved pieces was brought to America around 1802. It was probably purchased in Canton by an employee of Robert Waln (1765-1836), a prominent importer from Philadelphia who was in business with China with at least twelve ships trading with Canton between 1796 and 1815. On the silk brocade that covered the box is written "F. Waln 4 April 1802" and the puzzle may have been a gift to Francis Waln (1799-1822), the fourth child of Robert and Phebe Waln.
Other Western merchants who operated in Canton also brought home Tangram Chinese puzzles and even books on the subject, and soon throughout Europe and America the Tangram-mania broke out and raged. During 1817 and 1818, books on Tangram were published in England, France, Switzerland, Italy, the Netherlands, Denmark, Germany, and the United States.


Detail from a print by Wu Youru, 1892

In China, the popularity of Bi Wu Jushi and Sang Xia Ke's work created many new tangram enthusiasts and entrepreneurs who commercialized this game. They created additional tangram figures and published their own collection of diagrams. During the last half of the Qing dynasty, Tangram enjoyed great popularity among ordinary people, scholars, and the rich, including the Imperial Family. Imaginative tangram sets were also produced in the laboratories of Canton for sale to foreign merchants eager for curiosities to take home to their families, friends and sponsors.

Tangram tables

Between the middle and the end of the Qing dynasty, series of high-quality tangram-shaped wooden tables were created and sometimes enriched with inlay or briarwood or with marble shelves. While it is certain that Tangram puzzle descends from the Huang Bosi's Banquet Tables of and the Ge Shan's Butterfly Tables, there is no evidence to show that Tangram tables preceded the Tangram puzzle or vice versa.
There are two places in China, where sets of ancient tangram tables are still on display to the public. Suzhou in Jiangsu province is well known as an ancient artistic, teaching and cultural center. There are also many famous gardens, including the Persistent Garden (Liuyuan). Inside one of the pavilions of this garden are those that at first appear to be two square gaming tables with removable wooden covers. On a cover we find a table for playing Chinese Chess (Xiangqi) and on the other a board for Go (Weiqi). But by removing the two covers you discover a complete set of Tangram tables. Under a coverage there are two large triangular tables, and under the other there are tables in the shapes of the five small pieces of Tangram. The tables are made of Blackwood wood (hongmu) in the typical Suzhou style, with marble shelves inserted and the space between the table legs, at the bottom, is worked with textures called "crushed ice".


Tangram Tables of the Persistent Garden, Suzhou.


Tangram Tables at the Summer Palace, Beijing.

Beijing hosts as well a collection of Tangram Tables. Even these tables are a bit hidden, because they are in a closed building and you can see the whole set only by peeking through windows. Fortunately the building, called the Hall of Dispelling Clouds (Paiyundian), located in the Summer Palace (Yiheyuan) is accessible to the public. The Hall of Dispelling Clouds was built in 1750, rebuilt in 1890, and was the hall where the Empress Dowager Cixi's birthday party was held every year. Four complete sets of Blackwood (hongmu) wooden tables are on display, twenty-eight in total. There are two complete series with ten tables arranged to form a large hexagon and four tables arranged in two pairs. There are also two sets of smaller Tangram Tables, arranged in a group of ten and another group of four.

Tangram Condiment Dishes

piatto_tangram_portacondimentiDuring the XIX and early XX centuries, Tangram was so popular that condiment sets were made in the shape of the seven tangram pieces. The seven plates were generally placed in a square box with a specially made lid and were used to serve guests during Chinese New Year and other special occasions. Chen Liu, a civil servant and porcelain collector, described the Tangram Condiment Dishes in 1910 as follows, in a book that is a landmark on the theme of porcelain Notes on Porcelain (Tao ya):
"rice cakes and condiments, commonly called 'pastries' and also known as 'cold food' are distributed in tangram-shaped porcelain dishes, and are therefore called 'split plates', more commonly known as 'condiment dishes'... Some pieces are made in colored exported porcelain with flowers and birds, an unsurpassed craftsmanship".

piatto_tangram_portacondimenti2The furnaces of Jingdezhen, the Chinese capital of porcelain, produced tangram condiment dishes sets, painted plates and cups in a remarkable variety of sizes and styles. The plates may be large or small, deep or plain, and their parts may be vertical or non-vertical.

Small Jingdezhen tangram condiment dishes, Jiangxi; Qing Dynasty, Daoguang Kingdom (1821-1850) 19.7 x 19.7 x 2.1 cm


Large Tangram Condiment Dishes
37.0 x 37.0 x 10.0 cm

The clearest example of tangram dishes popularity is the wide variety of themes and patterns with which they have been decorated. Examples include scenes of stories and works, butterflies, birds and flowers, landscapes, mythical creatures, and calligraphic forms.


Tangram condiment dishes have been produced in a wide variety of decorative styles.
Tangram condiment sets, varnish plates and trays were also made of Yixing clay, lacquer, wood and Canton glaze.


il_giocoIt consists of seven tablets of the same material and color (called tan) which are initially arranged to form a square:

1. 5 triangles (2 large, 1 medium, 2 small)
2. 1 square
3. 1 parallelogram

The aim of the game is to form complete figures. The rules are quite simple:

1. Use all seven pieces in composing the final figure;
2. Do not overlap any.

Another use, inverse of the previous one, is to reproduce (solve) a composition of those present in the instruction booklet accompanying the game. The difficulty is due to the fact that the image of the composition is not of the same scale as the game tablets and that the sides of the individual pieces are not marked within the image since these are, unlike as illustrated in the side figures, of the same color and in adjacent places.

Tangram is known as "The Seven Stones of Wisdom" because it was said that mastery of this game was the key to gaining wisdom and talent.
Little or nothing is known about the origins of the game; even the etymology of the name is not clear.
There is a legend about the origin of the game, which tells of a monk who gave a disciple a square of porcelain and a brush, telling him to travel and paint on the porcelain the beauties he would have encountered on his journey. The disciple, excited, dropped the square that broke into seven pieces. In an attempt to recompose the square, he formed interesting figures. From this he realized that he no longer needed to travel, because he could represent the beauties of the world with those seven pieces.
By appropriately changing the pieces of Tangram, it is possible to obtain an almost infinite number of figures, some geometric, others that recall objects of common use, etc.


Photo 1: Running man
Photo 2: Rabbit

Educational aspects of the game

This application makes it possible to initiate, through a concrete experience, the intuition of concepts such as area conservation and area comparisons.
In the game there are several figures to compose.
Any figure made by Tangram must be made up using all seven pieces. The tiles can be moved to obtain figures with different but fair-extended shapes.
The tutor's task will be to solicit to recognize and highlight the equivalence of the figures, comparing the different forms previously obtained.
The rigid movements to be applied to the figures are:

- translation (hold down the left mouse button and drag the figure);
- 45° hourly rotation;
- overturning (only of the parallelogram).

Educational objectives

- represent through geometric shapes
- operate with flat figures
- recognize flat geometric figures, even if differently oriented in the plane
- compare surfaces
- experiment with surface conservation phenomena
- recognize fair-extended plane figures
- perform translations, rotations and overturns
- making compositions of isometries

There are various geometric relationships between the pieces of the Tangram.
Relations between areas:

- the large triangle has an area that is twice that of the medium triangle
- the medium triangle, the square and the parallelogram have the same area
- the medium triangle has an area that is twice that of the small triangle.

parallelogramma Angle measurements: 

- the square, logically enough, has four 90° corners
- the parallelogram has two 45° angles and another two of 135°
- the five triangles are isosceles rectangles, so they each have a 90° angle and two of 45° 

Relationships among sides:

- the cathetus of the large triangle has the same length as the hypotenuse of the medium triangle
- the hypotenuse of the small triangle has the same length as the long side of the parallelogram
- the cathetus of the small triangle has the same length as the side of the square and the other

These ratios, between the lengths of sides and the measures of angles, are those that make it possible to build thousands of different shapes through many combinations.

