## The Histories of Common Forms of Tiling Puzzles

The Many Forms of Tiling Puzzles
Tiling puzzles are a sort of puzzle that involve rearranging shapes, either solid or cut out of a specific geometric shape, like a square, and reforming them together to form a completely different silhouette, for instance a triangle, without overlaps or gaps. Common forms of tiling puzzles include jigsaw puzzles, dissection puzzles, and Tangrams. The combined histories of all of these types of tiling puzzles, and the extra less-known forms, is incredibly vast and differs between each type. Altogether, tiling puzzles have existed for centuries, the first probably being the dissection puzzles in the works of ancient mathematicians like Archimedes when he made a mathematical treatise called Ostomachion in the form of a geometric dissection.

Jigsaw Puzzles
One of the best known forms of tiling puzzle is the jigsaw puzzle. Jigsaw puzzles require the player to assemble a picture by placing one of many interlocking shapes together in the correct pattern. The shapes are often strangely cut and involve tessellation, the act of placing down shapes so that no shapes overlap or create gaps as they fill a plane. Originally, these puzzles were created by painting a picture or scene onto a rectangular wooden block, and then cutting the block into pieces with a jigsaw, hence the name (Encyclopedia Britannica).

Around 1766, a man named John Spilsbury was credited with the invention of the first jigsaw puzzle (“Jigsaw Puzzle, 1766”). Spilsbury, a teacher and mapmaker in England, and an apprentice to the Royal Geographer to King George III, Thomas Jefferys, invented the first jigsaw puzzle as a means to teach geography. The puzzle was handmade, carved out of wood and painted, and displayed a map of England and Wales with each piece representing a county that once all fitted together formed the full country (Ament).

Over a century later, Milton Bradley published their first jigsaw puzzle for children in the 1870s, called “The Smashed Up Locomotive” (“The Smashed Up Locomotive”). The pieces of the puzzle were different parts of a train, and the smashed up effect took place when the child first opened the box and noticed a pile of jigsaw pieces with train parts painted on each one (Ament).

Jigsaw puzzles typically come in a variety of sets, their difficulties being directly related to the amount of pieces that comprise each puzzle. Different sets include 300-piece, 500-piece, 750-piece, and 1000-piece, and the most common types of scenes that a jigsaw puzzle creates include castles, mountains, forests, or fantasy landscapes. Children’s jigsaw puzzles tend to have larger pieces for an easier method of play, and are often used as learning exercises. There are also 3-D jigsaw puzzles that add an extra layer of complexity such as the puzzle globe, a spherical jigsaw puzzle usually representing the Earth or the moon (Encyclopedia Britannica).

Today, jigsaw puzzles remain as one of the simplest, yet most time-consuming, tiling puzzles that after all these years have not completely changed design. Many innovative offshoots and styles of jigsaw puzzles have popped up, but this rawest form of tiling puzzle has remained.

Dissection Puzzles
A dissection puzzle is one of the earliest forms of tiling puzzle where the objective is to form a unique silhouette out of specific shapes given. Geometric dissections have been used as mathematical proofs for concepts such as the Pythagorean Theorem, the Quadrilateral of Omar Khayyam, the Fibonacci Sequence or “Golden Ratio,” Euclid’s Windmill proof of the Pythagorean Theorem, and Liu Hui’s proof of the Pythagorean Theorem (Wikipedia). Around the 10th century, Arabic mathematicians displayed geometric dissections while commentating on Euclid’s Elements, and Dai Zhen, a Chinese scholar from the Qing Dynasty in the 18th century approximated the value of pi using dissections (Tiwald). Common dissection puzzles today include the ancient Chinese game of Tangrams and dozens of American variants of Tangrams.

The first dissection puzzles appeared in the works of ancient mathematicians like Archimedes when he made a mathematical treatise called Ostomachion in the form of a geometric dissection. Ostomachion is also referred to as the stomachion, its original name which traces its meaning from the Greek word for “stomach,” “Loculus of Archimedes” (which means Archimedes’ box), and in Latin as “syntemachion” (Chung and Graham).

Ostomachion is a fourteen piece dissection puzzle which forms a square and the vertices of each piece rest on a 12×12 grid (“Stomachion,” Weisstein) as shown in fig. 1. The design is similar to Tangrams where the player may compose silhouettes of animals, ships, weapons, or people.

Bill Cutler, in 2003, solved the stomachion by revealing all 536 arrangements of the pieces as a square, where solutions rotated or reflected are not counted (“Stomachion,” Weisstein).

Fig. 1. Stomachion puzzle, Wolfram Research, Inc.

In the late 19th century, dissection puzzles saw a rise in popularity as newspapers began running them regularly. Sam Loyd from the United States and Henry Dudeney from the United Kingdom were two of the most published dissection puzzle authors at this time (Wikipedia).

In 1907, Henry Dudeney presented the haberdasher’s problem (fig. 2), a dissection in which four pieces could form both a triangle and a square (Wikipedia). It is one of the most classic examples known of a dissection puzzle.

Fig. 2. The haberdasher’s problem, Wikipedia.org.

Dudeney also mentioned a variant of his puzzle in which the pieces can only swing on a hinged joint. This interesting spin on the haberdasher’s problem was introduced alongside the puzzle in Dudeney’s 1907 book The Canterbury Puzzles. It demonstrates how the triangle and square silhouettes can be achieved with the four original shapes even if limited to rotating each piece on an imaginary hinge on each connecting corner as shown in fig. 3 (Frederickson).

Fig. 3. Greg Frederickson, Hinged dissection puzzle, Cambridge [UP].

