Contingencies Online - July/August 2011

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are Pythagoreans, and so we cannot
conceive of any irrational number,”
Philomath scolds. “It’s a holdover from
the old man’s musical obsessions. You
know: thirds, fifths, etc. But the square
root of two is right out.”
Socrates sits in thought for a mo-
ment. Then he asks, “So, it’s probably
tricky, puzzling even, how to construct
two right triangles with integer sides but
the same area.”
“Right you are, Socrates,” says
Philomath. “But if you manage to find
four right triangles, with integer sides
of the same area, we’ll let you in.”
Of course, having more tools at your
disposal, you just might find a shocking-
ly large number of such quadruples of
triangles. To keep it simple, I would like
the four smallest such triangles.

Solution
leg 1
518
280
231
111
leg 2
1320
2442
2960
6160
hypotenuse
1418
2458
2969
6161

Many people asked about how to get to
an analytical solution. I have no idea—
brute force was expected of you.

Solvers
Robert Bartholomew, Andy Blough,
Jason Boice, Colin Bracis, Bob Byrne,
Samantha Casanova, Joshua Davis,
Russ Davis, Andrew Dean, Matthew
Deveney, David Engelmayer, Mark
Evans, Mark Fowler, Mike Giampa,
Rui Guo, Leigh Halliwell, Greg Hansen,
Daniel Henry, Steve Johnson, Ben
Kester, Doug Levy, Orin Linden, Lee
Michelson, Ryan Morgan, Paul Navratil,
Jay Rine, Anthony Rubiano, Frank
Santasiero, Colin Scheriff, Craig Schmid,
Joe Schmuecker, Lenny Shteyman,
John Snyder, Al Spooner, Doug Szper,
Michael Vech

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