The Pythagorean Theorem


Tangram helps to introduce the basic Pythagorean Theorem in a very visual way and by playing with its constituent pieces one can approve the proportion between catheti and hypotenuse that subsists in a right-angled triangle.
The theorem reads as follows: the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

The visual demonstration of this statement consists in the fact that in one of the small squares of the picture, the square Tangram piece enters perfectly and in the other small square both Tangram small triangles can be placed. In the square built on the hypotenuse the medium triangle and the two small triangles exactly enter.
To confirm the fact that the sum of the small squares is equal to that of the large square, it is sufficient to recall the second point described in the relationship between areas: the medium triangle, the square and the parallelogram have same area.

Thus, given the composition of the squares and the equality of the areas, we affirm the Pythagorean Theorem.


Cuento. The Argentine fairy tale:

In a beautiful house  casa a boy lived ragazzo with the dog canethis guy was very cheerful and loved to dance ballarebut one day the dog got lost and the boy became very sad tristeHe drew a portrait of his dog and showed it to all his acquaintances uomo; someone told him  visto to have seen him near the pier; the boy ran towards the pier molothe dog, when he saw the master, ran towards him cane2and they both happily decided to take a boat trip together  barca

More Tangram animations online:

Tangram Love

Tangram on Music


Chen Liu. Tao ya (Notes on Porcelain). 1906.
Ge Shan. Dieji pu (Butterfly Table Diagrams). 1617.
Huang Bosi. Yanji tu (Banquet Table Diagrams). 1194.
Jean Gordon Lee. Philadelphians and the China Trade, 1784–1844. Philadelphia, 1984.
Bi Wu Jushi and Sang Xia Ke. Qiqiao tu hebi (Complete Tangram Diagrams). 1813.
Jerry Slocum. The Tangram Book. New York, 2003.
Adapted from:


Solitaire is a mind game. The inventor is not certain but several sources attribute the origin of the game to a Bastille prisoner. It is known that it was very popular and widespread in Europe in the 1800s, it was known as the "solitary peg" as it was played on a pegboard where small wooden pegs were moved and inserted.

The game consists in moving a pawn at a time along the horizontal or vertical lines, so as to "jump" the nearby pawn, which is thus eliminated from the game board. The "jump" of the pawn can be performed only if the destination place is free.
The game ends when you reach a point where you cannot perform other moves. If there is only one pawn on the board, the player win. An additional challenge is to finish the game with the last piece placed in the central position of the board.

Educational objectives: 

- run routes under pre-established rules
- determine sequences
- establish strategies

Victorian Solitaire

solitario_vittorianoThe aim of the game is to move all the blue pawns to the right and the red ones to the left. The pawns must be moved one at a time from one square to another adjacent which is free, horizontally, vertically or diagonally. It is also possible to move the pawns by skipping another.

To train online: Peg Puzzle

Fox and Geese (HALATAFL)

halatafiGrettis Saga, probably written by an Icelandic monk around 1300s, refers to a game called Hala-tafl which, from what we understand, corresponds to a type of game widespread then in the rest of Europe with the name of "Fox and Geese".

The players are two: the first plays with a pawn (the Fox), the other plays with thirteen pawns (the Geese); they move the Geese first with a pawn that will occupy any of the adjacent squares, as long as it is free.
At each following move, the Geese pawns may advance horizontally or vertically, but not diagonally.
The Fox, on its turn, moves one box at a time, but in any direction. The Fox catches the Geese by jumping the pawn and going to occupy the vacant space behind it.
The captured Geese are removed from the game. Geese cannot capture the Fox, but must try to immobilize it by preventing any move. In this case they won.
The Fox wins if it manages to capture so many enemy pawns to make the opponent harmless, that is unable to block its moves.

Online games are property.


pietra_molareThe origins of the game and name 

The name "Burr Puzzle", also known in Italy as "Master Cross", is the original English name which was initially known thanks to the English version of this game "Six Piece Burr", that is "Six pieces Cross".
Actually, we do not know the origins of this game for sure, even if we believe in an ancient story started in China from wooden cubic boxes whose faces fit together with the help of carved notches. For this reason some producers have called it "Chinese Puzzle". 

Known in China as "Lu Ban Blocks" (Lu Ban suo 鲁班 锁) or "Kongming Blocks" (Kongming suo 孔明 锁).
The interlocking puzzles form a variety of geometrically pleasing structures and are traditionally made of wood, bamboo or ivory.
Lu Ban (507-440 b.C.) lived in the "Spring and Autumn" period (771-476 b.C.) and was credited with the invention of the saw, the carpenter's board and a tool for marking straight lines. In China he is considered the Patron Saint of carpenters and the first Master of carpentry. Kongming was the brilliant strategist Zhuge Liang (181-234), the Prime Minister of Shu Han in the Three Kingdoms period (220-280).

Substantial traces are fairly recent and date back to 1917, when the first patent was registered in the United States, although in 1803 the Six Pieces Game appeared on the German catalogs of Bestermeier toys.
But it was in 1928 that the Six Blocks Puzzle appeared in the book "Puzzles in Wood" by the English researcher Edwin Wyatt and from here it began to be spread all over the world.
The origins of his Italian name "Croce del Maestro" remain unknown, although its three-dimensional cross shape is evident. The reference to the "master" probably refers to a carpenter capable of producing the pieces with recesses and fitting them together to form a compact cross. Some resources speak of an initiation ceremony that every apprentice of the legendary master had to face instead of the final exam.
Other theorists support the explanation of the Italian name for the fact that a component piece of the puzzle, without any indentation, is called "the master" as it is the first that allows the disassembly of the game but is the last to be inserted to form the so-called cross.


It is true that the interlocking puzzles share some characteristics with traditional Chinese carpentry, which was used first for the construction of buildings and subsequently for the production of furniture. The components of these puzzles fit together with hidden joints, and stick together without the use of glue or nails, allowing them to be easily disassembled and reassembled. However, there is no clear evidence that attribute the invention of these puzzles to Lu Ban or Kongming.


Interlocking Carpentry 
Shandong province, 19th–20th c.



The game is apparently simple, but it is not once it is disassembled. Looking at the six pieces of the puzzle, we will find some with which we do not even know how to start assembling it again.
The cross consists of six pieces modeled with complicated cubic cuts, which intersect without leaving any empty space between them in order to compose a three-dimensional cross. In the general assemblage, the pieces must be arranged parallel to each other in pairs. Three pairs intersect perpendicularly in their central section, so they are oriented towards each of the three orthogonal axes.
Once the puzzle is assembled, neither the assembly sequence nor the pieces can be seen.


The assembling

The assembly and disassembly process is perfectly symmetrical.
To fully understand and remember the process it is recommended to disassemble the puzzle, studying each movement and each piece with the utmost attention, trying to memorize the shape of the pieces and the sequence of the steps. If it is well analyzed, the assembly will be nothing but repeating the process in the reverse direction.
In this way we discover the crucial central intersection of the 3 pieces around which the remaining ones are assembled, in order to conclude the assembly with the "master" piece.
The fact that the cuts of the pieces are all different increases the difficulty of the game, but at the same time it helps to memorize them. Furthermore, it must be remembered that there must be no empty spaces inside the construction.
The assembly instruction is not the only possible sequence. Each player creates his own sequence of steps and memorize them according to his predisposition.

