He fell in love with math puzzles because of the tangram, and now he has solved a century-old problem

At the end of May this year, a Finn who fell in love with solving puzzles and then creating puzzles because of tangram puzzles in his childhood solved a century-old mathematical puzzle. Before him, a Chinese origami and puzzle-solving master also made important contributions to solving this difficult problem.

Written by | Jiawei

It seems that humans love jigsaw puzzles. In different civilizations, various jigsaw puzzles have been invented repeatedly by people of different times. According to the famous Archimedes Palimpsest, Archimedes once decomposed a square into 14 pieces and thought about how to reassemble the pieces in different ways to form a square. Tangram, which originated in China, is an educational toy that brings endless fun to children around the world. At the end of May this year, a Finn who fell in love with solving puzzles and then creating puzzles because of tangram in his childhood solved a century-old mathematical puzzle.

Archimedes' Ostomachion puzzle, a 14-piece square puzzle.

Splitting the puzzle

The popularity of these puzzles grew significantly in the late 19th century when newspapers and magazines began to fill their pages with various puzzles. American puzzle creators Sam Loyd and Henry Dudeney in the UK were the most popular. Since then, jigsaw puzzles and related derivative puzzles have been used for entertainment and mathematical education.

Lloyd once challenged the public: How many pieces (the minimum number of pieces) does a carpenter need to cut into a bishop's crown-shaped board (a square cut off by 1/4, that is, after removing an isosceles right triangle) so that it can be reassembled into a small square? Lloyd later gave his own answer, but unfortunately, his construction was not correct. Lloyd believed that it was enough to divide it into 4 appropriate small pieces.

The thing the man in the picture is holding is the so-called mitre.

In mathematics, decomposing regular polygons and other simple geometric shapes into several pieces and then reassembling them into another shape is called plane area dissection. Dissection can be said to be a professional upgraded version of the jigsaw puzzle game: players upgrade from completing the puzzle with the help of known pieces to designing and cutting appropriate pieces by themselves in order to reassemble the shape.

Dissection later became the subject of Martin Gardner's "Mathematical Games" column published in Scientific American in November 1961. In the column, he once again introduced Lloyd's problem to the public - the "Miter-Dissection Puzzle". Although readers participated enthusiastically, no one could come up with a 4-piece puzzle. People had to decompose the original figure into at least 5 pieces to reassemble them into a square.

It wasn’t until May 27, 2024, more than a hundred years after Lloyd’s death, that a user named Vesa Timonen from the mathstodon community posted the following picture:

4-piece puzzle

Let me first explain what is in the picture.

The top layer is the problem posed by Lloyd in 1901: split the figure on the left and put it back into the figure on the right (a square).

The second layer from top to bottom is the solution given by Lloyd himself. From the picture, we can see that he tried to construct a square by using the technique of staggered steps. However, a simple calculation shows that a square cannot be obtained.

The third layer is the five-block structure given by Henry Dewdney in history. Of course, in addition to Dewdney, there are others who have also given their own five-block structure, including a domestic origami and puzzle master named Fu Wei. Even in the references listed by Vesa Timonen later, there is an article published by Fu Wei on WeChat public account, "Origami Ideas New Solution to Centennial Mathematical Problems" (that is, reference 2 at the end of the article, the author gave a novel five-block solution).

Timonen believes that Fu Wei's article is the best document on this issue before him. Although he cannot read Chinese, he read the whole article with the help of translation software. Interested friends can find it and read it. There are also very detailed calculations in it, which can explain why Lloyd's method does not work. At the same time, because the four-piece split has not been found, for a period of time, people tend to believe that there is no four-piece solution...

As for the fourth layer, it is the 4-piece decomposition method discovered by Timonen himself.

Coming up with a method to split the numbers is difficult, but validating an existing method is quite simple. After initial testing by the mathematics community, some people were wondering: Who is Vesa Timonen? How did he do it?

Dual identity

Vesa Timonen has a dual identity. During the day, he works as an embedded software engineer, which he hates. At night, he is a talented designer of puzzles for intellectual toys. Although he is not well-known in the mathematics circle, Timonen is one of the most outstanding educational toy designers in Finland and one of the few puzzle designers. He is also well-known in the domestic circle of intellectual toy enthusiasts (such as the Qiaohuan and Luban Lock).

Vesa Timonen at work

He was interested in magic since he was a child, but as he got older, he preferred to solve magic. Later, his uncle taught him to play tangram, and they began to solve tangram puzzles one by one. As an adult, he began to design puzzles by himself. Many of his works are included in the Hanayama Cast series. Hanayama is a very influential Japanese toy company for puzzle lovers.

Timonen believes that anyone can create a unique puzzle by constantly trying and analyzing their failures. He also emphasizes that failure is part of the creative process, and each failure brings new revelations and inspiration.

The notebook of the puzzle designer is full of geometric shapes and mathematical calculations.

This time, Timonen wrote a software that he hoped could systematically solve various decomposition problems with the help of modern computer computing power. The first problem he chose was the Bishop's Crown problem, and he found the answer quite easily. In fact, if you slide the boundary, you can construct an infinite number of 4-piece decompositions.

