The Mechanics of Tangrams

From rock 'n' roll in the '50s to violent video games in the 2000s, society has always found a new craze to blame for the perceived downfall of a generation. It may seem amusing now, but in the early 19th century, tangrams—Chinese puzzles requiring the rearrangement of geometric tiles—were once viewed as an obsession. Their popularity in Europe was so immense that a French newspaper cartoonist mocked the phenomenon with an illustration of a young "bourgeois" couple fixated on a tangram puzzle while their baby wailed for attention [source: Slocum]. Today, tangrams serve as engaging educational tools in classrooms or enjoyable brainteasers for puzzle enthusiasts. While no longer seen as a societal menace, they remain a beloved pastime across generations.

Tangrams belong to a category of puzzles known as dissection puzzles, a term used by mathematicians and puzzle experts to describe puzzles made from cut-out shapes that can be rearranged to form new figures. The best-known dissection puzzle is the classic jigsaw puzzle. However, tangrams differ in that they consist of only a few pieces that can be configured into an extensive variety of designs. A standard tangram set contains seven geometric shapes, or tans, all derived from a single square. These pieces include:

Tangram puzzles take the form of silhouettes or outlines, created by strategically combining all seven tans. The goal is to reconstruct these shapes by determining the correct placement and orientation of each piece. In puzzle books, solutions reveal the individual outlines of the tans, whereas the puzzles themselves do not. The challenge lies in arranging the tans while adhering to three fundamental rules: every completed tangram must include all seven tans; no tans may overlap; and the final shape must be continuous, with each tan touching at least one other—though contact by a single corner tip is allowed.

These straightforward rules create a puzzle game that is both deceptively simple and infinitely intricate. Turn the page to explore the origins of tangrams.

Tangram History

The true origins of tangrams remain a mystery. Historians are uncertain about their exact inception. The earliest known reference to the puzzle dates back to approximately 1796 in a book mentioned in historical records, though the book itself has never been recovered [source: Danesi]. Existing tangram sets have been traced to as early as 1802, and a Chinese book containing tangram problems from 1813 has also been uncovered [source: Slocum].

Regardless of when the modern tangram emerged, its foundations lie in Chinese mathematical tradition spanning centuries. As far back as the 3rd century B.C., Chinese mathematicians explored geometric principles by manipulating cut-out shapes. In fact, this technique was instrumental in deriving what Europeans later termed the Pythagorean theorem, which defines the relationship between the sides and hypotenuse of a right triangle. Scholars believe tangrams likely evolved from this hands-on problem-solving method [source: Slocum].

Regardless of the true origins of tangrams, the myths surrounding their history are far more fascinating. A popular story suggests that a mythical god named Tan created the shapes, using them to tell a creation tale on parchment made of gold. These and other fabrications can be traced back to Sam Loyd, a writer and puzzle inventor. His 1903 book, "The 8th Book of Tan," spun various tall tales about the puzzle’s past. While Loyd invented these stories, likely with the expectation that readers would recognize the humor [source: Slocum and Hotermans], elements of his fabricated history continue to appear in otherwise factual accounts.

Loyd’s book coincided with the global craze for tangrams during the early 19th century. As tangrams spread from China to Europe and the United States around 1818, they quickly became a sensation. Puzzle books and sets made from polished wood or intricately carved ivory became highly sought after in Germany, France, England, Italy, and the United States.

Much like the puzzle itself, the origin of the name "tangram" is shrouded in uncertainty. Initially, it was simply known as "The Chinese Puzzle." The term "tangram" appeared later. Several theories about its origin exist, including the possibility that it came from the English word "trangam" (meaning "trinket"). Others suggest that it is a blend of "Tang," the name of a historical Chinese dynasty, and "gram," which refers to a figure or drawing [source: Grunfeld].

Tangrams and Mathematics

Tangrams have retained their popularity for so long, partly because of their simplicity paired with their complexity. The individual tans are simple shapes, yet they offer nearly endless combinations. In fact, there are over 1 billion possible arrangements that can be made using the seven tans [source: Cocchini].

The tans themselves are based on fundamental geometric concepts. Each tan consists of several smaller triangles, all of which are right isosceles triangles with a hypotenuse measuring √2 units, and two equal sides of 1 unit. (The unit can be any measurement, such as inches, centimeters, or even a custom unit, since the shapes rely on proportional relationships rather than specific numerical measurements).

For example, the small triangles in the set are created by placing two base triangles side by side. The square is formed by joining two base triangles along their hypotenuses, and so on. To create a set of tangrams, draw a square, overlay a 4x4 grid on it, divide each square into two triangles, and trace the shapes along the triangle borders to match a tangram template. The grid’s units don’t matter, as long as the squares are perfectly equal.

Tangram puzzles often take the form of recognizable shapes like a cat, person, or sailboat. For these freeform shapes, the possibilities are practically endless (especially when considering abstract or nonsensical shapes that don’t resemble anything specific). However, there are certain mathematical categories of figures that follow clear rules, making them easier to define and count.

Mathematical figures are those whose base triangles can be arranged to fit a square grid. In other words, each shape is positioned such that at least one of its sides aligns perfectly horizontally or vertically [source: Koller]. Fully matched figures require each tan to have at least one edge and one corner, or vertex, aligned with another tan, meaning there are no isolated pieces with easily identifiable outlines. There are also fully aligned figures that may have dangling pieces, but at least one edge of the dangling tan must form a continuous line with the figure's border [source: Cocchini].

A specific category of fully matched figures that mathematicians focus on is convex figures. These are convex polygons, shapes where all interior angles are less than 180 degrees. To determine if a polygon is convex, draw a line between any two angles. If all of these lines either fit completely inside the shape or align perfectly with its borders, the shape is convex. Surprisingly, only 13 convex polygons can be formed with the seven tans [source: Wang]. In contrast, the tans can create over 10 million fully matched shapes [source: Cocchini].

Tangram Shapes and Patterns

Without the specific rules that define some of the mathematical tangram patterns, the possibilities seem endless. Since the earliest puzzle books from China, tangram problems have taken on whimsical shapes, imitating a wide variety of objects. Tangram patterns can resemble animals, buildings, household tools, people, and vehicles. Even if it takes a bit of imagination to recognize the cat staring back at you from a jagged, triangular outline, that’s part of the enjoyment.

The only reliable method for solving tangram puzzles is through trial and error — shifting the pieces around in various combinations until the correct solution presents itself. However, there are a few tips available in puzzle books, and nowadays, in online problem collections, that can help you along the way.

To start, it’s generally best to focus on the dangling pieces — those tans whose edges are completely exposed, or exposed enough that no other tan could fit in that spot [source: Koller]. Some tans are interchangeable, such as the two triangles, which can form the same shape as the parallelogram or square. So, the dangling tail of a cat, for instance, might not be as easy to fill in as it seems. It's also helpful to look for corners that extend beyond the figure, as an exposed triangular edge, for example, would prevent the square from fitting into that area.

The hardest puzzles are those with regular edges and no exposed corners or edges [source: Koller]. For instance, the convex polygons mentioned earlier are particularly challenging. Probably the toughest puzzle is creating a perfect square [source: Koller]. Since most tangram sets are sold as pre-assembled squares, players face this challenge each time they return their tiles to the box. Representational figures, such as animals or buildings, are typically easier since they have more protruding pieces, like ears, legs, and chimneys. Keep reading for more insights on tangrams and to discover websites where you can create and solve your own puzzles.