Combinatorial analysis

Burr Puzzle and its six simple pieces is a game that has triggered mathematical research that provided scientists with a strong resource for developing the Combinatorial Analysis that is now used in every computer on the planet. Analyzing the cross and using a binary numbering system, it is possible to create 369 pieces with or without indentations that can form the known shape of the three-dimensional cross in 199.979 ways without any empty space inside.
It remains to be said that many modern studies in aerospace, medical and mechanical fields are unthinkable without the application of this highly sophisticated branch of mathematics: Combinatorial Analysis on the binary basis. Today's powerful computers automatically calculate all the combinations of "component parts" in all forms to find all the possible ways to build the final product. The optimization system concludes the work, choosing the best way.
In fact, there are Burr Puzzles composed of more than 6 pieces, for example of 24 pieces. Assembling such a puzzle without a Combinatorial logic would be unthinkable, because it may be necessary to try out thousands of combinations of crossing pieces to find the right one.


The Japan Pavilion at the 2015 Milan Expo realised by Prof. Atsushi Kitagawara

Interlocking Burr Puzzles and Chinese Joinery:



The first airplane, properly so called, was built in 1903 when in the United States the Wright brothers succeeded in flying with a sort of glider with a 16 horsepower engine. This first flight lasted 12 seconds, reaching a height of about 40 meters.
Most of the scientific and aeronautical community considers the French Santos Dumont as the "Father of Aviation" due to the fact that his aircraft took off thanks to the driving force of the propeller, while the Wright brothers' airplane was simply thrown.
However, in Italy the first aircraft was built in 1908.
Initially the plane was considered a simple curiosity for enthusiasts but, little by little, its capacities began to be recognized and were built models capable of performances considered impossible relatively short time before: flying over the Alps, fly over the English Channel or simply reaching ever higher heights and speeds.
For this reason the beginning of the development of aeronautical technology is linked to sporting events that aimed to set new records. In these early years airplanes were driven by piston engines connected to a propeller and the structure was biplane, or with two wing planes.


This original puzzle is an excellent educational resource that, through the assembly of the plane, develops the capacity for Combinatorial Analysis and three-dimensional imagination.
During disassembly, the child is taught the names of each piece and its correct position on the aircraft. Knowing them well, the game is no longer a difficult undertaking.

The pieces:


Wing position

According to the position in comparison to the fuselage, the wing can be:

- High: placed above the fuselage
- Medium or transversal: placed near the median of the fuselage (as in our game)
- Low: below the fuselage.


The position of the wing is an important factor of stability. A high wing makes the aircraft more stable, because this is "hanging" on the wings: its center of gravity is lower than the point of application of the lift, so the aircraft tends to return to a stable position.



The name of the game "Magic Numbers" has a mathematical basis and is also known as the "Magic Square".
The oldest Magic Square dates back to Ancient China, at the time of the Shang dynasty in 2000 b.C. when, according to legend, a fisherman found a turtle along the banks of the river Lô, a tributary of the Yellow river, an animal considered sacred, bearing strange geometric signs engraved on its shell. The fisherman brought the turtle to Emperor Yu and the mathematicians who were at his service, studying those signs, discovered an unpredictable structure: a square of numbers with the constant sum of 15 on each row, column and diagonal. This numerical square was baptized "Shu" and became one of the sacred symbols of China, a representation of the most arcane mysteries of Mathematics and the Universe.


The signs on the shell of the turtle and their translation into numbers:


This magic square, called Lo-shu, meaning "The sage of the river Lo", was made not with figures, but with small circles inside each box. With that type of graphic (see picture above on the left) the Lo-Shu subsequently became also a form of ornamentation in many areas of Asia, taking on a symbolic and propitiatory value linked to the belief that such a magic square, engraved on a precious metal plate or leather and worn around the neck, could protect against serious illness and calamity.
This tradition continues today in some countries of the East, where these symbols are also engraved on everyday tools such as bowls and containers for storing herbs or medicinal potions. The square of Lo-shu, in the picture above on the right with digits rather than circles, has 15 as a constant (every sum at the end of a line, column or diagonal is equal to 15).

elementiChinese attributed a mystical meaning to its mathematical properties, so much to make it the symbol that brought in itself the first principles that formed things, men and the universe together and that are still present in it. Thus the even numbers symbolized the feminine principle of the Yin, while the odd ones the male of the Yang. In the middle there is number 5 which belongs to the two diagonals, to the central column and line: it represents the Earth. The four main elements are distributed all around: Metals symbolized by 4 and 9, Fire indicated by 2 and 7, Water from 1 and 6 and Wood from 3 and 8.


The mathematician Cornelio Agrippa (1486-1535) devoted himself to the construction of magic squares of order higher than two, in fact he built magic squares of order 3, 4, 5, 6, 7, 8, 9 and attributed them an astronomical meaning: they represented the seven planets then known (Saturn, Jupiter, Mars, the Sun, Venus, Mercury and the Moon).
One of the most famous magic squares is certainly the one that appears in Dürer's engraving, Melancolia I, where a thinking scientist from the Renaissance era is depicted and there is a magic square of order 4 in the right-hand corner of the image.


Frenicle de Bessy (1605-1665), a French mathematician friend of Descartes and Pierre de Fermat, in 1663 calculated the number of perfect magic squares of fourth order: 880, with constant sum 34, on rows, columns and diagonals. Only in 1973, thanks to computers, it was possible to extend the result to higher orders: the magic squares of order 5 are 275.305.224.
The precise number of magic squares of order 6 is not known, although many are committed to their determination. According to some surveys, their number is in the order of 1.7754×1019. However, the main problem of finding the rule that allows to determine the number of magic squares of order n remains.
Very similar to magic squares is the magic cube, built in Europe for the first time only in 1866. The first perfect cube of order 7, and therefore containing the first 73 = 343 positive integers, was obtained by a missionary passionate about mathematics. Later the search was extended to hypercubes of size m and order n, each consisting of nm integers.

The article Magic of numbers has been prepared thanks to material extracted from the websites listed below, where more information can be found:


The story of the puzzle:


This puzzle is also known as "Chinese Rings" or "The Devil's Chain".
It is a very popular puzzle even in modern China, sold everywhere as national entertainment.
The goal is to untangle all nine rings and the solution requires 341 moves, so a lot of patience is needed. But there is a method for the solution and once you learn to solve the puzzle it's hard to forget!

anelli_cinesi2The challenge launched by this very ancient puzzle is more difficult than it seems. To solve it you require good concentration and exceptional patience, because in this continuous sewing and unstitching, the minimum number of movements required doubles for each new ring added.


Although it was not established when the Rings and Swords puzzle was invented, the concept of untangling linked rings was incorporated into Chinese culture, at least from the Warring States period (475-221 b.C.), when the philosopher Hui Shi (380-305 b.C. aprox.) stated, "the connected rings can be separated". Hui Shi's solution has been lost, but this paradox has been passed down to us by other writers.
A history belonging to the China Warring States period, dating back to the time of the Han Dynasty (206 b.C.-220 a.C.), contains an episode involving King Zheng of the Qin kingdom, the man who would later become Qin Shi Huang, the first emperor of China. King Zheng sent an emissary to present a series of jade rings linked to the widowed Empress of the Qi kingdom. The king's message said: "Qi people are smart enough, but could them untangle these rings?" The Empress showed the rings to her ministers, but none of them managed to untangle them. The Empress then took a hammer and broke the rings, thanked the emissary of Qin and said: "Now they are untangled!".


A painting by Louis-Émile Pinel de Grandchamp (1820-1894) shows two Parisian girls in a luxurious veranda playing with the nine bound rings puzzle while the others watch.


