 The Asahi Shimbun

As brainteasers go, the problem seems easy enough to understand.

Is it possible for two triangles of different shapes to have the same perimeter and occupy the same area?

Two young graduates of Tokyo's Keio University have just proved a theorem to solve the conundrum.

Yoshinosuke Hirakawa and Hideki Matsumura said in their published article that although the concept is "quite elementary," finding the proof involved "sophisticated theory of modern arithmetic geometry."

Hirakawa, 28, and Matsumura, 26, both postgraduate students in advanced mathematics at Keio's Faculty of Science and Technology, embarked on the challenging proof question last December.

Their enthusiasm for the project centered around their hopes of encouraging others to "enjoy the profoundness and interest of mathematics."

The vexing question they grappled with was whether a pair of triangles with integral sides, a right-angle triangle and an isosceles triangle, can have the same perimeter and area.

No one had succeeded in proving the existence of such a pair.

The mathematicians converted the geometric question involving numerous triangles into an algebraic equation in their search for an answer.

They then applied diophantine geometry, a modern technique of mathematics, to the task to prove their theorem that “only one pair of such triangles exists.”

The new theorem, to be called the “Hirakawa-Matsumura theorem” from now on, states that there exists a unique pair of a rational right triangle and a rational isosceles triangle, which have the same perimeter and area. The unique pair consists of the right triangle with sides of lengths (377, 135, 352) and the isosceles triangle with sides of lengths (366, 366, 132), excluding pairs of similar triangles.

Hirakawa and Matsumura noted that in ancient Greece, mathematicians were preocuppied with geometric questions such as Pythagorean theorem.

“It seems likely that the theorem we proved was something the ancient Greeks also tried to figure out," Hirakawa said.

"In the end, after thousands of years, advanced modern mathematics has proved the theory," he added. "That is rare and what makes mathematics so exciting."

Matsumura chipped in: “It was my dream to prove a simple theorem like Fermat's last theorem. The fact we achieved what we did is very rewarding.”

Their research results were posted on the website of Journal of Number Theory, a U.S. specialized magazine, on Aug. 24.

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