Maths get proof of 122-year-old triangle-to-selfier puzzle
A long -standing figure mystery has finally been solved
Size transfer of polygon.
About a decade ago Tonon KamataNow a mathematician at Japan Advanced Institute of Science and Technology (JAIST) was mesmerized by the original performance of a Mathematics Museum. It had a triangular tile cut into four pieces that were associated with small hinges. With a simple swing, the pieces rotate around to convert the triangle into a square.
The exhibition detects its origin for a mathematical puzzle published in a 1902 newspaper. Henry dudeniA self-centered English mathematician and puzzle columnist, asked its readers to dissect a sambahu triangle into pieces, which could be re-located in a class. Two weeks later in his next column, he stated that “Mr. CW McClayroy of Manchester”-Charls William McLaroy, a clerk, often wrote to Dudeni with a puzzle solution-a four-tukra solution. After two more weeks, Dudeni reported that none of other newspaper readers had resolved, and since then, the record has been standing. However, it remained unproven, whether a solution was present with low pieces.
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The puzzle came to be known as “Dudeni’s dissection” or “problem of habaradasher”, and was also painted scientific AmericanBack of June 1958. A mathematician for the magazine and long -standing columnist Martin Gardner wrote about the Quality.
,Try scientific AmericanMathematics puzzles,
Now, more than 122 years after the first time it is proposed, the lower and two other mathematicians have finally proved that a solution with low pieces is impossible. The results were posted on the server Arxiv.org in December 2024, titled “Dudene’s dissection is optimal,
“I believe that many people who appreciate mathematics will agree that an unresolved problem appears, the more deeply attracted to those who love mathematics.”
With the Massachusetts Institute of Technology Mathematician Eric Demine and Zest Mathematic Royuhe Uhra, it was developing a new approach to deal with Origami-folding problems using the Kamata Graph theory. In graph theory, a graph is essentially a collection of lines, or edges, and corners, they find points where the edges are found. The edges and corners of a graph can be compared with another to detect deep relationships between the two structures – one approach that Avata thought could help solve the dissection of Dudeni.
A part of the problem is quite simple: a two-tukra solution can be denied by thinking about the obstacles of the problem. For the beginning, the triangle and the square must have the same fields as the pieces are the same. For a square, its longest possible cut is beyond the diagonal. A bit of pen-end-paper mathematics suggests that, unfortunately, the length of the diagonal is very low for the side of its similar-region triangle, which controls the two-tukra solution.
To prove that there are no three-tukra solutions, however, it is very difficult, and that is the cause of a century-delay. Although it is a simple three-tukra puzzle, there are infinite numbers to cut the triangle, says Demine. “Each of those pieces can arbitrarily occur to several edges, and the coordinates of those cuts begin at arbitrary points,” they say. “You have these constant parameters, where there are lots of potential options and lots of roofs that bother it very much. You can’t only cruel it with a computer.”
To deal with the problem, the group classified the potential dissection of a synchronized triangle of how the cuts cut the edges of the triangle. First, the researchers sorted the infinite of cutting ways to cut the triangle into five unique classifications. Then he repeated exercise for a class and got 38 separate classifications.
Subsequently, the researchers tried to match a triangular graph with a square by detecting all possible paths in each size and comparing the resulting collection of the edge and angles. If one of the sections of the classes matched a triangle, it would mean the researchers discovered a three-tukra solution.
The approach turned the continuous problem into a discrete – almost. Within each classification, there are still infinitely many places that can go all these corners, “says Demine. Finally, the group obtained a collection of complex lemon, or intermediate stages in a theorem, which, combined with categories, used evidence by contradiction to find a matching path.
Smith College computer scientist Joseph O’Rurke, who has Wrestling with puzzle For decades and closed and current work was not involved, thinks the evidence of the group could possibly be simplified. He notes that a messy collection of highly specific lemon was taken to eliminate all possibilities, “many other researchers would have left.”
Importantly, if the author can simplify his evidence, the Milan-Dygram technology can highlight a group of original open questions related to the technology. “These problems remind us how much to discover so far,” Cama says. “Anyone can become a pioneer in this limit.”