During the Ming dynasty (1368-1644) the poet Yang Shen (1488-1559) wrote that the story of the widowed Empress breaking the rings with a hammer was all an invention: "If this had been true, she would have simply been a stupid woman who thinks she could outsmart the Qins like that. The rings were an ingenious idea of jade artisans. There are two rings tied in one piece, but it can be untangled in two". Then he continues: "Nowadays, we also have an object called Nine Chained Rings. It is made of brass or iron instead of jade. It is a toy for women and children." This reference dating back to the XVI century is the oldest Chinese mention of the puzzle of the nine bound rings known.

The game in Europe

pacioliThe first western description of a connected ring puzzle known is by the Italian mathematician Luca Pacioli (1445-1517), who was a friend of Leonardo da Vinci. This description appeared in the manuscript of Pacioli De Veribus Quantitatis, written around 1510. Pacioli states that "it can be composed of three rings or even many others, as many as you want" and includes a solution for the case of the seven rings. Pacioli's description goes back a few years before Yang Shen's, so this raises the question of whether puzzles with linked rings originate in the East or in the West. Without further proof, it is impossible to say.


The nine-ring puzzle came into the Palace!

In 1713, during the Qing dynasty (1644-1911), a nine rings linked puzzle was donated to the emperor Kangxi (who reigned from 1662 to 1722) for his sixtieth birthday by the third daughter of his seventh son, Prince Chun.

Pu Yi (1906-1967), who took the throne in 1908 at the age of three to then become the last emperor of the Qing dynasty, had a similar game in silver with jadeite rings.
This game is mentioned in the most popular novel of Chinese literature, Dream of the Red Chamber, written by Cao Xueqin (1715-1763) around 1760 and published in 1791. It contains a passage involving the two main characters, in which Daiyu was in the room of Baoyu trying to untangle the nine rings tied together.

The songbook Echoes From The White Snow, written by Hua Guang Sheng in 1804, contains a song that refers to the nine bound rings puzzle:

"My lover gave me nine bound rings.
With my two hands I couldn't untangle them, I couldn't unravel them.
My lover, please, untangle my nine bound rings, nine bound rings.
I will marry you and you will be my man".

The painter Yu Ji (1738-1823) was born in Hangzhou and gained fame in Beijing for his portraits representing elegant ladies. In 1807 he painted a lady holding a puzzle with nine tied rings. This portrait was purchased in Yangzhou in 1893 by the German sinologist Friedrich Hirth, who believed it was the copy of a painting by the master Tang Yin (1470-1523) of the Ming dynasty.

Around 1821, a writer who called himself Zhu Xiang Zhuren published six volumes of activities for girls and young women entitled Fragments of Wisdom. It included an illustration of a nine bound rings puzzle and two graphs showing the recursive nature of the puzzle solution.


Nine Bound Rings solution in Fragments of Wisdom by Zhu Xiang Zhuren, 1821

A puzzle with many names


In the book by Ch'ung-En Yù we talk about the Nine Rings puzzle, since this is the number of rings in the traditional version. This version is the most complicated known.
In China, at the beginning of the XX century, the puzzle was called Lien Nuan Chuan, meaning "rings of intertwined rings". In Europe, the puzzle took the name of Baguenodier, a French term that indicates a person who likes to waste time, such as loitering and browsing.

Perhaps this refers to the time it takes to solve the puzzle. It is also known as the Devil's Chain, because the act of separating the bolt from the rings may become something diabolical. It could also be maned as the Cardano's Rings puzzle, from Cardano's quote in his De Subtilitate. And finally, like the Chinese Rings puzzle.

This puzzle also became very popular in Scandinavia, where it was used as a padlock. In Norway it has had this function for centuries and in the National Museum of Finland it is exhibited as a traditional toy.


A formula to calculate the number of steps.


The Chinese Rings puzzle could be simplified with the elimination of some rings, or make more complicated with the addition of others. The more rings there will be, the higher the number of passes required to solve the game.

How many steps does it take to solve the nine-ring puzzle from the starting position?
Here is the formula for the calculation, where X is the minimum number of steps necessary to solve the snap break and n is the number of rings. This is really the minimum number, as it takes an astronomical concentration to solve the nine-rings game without losing the way at least once.
The game sold by LOGICA GIOCHI has nine-rings and it is packaged with an original Chinese box.


VERSION 1 (metal)
VERSION 2 (wood+metal)


Read more about Chinese Nine Linked Rings Puzzle



Gordius, in Greek mythology, was one of the kings of Phrygia.
But it must be borne in mind that in the mythology the kings of Phrygia were alternately called Gordius and Midas.
It is also the eponymous name of a Phrygian city (inhabited from the VIII to the II century b.C. and located in the current village of Yassihüyük in Turkey), linked to the famous anecdote of the intricate Gordian knot dissolved by Alexander the Great.

Gordius, king by chance. 
In mythology, the first Gordius was a factor. When an eagle landed on his plow Gordius interpreted the fact as a sign that he would one day become king. The oracle of Sabatius (identified by the Greeks with Zeus) confirmed its future destiny in the following manner: the Phrygians, finding themselves without sovereign, consulted the oracle and they had as response that they should elect as king the first man who had risen to the temple with a cart. Thus the Gordian factor appeared, on his ox-driven cart.

The eponymous founder. 

Gordius founded the homonymous city of Gordius, which became the capital of Phrygia. His cart was kept in the city acropolis. His yoke was secured with an intricate knot called "Gordian knot" or "king Gordius' knot".
Legend had it that whoever managed to untie that knot would become lord of Asia or of the then territory of Anatolia. Instead, in 333 b.C., Alexander the Great cut the knot in half with his sword.
Since then the expression "Gordian knot" designates an insurmountable difficulty, which can only be solved with extreme determination (as Alexander did, which instead of unfastening it broke it with a slash).

Full article: Wikipedia, the free encyclopedia



A long time ago there was no one on Earth and the Gods reigned over an empty world. They lived on Mount Olympus, in rooms made of clouds and sunbeams. When they looked down, they saw oceans, islands, woods and mountains, but nothing moved because there were neither animals, nor birds, nor men.

One day Zeus, King of the Gods, ordered Prometheus and his brother Epimetheus to make living beings and sent them both to Earth.
Epimetheus made turtles and gave them shells. He made horses and gave them a tail and a mane. He made the anteaters and gave them long noses and even longer tongues. He made birds and gave them the ability to fly.

Epimetheus was a very good craftsman, but his brother Prometheus was even more so.
While Epimetheus worked, Prometheus watched.
When Epimetheus finished creating all the insects, fishes and other animals, it fell to Prometheus to make the last living thing.
He took the earth, mixed it with water and molded the First Man with mud.

"I will make it like us, with two legs and two arms. And I want it to walk straight and not on four legs. All the animals look at the earth, but the Man will look at the stars!"

When he finished, Prometheus was very proud of what he had done. He looked for something to give to Man, but alas, there was nothing left.

"Give him a tail," Epimetheus suggested.
But all the tails had been distributed.
"Then a trunk," Epimetheus proposed.
But the elephant already had it.
"How about a nice fur coat?"
But even those had already been shared.

prometeo_fiammaSuddenly Prometheus exclaimed: "I found it! I know what to give!"

He went up to the sky, up to the Chariot of the Sun. He approached a burning wheel and stole a tiny flame. It was so small that it managed to hide it in a reed. Then he went back to Earth: nobody had noticed what he had done. But the secret didn't last long.
When Zeus looked again from the top of Mount Olympus, he saw something red and yellow sparkling under the column of gray smoke.

"Prometheus, what have you done?" He cried furiously.
"Did you give those mud beings the secret of fire? It was not enough for you to have made them like us? You also wanted to share with them what belongs only to the Gods. Are those beings of mud more important than us? I will make you regret have them made! I will make you regret being born!".

prometeo_legatoSo Prometheus was tied to a rock and Zeus decided that the eagles would catch him every day. In its place, anyone would have died.

