Lewis Carroll's last piece of mathematics was a problem that involved finding three right-angled triangles of the same area. His diary entry for
19 December 1897 reads:
Sat up last night till 4 a.m., over a tempting problem, sent me from New York, "to find three equal rational-sided right-angled triangles." I found
two, whose sides are 20, 21, 29; 12, 35, 37: but could not find three.
Rational-sided means that all three sides are whole numbers or fractions, while 'equal' means that they have equal areas. Lewis Carroll's triangles each have an area of 210 square units. He was closer than he knew. The smallest solution of this problem consists of the three right-angled triangles with sides:
40, 42, 58, 24, 70, 74, and 15, 112, 113.
Each has an area of 840 square units. The first two of theses have sides twice the length of those that Lewis Carroll found. It is now known that there are infinitely many solutions to this problem; another is:
105, 208, 233, 91, 120, 218, and 56, 390, 394.
-- from Lewis Carroll in Numberland by Robin Wilson
Many thanks to Sture Sjöstedt for correcting the error in the last sentence above. It should read:
It is now known that there are infinitely many solutions to this problem; another is:
105, 208, 233, 182, 120, 218, and 56, 390, 394.