Schematic diagram (it may look like the sides of the graph may be of different lengths, but in fact the error is very small)

Jin-Hoo Ahn, a mathematics PhD from Yonsei University in South Korea, provided a non-textual proof for Timonen's solution, which you can enjoy.

Image source: THE MITRE DISSECTION PUZZLE (vesatimonen.github.io)

By the way, Jin-Hoo Ahn is also a puzzle lover and can make puzzle boxes and other intellectual toys. Perhaps this is why he and the protagonist of this article have crossed paths.

More Mathematics

There is a flaw in Timonen's 4-piece decomposition: the green pieces on both sides are actually mirror-symmetric. That is, when the puzzle was put together, the green pieces did not distinguish between the front and back sides. Timonen tried to find a decomposition without mirror symmetry, but was unsuccessful. Instead, the program found more than 50 5-piece solutions without mirror symmetry. So, perhaps there is no 4-piece non-mirror decomposition, but we have not yet been able to prove it. This is also the last missing piece of this century-old puzzle.

In mathematics, we have the famous Wallace-Bolyai-Gerwien Theorem (1807): for any two polygons, one of them can be split into a finite number of small polygons, and then through translation and rotation, they can be pieced together into a second large polygon.

The above theorem guarantees the feasibility of the split, but when the "finite" is limited to a specific value (for example, today's problem is 4 pieces), it cannot be guaranteed that the piece can be successfully pieced together by translation and rotation alone. Flipping the small piece to get a mirror image may be a necessary operation.

In addition to limiting the use of flip-mirror, people sometimes impose stronger restrictions on dissection, such as the famous hinged dissection.

In geometry, an articulated mesh (also called a swing-hinge mesh or Dewdney mesh) is one in which all of its parts are connected by "hinge" joints. This allows rearrangement from one figure to another by a chain of continuous swings without (and without) severing any of the connections. Usually, it is assumed that the pieces are allowed to overlap during folding and unfolding.

The concept of hinged partitioning was popularized by the aforementioned Henry Dewdney, who introduced the famous hinged partitioning of a square into triangles in his 1907 book The Canterbury Puzzles.

However, whether the Wallace-Boey-German Vienna theorem can be extended to articulated partitions? In other words, can two polygons of equal area necessarily be split into each other through an articulated partition? This question has been unresolved.

It was not until 2007 that Erik Demain et al. proved that such an articulated subdivision must exist, and provided a construction algorithm for generating an articulated subdivision. This proof is valid even if the components do not overlap when swinging, and can be generalized to any pair of three-dimensional figures with a common cross-section. However, in three-dimensional space, there is no guarantee that the components will not overlap each other when reassembled. If they overlap on a two-dimensional plane, it is very easy to achieve physically - as long as you understand it as separating the upper and lower layers during the movement. However, if three-dimensional components overlap each other, it is physically impossible for rigid bodies to achieve this. They can only be regarded as mathematical non-entity objects.

Squaring the circle

The previous discussion has always stayed on the figures formed by straight lines. So, can the figures surrounded by curves also be divided and reorganized? Perhaps the ultimate division problem is to square the circle.

Around 450 BC, Anaxagoras, a philosopher, astronomer, and mathematician who believed that "reason rules the world," posed a now-famous mathematical problem while in prison for blasphemy: Using only a compass and an unmarked ruler, can you draw a square with the same area as a given circle?

The answer to this question came in 1882, when the German mathematician Ferdinand von Lindemann showed that this is a problem that cannot be solved with ruler and compass. He showed that pi is a special kind of number, called a transcendental number (which also includes e).

The story could have ended there. But in 1925, Alfred Tarski, one of the most important logicians in human history, re-posed the question by tweaking the rules. He asked if it was possible to cut a disk into a finite number of pieces and then re-assemble them into a square.

In 1988, Miklós Laczkovich answered Tarski's question head-on: the circle can be split and reconfigured into a square, and it is necessary to decompose the circle into about 10^50 pieces. But for quite a long time, the method of squaring the circle always involves some components that cannot be intuitively displayed and constructed: there are pieces whose area cannot be defined (Lebesgue non-measurable sets) and pieces whose area is 0 (zero-measure sets).

It wasn’t until a few years ago that mathematicians Andrew Marks of UCLA and Spencer Unger, now at the University of Toronto, provided the first fully constructive proof of squaring the circle: every piece has a definite area, without exception.

The cost is that they have to decompose the circle into 10^200 pieces, and although it is theoretically constructible, the process is too complicated to be demonstrated.

In February 2022, a paper published online by Andras Máthé and Oleg Pikhurko of the University of Warwick and Jonathan Noel of Victoria University added new content to this old problem. Although their work also divides the circle into about 10,200 pieces, the shape is simpler and easier to visualize. It can even be made into a demonstration video.

Mathematicians have had ideas for simplifying the puzzle pieces further, reducing the total number and the unevenness. Computer simulations Marks has done suggest — but not prove — that the decomposition can be done with no more than 22 pieces. He thinks the minimum number could be lower.

“I bet you could square the circle for less than $20,” he said. “But I wouldn’t bet $1,000.”

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