But Gods do not die and Prometheus was a God. He knew that his pain would never end, that eagles would never stop or the chains break. In his heart there was no hope and this made him suffer much more than the eagles.
Zeus was also enraged with Man because he had accepted the gift of fire, but he did not make him understand. Indeed, he prepared a wonderful gift for him.

With the help of the other Gods, he made the First Woman. Aphrodite gave her beauty, Ermes taught her to talk and Apollo taught her to play very sweet music.
Zeus called the First Woman "Pandora" and covered her head with a veil. Then he sent for Epimetheus, who was not smart enough to suspect a trap.

"Here is a bride for you", said the King of Gods.
"I want to reward you for making all the animals. I also brought a wedding gift for you both. But I warn you: never open it!"

The gift was a box closed with a padlock.
When he arrived home, at the foot of Mount Olympus, Epimetheus put the box in a dark corner, threw a blanket over it and forgot about it. After all, with a beautiful wife like Pandora, what more could he desire?


At that time the world was a beautiful place. No one was sad, no one grew old or sick. Epimetheus and Pandora were married and he gave her everything she wanted.
But sometimes, when her eye fell on the box, Pandora said: "What a strange wedding present. Why can't we open it?"
"It doesn't matter. Remember well: never touch it", Epimetheus always replied decisively.
"Never, ever. Have you understood correctly?"
"Of course. I will never touch it. It's just an old box... What do you think is in it?"
"It should not interest you".

Pandora tried, but one day, while Epimetheus was out, the box came back to her and, who knows why, she went to look at it.

"No!" she said to herself. "I promised Epimetheus that I would never open it."
Then she returned to the housekeeping.
Suddenly ... "Let us out!"
"Who spoke?"
"Let us out, Pandora!"
Pandora looked out of the window. But in her heart she knew that the voice came from the box. She pushed aside the blanket that covered it with trembling hands.
The voice grew louder: "Please, oh, please, let us out, Pandora!"
"I can't. I don't have to," Pandora said sitting next to the box.
"Instead you have to. We want you to do it. Help us, Pandora!"
"But I promised!" she exclaimed, as her fingers brushed against the box.
"It's easy," said a little voice that resembled a cat's meow.
"No! No! I don't have to!" Pandora said.
"But you want, Pandora. And why shouldn't you? This is your wedding present ... Anyway, if you really don't want to, forget it. But a single glance ... what harm can it do?"
Her heart was pounding. She opened the box and Pandora was thrown to the ground by an icy wind.
In an instant the wind invaded the room howling. The curtains tore. And, after the wind, disgusting creatures came out of the box, roaring and snarling and having sharp claws and frightening faces. They were bad and horrible.


"I am Illness", said one.
"I am Cruelty", said another.
"I am Pain and that is Old Age."
"I am Disappointment and that is Hate."
"I am Jealousy and that is War."
"And I am Death!" Said the little voice that resembled a cat's meow.

Trembling like a leaf, Pandora violently closed the box but someone remained inside.
"No, no, Pandora! You are making a mistake in closing me here. Let me go!".
"No way! I don't buy it anymore", sobbed Pandora.
"But I am Hope!", the creature whispered.
"Without me the world will not be able to bear all the unhappiness you released!"

Pandora reopened the box and a little white thing, small like a butterfly, fluttered out and was tossed here and there by the wind that kept whistling. It was Hope, who flew out of the window and immediately a pale sun came out of the clouds that illuminated the devastated garden.

Chained to the cliff, Prometheus could do nothing to help the mud beings he made. He pulled with all his strength, but he could not free himself.
The men's cries of pain rose to him. Now that those evil creatures had been released, men and women would no longer have happy days and peaceful nights. They would become rude, suspicious, greedy and unhappy. And, one day, they would die and come down into the cold and dark Afterlife.

Thinking about all this, Prometheus's heart hardly broke.
But here... a little white light sparkled in front of his chariots. A little thing as light as a butterfly touched his chest.
Hope landed on his heart. Prometheus felt stronger as his courage returned. His heart wouldn't break.
"A lot of bad things have happened today, but it doesn't matter. Maybe tomorrow will be better," he said to himself.
"One day someone will pass by here, he will take pity on me and break these chains. One day it will happen!"

The eagles tried to catch the little white light, but they weren't fast enough and Hope flew off to go and bring her little flame into the world.

Pandora Box Puzzle >>>



The puzzle as we know it today was invented in 1883 by the French mathematician Edouard Lucas D’Ameins, famous for his studies on prime numbers and for analyzing the Fibonacci sequence. Lucas, to make his game even more fascinating, reported the curious legend of the Tower of Brahma (as the game is also called) and commercialized the puzzle concealed under the pseudonym of N. Claus de Siam, mandarin of Li-Sou- Stian college in Tonkino (Northern Vietnam).
We also see his passion for games from this particular joke: N. Claus De Siam is actually the anagram of his surname, and Li-Sou Stian is the anagram of the city where he taught, Saint Louis.



The legend tells that, at the beginning of time, Brahma (The God Creator of the Indian Sacred Trimurti, a trinity that also included Shiva and Vishnu) led to the great Kashi Vishwanat temple in Varanasi (Benares), under the golden dome set in the center of the world, three diamond columns fixed to a bronze plate and sixty-four gold disks, placed on one of these columns in descending order, from the smallest at the top to the largest at the bottom. It is the sacred Tower of Brahma that engages the priests of the temple day and night in transferring the disk tower from the first to the third column.

They must not bend precise rules, imposed by Brahma himself, which require you to move only one disk at a time and that there should never be a disk on a smaller one.

When the priests will complete their work and all the disks will be rearranged on the third column, the tower and the temple will collapse and it will be the end of the world.


If we calculate the number of steps needed to move the disks, with the formula given in the text, 264-1, we get 18.446.744.073.551.615 movements.

In the event that the priests would use a second for each step, it will take more than five billion centuries (according to the calculations of Lucas himself) to transport all the disks from one column to another.
Therefore we are safe, at least from this point of view, for our future.

In other words, even assuming that the monks would make one step to the second the world will end between 5.845.580.504 centuries, so long that when the sun becomes a giant red ball and burns the Earth, the game will not have been completed.

The general solution is given by the following algorithm:

Recursive algorithm

The basic solution of the Hanoi Tower game is formulated in a recursive way.
Let be the poles labeled with A, B and C, and the disks numbered from 1 (the smallest) to n (the largest). The algorithm is expressed as follows:

1. Move the first n-1 discs from A to B. (This leaves the disc n alone on pole A)
2. Move disc n from A to C
3. Move n-1 disks from B to C

In order to move n disks, it is required to perform an elementary operation (displacement of a single disk) and a complex one, that is the displacement of n-1 disks. However, even this operation is solved in the same way, requiring the movement of n-2 disks as a complex operation. Iterating this reasoning reduces the complex process to an elementary one, that is the displacement of n- (n-1) = 1 disk.

This is a recursive algorithm of exponential complexity.

It is interesting to note that the puzzle could be solved for any "n", with a demonstration by recurrence: let suppose we have a tower in A composed of N disks, and suppose that N is the maximum number of disks allowed to solve the game. At the end of the displacement of the tower from A to B, we add an additional disk to A, of size equal to N + 1, and we assume that this disk has been stopped all the time under the others. At this point, let simply move the disc from A to C, and we will certainly be able to move the tower from B to C, following the same steps that were necessary to move it from A to B. Having shown that a tower of N disks is displaceable from one column to another, it is shown that you can also move a tower of N + 1 disks.

Psychological aspects

This puzzle is used in psychological research, in particular by solving problems. It is also used as a neuropsychological test.