Dissection puzzles are both a form of entertaining tiling puzzle and a solid mathematical tool, with the creation of dissection puzzles acting as puzzles themselves to mathematicians and others alike. They are very useful educational resources when teaching geometry and when demonstrating higher concepts of mathematical principles.

Tangrams
The most recognizable dissection puzzle today is a game so old that its origin is undocumented and persists only in legend. The ancient Chinese game of Tangrams (or just the Tangram) has survived for untold centuries. Tangrams, literally translated to “the seven clever pieces”, involves seven geometric pieces called “tans” that are placed together to form silhouettes (Rob). The seven geometric pieces are: two large right triangles, one medium right triangle, two small right triangles, one square, and one parallelogram. The parallelogram is a special case in that it is the only piece that must be flipped in order to achieve its reflection, as it has no reflection symmetry but only rotational symmetry (“Tangram,” Weisstein).

Fig. 4. Image created by the author, Tangrams set.

Tangrams’ popularity has made it one of the most well-known dissection puzzles in the world.  A fictitious history of Tangrams popularized the game when Sam Loyd published his book The Eighth Book Of Tan. Loyd’s book claimed the game was invented over 4000 years ago by the god Tan and the book included over 700 silhouettes that the Tangrams set could complete (Rob). In reality, the game was invented in China at an old yet unknown time (disputes argue over whether it was more near the late 1700s and early 1800s or thousands of years ago), and was first brought to America in 1815 by Captain M. Donaldson on his ship, the Trader (The Tangram Book, Slocum). Tangrams would soon reach England where a craze for the puzzle spiked and spread across Europe. The Tangrams fever spread to Denmark, where at the Copenhagen University, a student wrote the book Mandarinen, which documented the history and popularity of the game. China exported a great number of Tangrams sets during this time. In Germany, around 1891, the industrialist Friedrich Richter began producing the Anchor Stone Puzzles, silhouette-matching puzzles based on Tangrams. And in World War I, the game grew even more popular on both sides of the battlefield, occasionally going under the name “The Sphinx” (Rob).

Over 5900 different Tangrams silhouettes have been compiled from texts from the 19th century, with more appearing every day (The Tao of Tangram, Slocum). However, there is a limit. In 1942, Chinese mathematicians Fu Traing Wang and Chuan-Chih Hsiung proved that there exist only thirteen convex figures which can be constructed with Tangrams, as displayed in fig. 5 (Rob).

Fig. 5. The thirteen convex shapes in Tangrams, Wikipedia.org.

Lesser-known Tiling Puzzles
While jigsaw puzzles, dissection puzzles, and Tangrams are at the forefront of the tiling puzzle category, there are dozens of less-known puzzles out there. The Conway puzzle, named after its inventory John Conway, demands the solution of packing rectangular boxes into a 5×5 hollow cube (Weisstein). Domino tiling is another form of tiling puzzle in which 2×2 dominoes are arranged in order to fill a square or rectangle in a tessellated pattern. Sliding puzzles are yet another type of tiling puzzle in which flat pieces are moved across a board on established routes in order to achieve a perfect configuration.

Certainly, a very interesting yet widely unknown tiling puzzle, the Eternity puzzle, created by Christopher Monckton in 1999, was a 209-piece puzzle in which a large assortment of randomly cut shapes formed a circle. It was considered unsolvable, and its creator held a one million cash prize for anyone that could solve it within four years of its release. In 2000, two mathematicians from Cambridge won the reward with their solution (Richer).

Conclusion
The genre of tiling puzzle covers a wide expanse of puzzle types that appear seemingly unique in their own right. The history of tiling puzzles in general is a combined effort of multiple histories and multiple puzzles that all co-exist under a greater definition. Jigsaw puzzles have survived for years and still have that same simple enjoyment from when they were first invented. Dissection puzzles have long been used for mathematics and as teaching aides, and to this day still intrigue many mathematicians and people interested in geometry. Tangrams, one of the oldest dissection puzzles, still survives to our time and has largely remained untouched. And of course, all the dozens of variations and alternate tiling puzzles that are being invented every day add to the flavor of the exclusively unique puzzle category of tiling games.

Sources
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Chung, Fan, and Ron Graham. “A Tour of Archimedes’ Stomachion.” Department of Mathematics. UCSD. math.ucsd.edu. n.d. Web. 07 Mar. 2011.
“Dissection Puzzle.” Wikipedia: The Free Encyclopedia. Wikipedia.org. 20 Jan. 2011. Web. 10 Feb. 2011.
Frederickson, Greg. Hinged Dissections: Swinging & Twisting. Cambridge UP. 15 Sept. 2009. Web. 10 Feb. 2011.
“Jigsaw puzzle.” Britannica. Britannica.com. 2011. Web. 10 Feb. 2011.
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Rob. “Tangrams and Anchor Stone Puzzles.” Rob’s Puzzle Page. Comcast.net. n.d. Web. 07 Mar. 2011.
Slocum, Jerry. The Tangram Book. New York, NY: Sterling Pub., 2003. Print.
—. The Tao of Tangram. New York: Barnes & Noble, 2007. Print.
“The Smashed Up Locomotive.” Icollectpuzzles.com. n.d. Web. 07 Mar. 2011.
Tiwald, Justin. “Dai Zhen.” Internet Encyclopedia of Philosophy. Iep.utm.edu. 22 Sept. 2009. Web. 07 Mar. 2011.
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—. “Stomachion.” Wolfram MathWorld. n.d. Web. 07 Mar. 2011.
—. “Tangram.” Wolfram MathWorld. n.d. Web. 07 Mar. 2011.