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Rights in dispute

Rights in dispute

As a puzzle maker, I sometimes receive emails asking me why this or that solution is given a prize when, in his opinion, his solution was as good as the one he took. I may speak of a mathematical problem in which the one who took the prize followed the custom of taking the result only to the third decimal place, while the one who writes to me complains that he was breaking the horns until he reached the tenth, giving clearly what he considers a better answer.

Keep in mind that I write since the 19th century and there are no calculators, so the good man may have spent more than twelve pages in finding his solution, while the winner approaches the solution by methods that anyone could understand and whose resolution does not take more than half a page, but proves to have understood the principle of the riddle well and that it could take the answer to any number of decimals if a prize were given to patience and endurance.

An arbitrator may not always respond to the reasons that lead him to give one or the other a prize, but it may be that the winning answer came several days earlier, or that it was clearer, or more intelligent and sharp than the others. I tell you all these things to encourage you to be clear and concise when solving puzzles. Stay away from mathematical terms. The one that has to be clear is the solution, not the explanations or the arguments ...

In the illustration you can see some miners discussing their land. It seems that they had obtained permits on some farms of the same size. Each farm has the shape of a right triangle, all with the same surface, but of different dimensions. A triangle has a base of 140 feet, a height of 48 feet and a hypotenuse of 148, another has a base of 84 feet, a height of 80 feet and a hypotenuse of 116 feet. Both triangles have an area of ​​3,360 feet.

¿What dimensions does the third triangle have, assuming it has the same surface as the other two and that all three sides are integers?

Solution

Finding the third triangle with an area of ​​3360 feet is so complicated that it is said that renowned mathematicians such as Euler and Laplace said it was impossible to discover a fourth triangle.

The dimensions are: Base 224 feet, height 30 feet, hypotenuse 226 feet.