This test is able to detect malfunctions of the frontal and prefrontal area and allows to evaluate executive functions such as planning, work, memory and inhibition. The solution of the Tower of Hanoi game depends on the potential for inhibition, on the "working memory", that is the use of short-term memory, on procedural memory and fluid intelligence.
This test is similar to that of the Tower of London, as well as that of the Toronto Towers, used primarily to assess strategic decision-making and problem solving skills in children aged 4 to 13 and to study the effects of aging on resolution of simple problems.

The Tower of Hanoi puzzle is very much played online, you can find many forms of this game, both in Flash and in Java.

Tower of Hanoi Puzzle >>>


tetris-3dThe Soma Cube, one of the funniest puzzles born from the cube, was invented in 1936 by Piet Hein, the Danish mathematician-poet, with a passion for mathematical games. His is another nice game, the Hex, rediscovered and studied, in its mathematical properties, by John Nash.


Piet Hein, who died in 1996 at the age of ninety-one, more than for mathematics is famous for his poems, published under the pseudonym of Kumbel. When Hitler occupied Denmark in 1940, Hein was elected president of the Anti-Nazi Union and became popular with his epigrams against Nazism.
Those that follow are two of his best-known poems.


Naive you are
if you believe
life favours those
who aren't naive.

The road to wisdom 

The road to wisdom?
Well, it's plain
and simple to express:
and err
and err again
but less
and less
and less.

Hein had the opportunity to work for some years with Albert Einstein and his most important contribution to mathematics was the discovery of a particular family of curves, the Superellipses, defined by equations similar to those of ellipses, but with exponents greater than two. Some of these curves are shown in the picture below and one of these is the one surrounding Piet Hein's face in the photo. They are curves similar to both the ellipse and the rectangle, which have a particular aesthetic value and which have been adopted as models for art objects, lamps, furniture even in their three-dimensional shapes and even for a big fountain located in central Stockholm.


One day, in 1936, Piet Hein was following a lesson in quantum physics by Werner Heisenberg and while the great physicist described a space divided into cubic cells, it came to him to ask himself which figures could populate this space, built with cubes all the same, having at least a face in common. It is the three-dimensional idea of polymins.
If you use 1 to a maximum of 4 cubes, the possible shapes are 12 and they are those shown in the picture below.

Instead, the possible pentacubes are 29 and their number, being a prime number, tells us that it is not possible to build parallelepipeds using all the pieces. But we can choose 27 pieces in order to try to build a new piece having the shape of one of the two discarded, three times higher.


A reader of good will, after having solved this problem, could continue the game in search of the hexacubes, the shapes that can be made of six cubes and that are 166, according to Martin Gardner.

After going further into the problem, Piet Hein came to identify a set of particularly interesting pieces from the study of the twelve simplest forms, and enunciated a precise "theorem":
"If we consider all the non-linear shapes that could be built with less than four cubes, all of same dimensions and joined at least on one face, it is possible to combine them in a 3 x 3 x 3 cube".


aldous_huxleyOf the 12 possible forms that could be built, at most with 4 cubes, we discard the "parallelepipeds". The 7 non-linear forms remain, having at least a concavity or a recessed angle, as shown in the picture.

Piet Hein named the game Soma Cube, referring to the drug, called Soma, circulating in a hypothetical mechanized world of the future, described by Aldous Huxley in his novel Brave New World.

<<< Soma Cube, therefore, may be used as a medicine against frustrations of modern life. We're kidding, of course!
Soma Cube is an incredible object that stimulate our mind and exercise it in problem solving in three dimensions! >>>

There are 7 pieces in total, six made up of 4 cubes and one made up of 3, two of which are easily identifiable, mirror images. As we said, with these seven forms you can compose the 3 x 3 x 3 cube.



If we leave the piece made up of 3 cubes aside, with the other six pieces we can build a shape exactly equal to the one we discarded, of double height.
But besides the cube, there are thousands curious shapes we can build with the seven pieces of Soma Cube.

Only in 1970, Parker Brothers Corporation began to commercially produce the game that had immediate success. Even today, you found it in many game stores. A copy could easily be built using wooden or plastic cubes, like those of Lego, glued together.

In 1961, J. H. Conway and M.J.T. Guy established that there are 240 different ways of reconstructing the 3 x 3 x 3 cube, excluding symmetries and rotations. Few years later the computer confirmed their result.

If you build the seven pieces of Soma Cube by alternating black and white cubes, so that a cube of one color is never close to a cube of the same color, then there are only two ways to obtain the cube chess with the seven pieces.


The reader is invited to find some of the 240 solutions and the two of the chess cube. He may try to reconstruct the forms shown below and discover new ones, certain that the puzzle, apparently so simple but actually intriguing and varied, will capture him as the drug that captured the inhabitants of Huxley's world, but without damage, this is at least our opinion and that of Piet Hein.

Author: Federico Peiretti (

Logical-mathematical problems to be solved

Problem 1


How deep is the hole in the well?
One, two or three cubes?
Motivate the answer reasonably.

Problem 2


Based only on the observation of the figure, is the scale a possible or certainly impossible construction?
Motivate the answer reasonably.

Problem 3

We want to make the seven pieces of the soma cube.


We have available:
- pieces made up of 1 cube:1_cubetto
- pieces made up of 2 cubes:2_cubetti
- pieces made up of 1 cubes:3_cubetti

How many do you need of any kind?
It would be advisable to use as few as possible.

Problem 4

We want to make the pieces of a soma cube from a square wooden strip of 1x1 cm.


How much strip do we need, in centimeters?
We must bear in mind that at each cut, 1 mm of strip is consumed due to the thickness of the blade.

Problem 5

We want to build 25 soma cube.
We have 2x2 cm square section slats strips available.
How many meters of strip do we need?

Problem 6

The carpenter has a 3x3 cm square section strip, 3 m long.
How many soma cube can you get?

Problem 7

Is it possible to build the Mayan Pyramid with the Soma Cube?
Motivate the answer reasonably.


Problem 1

risposta_problema_1It is 3 cubes deep because the biggest cube is 3x3x3 = 27, from which the three cubes forming the stairs must be removed.

Problem 2

risposta_problema_2It is not impossible because it consists of three layers:
To demonstrate that it is possible, we report the solution.

Problem 3

There are several possibilities.


The picture illustrates 3 solutions for the L-shaped piece.

We used:
- 5 pieces of 1, 8 pieces of 2 and 2 pieces of 3 (15 pieces)


- 4 pieces of 1, 10 pieces of 2 and 1 piece of 3 (15 pieces)

Problem 4

Soma Cube is made up by 27 cubes, which together give a length of 27 units of measure, in this case 27 cm.
If we adopt the solution of 15 pieces of strip we have to make 15 cuts (excluding the first but not the last), which consume 1.5 cm.
Therefore, we need: 27 + 1.5 = 28.5 cm of wood strip in total.

Problem 5

In this case we require 27x2 + 1.5 = 55.5 cm of wood strip for a Soma Cube.
Therefore, for 25 Soma Cube we require 55.5x25 = 1387 cm of wood strip, neglecting the waste.

Problem 6

For a Soma Cube we require 27x3 + 1.5 = 82.5 cm of wood strip.
In 3 m, we may therefore obtain 3 Soma Cubes.

Problem 7

No, because the Mayan Pyramid requires:
5x5 + 3x3 + 1 = 35 cubes.

More details and shapes to play:



This logical game is more than 100 years old. In its long history it had several names: Fifteen Puzzle, Puzzle-Blocks, Gem Puzzle, Boss Puzzle, Game of Fifteen and Mystic Square.

Many sources assign the creation of the game to the American Samuel Lloyd, who lived at the turn of XIX and XX centuries. The year of the invention is 1891, but there are other testimonies of the fact that the game was actually invented a little earlier by another person, with the "16 edition", in which there were 16 wooden tiles to be placed in so as to obtain the sum of 34 horizontally, vertically and diagonally; but since the patent was filed in the name of Samuel Lloyd, the copyright is his.

