> 3 Pythagorean Triangles have the same hypotenuse, and
> areas A, B, and C. Can A = B + C?
No such triplets are known, but there is no proof that they cannot exist. If
anyone proves no such triplet of triangles exists, they qualify for a $100
reward from Martin Gardner.
Gardner offers a $100 reward for an example of a 3x3 magic square, with nine
distinct entries, such that each entry is a perfect square, or a proof that
no such square exists. Given such a square, one could find a triplet of
triangles satisfying Ed's conditions.
There are some relations among Pythagorean triples, 3-term arithmetic
progressions of squares, and 3x3 magic squares of squares. Some of these are
covered (along with relations to certain elliptic curves) in an article by me
in the October 1996 Mathematics Magazine.
The Pythagorean triple a, b, c, with a^2 + b^2 = c^2, corresponds to the
3-term arithmetic progression of squares A^2, B^2, C^2 that has A = b - a, B
= c, C = b + a. The area of the triangle corresponds to the difference
between terms of the arithmetic progression. Suppose there are three
Pythagorean triangles with the same hypotenuse, and the sum of the areas of
two is equal to the area of the third. It would then follow that there are
three 3-term arithmetic progressions of squares so that all three have the
same middle term, and the sum of the differences between terms of two of the
progressions is equal to the difference between terms of the third
Any 3x3 magic square (whether of squares or not) can be decomposed into three
3-term arithmetic progressions that have the same difference between terms,
and have corresponding terms in arithmetic progression (see my article,
above, or Martin Gardner, Riddles of the Sphinx, Mathematical Association of
America, 1987, page 137; Maurice Kraitchik, Mathematical Recreations, Second
Edition, Dover, NY, 1953, page 148; American Mathematical Monthly, Problem
E3440, Proposed Vol. 98, No. 5, May 1991, page 437, Solved Vol. 99, No. 10,
December 1992, pages 966-967). Let the term ap-array denote the arrangement
of these three progressions into a 3x3 array in the obvious way. For
example, the magic square
10 1 7
3 6 9
5 11 2
becomes the ap-array
1 2 3
5 6 7
9 10 11
If all the terms of an ap-array were squares, then the central column, the
central row, and the diagonal from the upper left to the lower right would be
three 3-term arithmetic progressions of squares that have the same middle
term and the sum of the differences of the terms of the first two would be
the difference of terms of the third.
One related question is how many terms of an ap-array can be squares. Andrew
Bremner has determined that any six terms of an ap-array can be squares.
Only one ap-array with seven squares is known (first published by Lee
Sallows): [[23^2, 205^2, 285^2], [373^2, 425^2, 222121], [527^2, 565^2,
360721]]. No ap-arrays with eight or nine squares are known. An example of
Ed's triangles would give an ap-array with seven squares, that would be
different from Lee's square.
Another question is whether there can be an ap-array so that each of the
terms in the left column, the top row, and the upper-left-to-lower-right
diagonal can all be squares. This would correspond to three Pythagorean
triples with the same difference between the shorter sides, and the areas of
two adding to the area of the third.
Work of Darmon, Merel, and Ribet (and Denes before them) shows that there are
no 3-term arithmetic progressions of powers greater than two, so much of the
above is hopeless for higher powers.