# Dissection Puzzles: Ostomachion, Egg of Columbus, Hexiamonds

TR Smith is a product designer and former teacher who uses math in her work every day.

## Tangram-Like Puzzles for Making Shapes and Patterns

While tangrams are the most popular dissection puzzle, there are many other shape-making or tile puzzles like tangrams. The Ostomachion (aka Stomachion and Syntomachion) is an ancient tile puzzle sometimes attributed to Archimedes, though it predates him. The Egg of Columbus is a tangram-like game with curved tiles. Hexiamonds are similar to tangrams but based on equilateral triangles. Broken heart tangrams are the most similar to classic tangrams, but with quarter and eighth circular sectors added.

Tangrams, hexiamonds, the Egg of Columbus, the Ostomachion, broken heart, and similar puzzles are great learning tools for teaching kids about geometry and improving their spatial reasoning skills. They're also fun for older kids and adults.

## Familiar Favorite: Tangrams

Tangrams are the most popular dissection puzzle game, consisting of a set of seven tiles based on isosceles right triangles. The puzzle originates from China, though its creator is unknown. With the seven pieces you can can form an endless number of shapes in the form of people, animals, and common objects. In play, you are given a silhouette or outline of the finished shape and your task is to figure out how the seven pieces can be arranged to form it. Tangrams are also useful in modeling quilt patterns.

## Broken Heart Tangram

The broken heart tangram puzzle, also called heartbreaker puzzle, is a modern variation of classic tangrams. It is based on isosceles right triangles and 1/8 sections of a circle. Since five of its nine pieces are circular sectors, you can make countless figures with curved edges. This is a fun variation of tangrams for people who have mastered the original game.

## Ostomachion: One of the Oldest Games

The Ostomachion has more tiles in more complicated shapes than Chinese tangrams, yet this game is actually older. It was first described by Archimedes, who lived during the 3rd centery BC, though he is not believed to be the inventor of the puzzle. In comparison, tangrams were invented during the Song Dynasty, spanning the 10th through 13th centuries AD.

The Ostomachion, also called Stomachion and the Loculus of Archimedes, consists of 14 pieces based on a dissection of a 12-by-12 grid. 11 of the pieces are triangles, two are quadrilaterals, and one is a pentagon. Curiously, there are 536 distinct ways to arrange the pieces into a 12-by-12 square, where reflections and rotations of an arrangement are not counted as distinct. Below are 104 such arrangements.

## Egg of Columbus

The Egg of Columbus is a clever variant of tangrams with 9 pieces, 6 of which have circular arcs and 3 of which are isosceles right triangles. The tiles of the Egg of Columbus are ideal for making a wide array of bird shapes.

The name of this game is taken from the story about Christopher Columbus making an egg stand on its tip. One day, many years after he had become famous for discovering the Americas, Columbus was having dinner with a group of nobles, one of whom said that if Columbus had not discovered the Americas they would have been easily discovered by some other navigator. (It did not occur to them that the Americas had been discovered tens of thousands of years earlier by the indigenous inhabitants.) In response, Columbus challenged his dinner companions to make an egg stand on its tip. After they had all tried and failed, Columbus took the egg, cracked its tip, and made it stand. The point of the demonstration with was to show that accomplishments look easy in only retrospect after you have seen how they are done.

The shape of the egg is formed from circular arcs as in the image below.

If the two largest circles on the outside have a radius of 2, then the medium sized circle in the middle has a radius of 1 and the three smallest circles have a radius of 2 - sqrt(2) ≈ 0.5857864.

## Hexiamonds

A hexiamond is a polygon formed by connecting six equilateral triangles connected at their edges. In general, a polyiamond of order N is a polygon formed by joining N equilateral triangles. The word is a back-formation from diamond, which is constructed by putting two equilateral triangles together, as "di" = two.

When N = 6, there are exactly 12 different hexiamond shapes, shown in the upper image. These 12 tiles can be used to make many geometric arrangements, such as the eight figures shown in the lower image. Hexiamonds are a lesser-known dissection puzzle, a bit more sophisticated than tangrams.

With a little math, it is easy to prove that certain shapes cannot be formed with the standard set of 12 hexiamonds. For example, it is impossible to arrange the tiles into the shape of an equilateral triangle or regular hexagon. The reason is simple. The total number of equilateral triangle base units in the set of 12 hexiamonds is 6*12 = 72. If a larger equilateral triangle is to be decomposed into K smaller equilateral triangles, then K must be a square number. If a regular hexagon is to be decomposed into Q smaller equilateral triangles, then Q must by 6 times a square number. But 72 is neither a square number nor 6 times a square number.

## Pattern Blocks

Pattern blocks are an educational toy invented by Elementary Science Studies in the 1960s as a way to introduce children to the concept of tilings or tessellations, and to show how larger shapes can be decomposed into smaller shapes. The standard set of pattern blocks consists of six different shapes: regular hexagons, squares, equilateral triangles, 60-120-degree trapezoids, 60-120-degree rhombuses, and 30-150-degree rhombuses. The edge lengths are all equal, allowing players to match up the sides of the shapes precisely to develop more complicated patterns. Traditionally they are made of wood and colored as shown above, but cheaper sets made of plastic can also be found.

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