Samuel Lloyd was born in Philadelphia, but soon moved with his family to New York. He wanted to become an engineer but he began to notice that his ideas yielded more. Chess puzzles already made him very famous. He invented his first quiz-puzzle game at the age of 14 and at 16 he was an editor of a monthly about chess. After starting with chess, he greatly expanded his interests.

Ordinary puzzles, in his hands, became more engaging and interesting. Thus 15 Puzzle became his best invention. With Lloyd's promotional ingenuity, this puzzle shook America to then cross the ocean like an epidemic and conquer the whole world. The popularity of the game was so big that the owners of Companies had to impose explicit prohibitions on their employees, because they played during work. In Germany, 15 Puzzle was played during the sessions of the Parliament and in France they named it "Taquin" (rooster) because it seemed more harmful than alcohol or smoke.

Sam Lloy awarded a $ 1.000 prize, huge for the time, for anyone who solved the riddle of the repositioning of 15 and 14 tiles, while all the other pawns were already settled. So many people rushed to look for the solution buying the game produced by Samuel Lloyd, of course. Thus began the so-called "fifteen madness".


The passion for 15 Puzzle spread very quickly throughout America, Europe, Australia, New Zealand and even in the countries of the Far East. The search for the solution of 15 and 14 repositioning seemed to be total madness. There was such an involvement that many people engaged in research to the point of forgetting to eat, sleep, study or work. The owners of the activities forbade bringing this diabolical game to work. Bakers forgot to open their shops, captains gave up, train drivers jumped the stations in passion for the game. They also tell of a famous priest who stood all night under a street lamp in order to remember how he had repositioned the 15 and 14 tiles. It was surprising that those who had succeeded in repositioning the numbers did not remember the exact sequence of the gambling.

"... in the last few weeks a toy puzzle had come into sudden favor ... all the populations of all the States had knocked off work to play with it, and that the business of the country had now come to a standstill by consequence; that judges, lawyers, burglars, parsons, thieves, merchants, mechanics, murderers, women, children, babies everybody, indeed, could be seen from morning till midnight absorbed in one deep project and purpose, and only one: to work out that puzzle successfully; that all gayety, all cheerfulness, had departed from the nation, and in its place care, preoccupation, and anxiety sat upon every countenance, and all faces were drawn, distressed, and furrowed with the signs of age and trouble, and marked with the still sadder signs of mental decay and incipient madness; that factories were at work night and day in eight cities, and yet to supply the demand for the puzzle was thus far impossible ..."
Mark Twain - "The American Claimant".


However it turned out that the enigma posed by Samuel Lloyd to win the stratospheric sum for those times had no solution. That puzzle could not be composed, because it had no solution. This puzzle belongs to the category "impossible". The 15 Puzzle would have been solved if the number of numerical pairs, in which the highest number precedes the minor one, was even. But since in the task posed by Lloyd it was necessary to reposition only a couple of numbers (15 and 14), the so called "the parameter of disorder" makes this task unsolvable. The author knew this from the beginning, but the public came to know it much later, when the madness had passed and the clever Sam Lloyd had already made a capital.
In the process of finding the solution for the repositioning of 15 and 14, other puzzles were developed. They are still very difficult and current, like almost a century and a half ago.

Mathematical description

The 15 Puzzle represents a classic task for the creation of heuristic algorithms. Usually this task is resolved with a number of steps and the search for Manhattan distance between each pawn and its position in the solved puzzle. For the solution we normally use the IDA algorithm.

It can be proved that exactly half of all possible 20.922.789.888.000 starting numbers do not lead to the resolution of the game.
Let's say that the tile with the i number is before the k tiles with the minor numbers at i. Let's consider that ni = k, that is that after the pawn with the i number there are no other numbers less than i, so k = 0. Let's also add the e- number, the number of the row with the free cell.

If the sum is odd, the solution to the puzzle does not exist.


For the 15 Puzzle with a number of pawns greater than 15, the dilemma of finding the shortest solution is и NP-full.

If, on the other hand, we have to turn the box 90 degrees, where numbers are upside down on the side, you could solve what was previously called unsolvable (and vice versa). If, therefore, instead of numbers you put dots on the pawns and we don't fix the position of the box, the unresolvable combinations would no longer exist.


Prison Escape belongs to a large family of games with scroll blocks, usually they are ten blocks, one of which must be moved from one position to another moving all the others. It is known worldwide with different names and some of these variants belong to the most ancient oriental traditions. Prison Escape is sometimes presented as a game of Thai origin, the Thailand name of the game is that of a famous imprisoned warrior who tried to escape "Khun Chang Khun Phaen".
The following variants basically have the same pattern and layout of the blocks, varying only in the name (human, animal, or other), and behind these names there is a story telling.

Huarong Dao

huarong_daoHuarong Dao (also known as Huarong Path or Huarong Trail, Chinese name: 華容道) is the Chinese variant based on a fantasy story in the historical novel of the Three Kingdoms on the Warlord, Cao Cao, retreating along the Huarong Passage (now Jianli County, Jingzhou, Hubei) after its defeat in the Battle of the Red Cliffs in the winter of 208/209 b.C., during the late Eastern Han dynasty. He met an enemy general, Guan Yu, who watched the road waiting for him. Guan Yu spared Cao Cao, who had been generous with him in the past, and allowed him to cross the Huarong Passage. The biggest block in the game is called "Cao Cao".

The Daughter in the Box (箱 入 り 娘)


The Daughter in the Box (Japanese name: Musume hakoiri) depicts an "innocent girl who knows nothing of the world" trapped in a building. The biggest piece is called "daughter" and, on the other blocks, there are the names of other family members (such as father, mother and so on).
Another Japanese variant uses the names of Shogi pieces.

The Red Donkey (L'Âne rouge)

In France, it is known under the name of "Âne rouge". It includes a Red Donkey (the largest piece) which tries to overcome a maze of obstacles to reach its carrots.

Khun Chang Khun Phaen


This is a Thai variant. Khun Phaen is a famous figure in Thai legend and the game is named after the epic Khun Chang Khun Phaen, in which the hero is imprisoned. The game describes Khun Phaen's escape from prison, evading the surveillance of his nine sentinels.
Khun Chang Khun Phaen (Thai name: ขุน ช้าง ขุนแผน) is a Thai epic poem originated from a Thai folklore legend and it is one of the most important works of Thai literature. Chang and Phaen are the male protagonists, and "Khun" was an inferior feudal title, typical for ordinary male people. The story is a classic love triangle, which ends in tragedy. 

Khun Phaen (dashing but poor) and Khun Chang (rich but ugly) have been competing for the beautiful Wanthong since childhood and for over fifty years. Their competition causes two wars, several kidnappings, a coup d'etat, an idyllic stay in the woods, two court cases, a severe trial, imprisonment and betrayal.
Ultimately the king condemns Wanthong to death for not having to choose between the two men.
The poem currently on the market is in English and was written in the early nineteenth century. The first serial edition was published in 1917-1918. Like many works with origins from folk tales, Khun Phaen is a story of rapid evolution and full of heroism as well, romanticism, sex, violence, crude humor, magic, horror and lyrical beauty. In Thailand, the story is known by all the inhabitants: at school children study it, poetry inspires songs, some phrases became popular sayings and everyday metaphors.

Other variants

There are also versions where the scheme is different, such as Pennant Puzzle and Ma's Puzzle, and a computerized version for Windows created by ZH Computing in 1991.
After the success of Taquin or 15 puzzle (15 1x1 squares in a large 4x4 square) in 1880, Dad's Puzzle or Pennant's Puzzle introduces 1x2 rectangles in 1909 and 1912 (two variants both with copyright registered LW Hardy in the United States). Subsequently, JH Fleming deposited the copyright in 1934 for this game that was known all over the world, under different names: Klotski (wooden block) in Polish, Hua Rong Dao in Chinese, Hakoiri Musume (daughter in the box) in Japanese, Forget-me-not or Mayor Migraine Maker in English. This game is found today under a lot of names, alone or sometimes with very different variations on different gaming platforms (iPhone, Ds): Block Puzzle, Path puzzle, Kwirk, Professor Layton, etc. The most known and closest variants of this game are Century, SuperCompo and Quzzle.

Technical data

In these game there are 65.880 different placements of the 10 pieces. There are 114.958 different steps between these placements, which corresponds to an average of about 3,48 movements per placement. These placements are divided into 898 different components and the two main ones contain 25.955 placements each. These two components are symmetrical to each other, in relation to a horizontal axis, because they are two. Then, each one has an internal vertical symmetry axis that makes it possible to pass from a positioning to its symmetrical (in relation to this vertical axis) following a path (a sequence of steps).

Read more about Huarong Pass on



Snakes and Ladders is a traditional board game, born in England and widespread especially in English-speaking countries. It is a very simple path game, rather similar to the Game of the Goose. As in the Game of the Goose, the outcome of a game is completely determined by the roll of the dice.


The origins

Snakes and Ladders's origin is found in India, in a game based on morality which in Indian language is called Paramapada Sopanam (the Ladder for Salvation).

Widely played since ancient times and known as Moksha Patamu, this hobby shows us how Indians conceived morals. Hindu spiritual masters used it to educate children about the effects of good and evil. The ladders are the virtues and the snakes the vices.

Moksha, or the Salvation of the Soul, may be achieved through good deeds, while behaving badly one obtains reincarnation in lower life forms (Patamu).
Ladders are few in this game, while snakes are many: the good is difficult to achieve, while the ways of evil are easy to follow. It is difficult to climb because the many snakes make you slide down.
Even the numbered boxes reached are significant: number 100 is called Moksha, that is Salvation.

Then we have Faith (51), Generosity (57), Knowledge (76), Ascetism (78).
The boxes of evil are: Disobedience (41), Vanity (44), Vulgarity (49), Theft (52), Lie (58), Drunkenness (62), Debt (69), Anger (84), Greed (92), Arrogance (95), Murder (73) and Lust (99).
In this game Indian religion and morals are important. The last opponent to defeat is Lust. It is not just the sexual one, in India means the greed to take possession of things that are not ours, it is also the worst form of envy that blinds and hides the path to Salvation.

Using this tool, Indian educators discussed with their pupils all the moral doubts and dilemmas that children faced during their growth.



snakesImported into England in 1892 with the current name of Snakes and Ladders, the game was in line with Victorian puritanism of the time. The names of some boxes were changed and thus Penance, Parsimonity and Industriousness raise the player with a ladder up to the Divine Grace, Satisfaction and Success box, while Indolence, Indulgence and Disobedience cause him to slip into the Poverty, Illness and Misfortune boxes. In this version the number of ladders and that of the snakes are the same.

Board and rules

The traditional board of "snakes and ladders" represents a path of bustrophedical form, usually consisting of 10 rows of 10 boxes. The path is made by a certain number of "ladders" and "snakes" that cross the board vertically, joining two boxes in different lines. The position of the ladders and snakes may vary. Similar to what happens in the Game of yhe Goose, the players proceed with the number of squares indicated by the throwing of a dice.
A marker that arrives in a box "at the foot" of a ladder is moved to the box at the top of the ladder; vice versa, a marker that arrives in a box with the mouth of a snake "recedes" to the tail. In most versions, a player who throw 6 has the right to play again.
The winner is who arrives first in the last box of the path. In some variants (not always), the last box must be reached with the exact throwing of a dice; any points in excess would lead the marker to reach the goal and then recede the remaining points.


Ludo (from the Latin ludus, "game") is a popular board game; is a modern and simplified variant of the Indian Pachisi. It was published for the first time in 1896 by the John Jaques & Son publishing house in London, to which many other "classics" are due, including Tiddlywinks game and Ladders and Snakes. In Italy there is a variant, called "Non t'arrabbiare", which allows the game up to six players.


Pachisi or twenty-five is a game born in ancient India, described as the "national game of India". It is played on a board in a symmetrical cross shape. The player's pieces move around the board according to the throwing of six or seven shells, the number of shells that remain with an opening indicates the number of boxes corresponding to the movement.
The name of the game comes from Hindi: pachis means 25, the largest score that can be achieved by the throwing of the shells. Usually palyed by 4 players, 2 per team, one team with yellow and black pieces, the other with red and green ones.

Pachisi may be very old, but so far we do not know its history before the XVI century. There is a representation, dating back to the VI or VII century, of God Shiva and Goddess Parvati playing Chaupar (a strictly connected game). In fact, this depicts only the dices and not the frame that distinguishes Pachisi.

There is a large XVI century garden version in Fatehpur Sikri Palace, in northern India at the time of the Great Mogul Akbar the Great (15 October 1542 - 27 October 1605).


The English philologist Irving Finkel writes about it:
"Pachisi was played by Akbar in a truly regal way. The playing field was divided into red and white squares and a huge stone on four supports represented the central point. It was here that Akbar and his courtiers played this game; sixteen young harem slaves, who wore the colors of the game, represented the pieces and moved into the boxes according to the throwing of the dice. It is said that the Emperor was so fond of this large-scale game that he set up a courtyard for Pachisi in each of his palaces, and traces of it are still visible in Agra and Allahabad.

So far, these great game fields are still the first solid evidence of the existence of this game in India. The importance of the game in Indian history remains to be studied. It is often said that Pachisi is a game of chanc,e that played such a significant role in Mahabharata, one of the greatest epic poems in India, but the descriptions, as they are, do not exactly coincide with the game in question and this conclusion is perhaps wrong”.


tavoliereThe board is made by four boxes at the corners, called "base boxes", and by a central path that follows the frame of a cross and ends in the middle of another large box. In each arm of the cross there is also a path of 6 boxes, which starts from a special box, the starting box, located on the frame of the cross, and continues up to a central box (the arrival box). Each player is assigned a column and a set of pawns (usually four).

The pawns and the columns of the different players are distinguished by different colors: generally red, green, yellow and blue.
The players' pawns make their entry into the board from the starting box and must travel the whole board; once they are back to the starting box, they will take the path to the arrival box.


Initially each player puts his pawns in his base box. The aim of the game is to get all your pieces into the path and, after a whole lap, get them to the center before the opponents.
To enter a pawn in the game, a throwing of dice must roll 6. A player throws a dice and moves his pawns according to the number that came out. For each die roll, only one piece could be moved, corresponding to the number of boxes.
If the player in turn gets 6 with the dice, he has the right to play again. Furthermore, he can choose whether to move a pawn already in play by 6 squares, or enter a new pawn on the board. If a pawn ends its movement on a box occupied by an opposing pawn, the latter returns to its base box, from which it can exit again only with a 6 of the die roll. Instead, when a pawn reaches a box occupied by a pawn of the same color, it "ride on its back" and the two pawns continue the race together: from this moment on, the two (or more) pawns cannot be overtaken by opposing pawns and cannot not even be sent back to the base.
If a player cannot make a valid move, he passes the turn.
When a pawn completed the lap of the board, from the initial box it proceeds to the final column which leads to the arrival box. From this moment on, the pawn could only be moved by the exact steps that would take it to the arrival box.
The winner is the first player to complete the path with all his pawns.

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