## Thursday, January 1, 2009

I used Heron's Law the first time I solved this and a buncha triangles the second time I tried. I kept getting 11 acres but, as jd mentioned in the other comment, the elegant solution must be available but it escaped me every time I tried.

The puzzle solution relies on the recognition of three pythagorean triangles. (How one came to these combinations was never quite clear to me, but hey ...)

5-7-√74 triangle would have the same length side as a square of area 74 sq. units.
4-10-√116 would have the same length side as the square of area 116
9-17-√370 triangle would have the same side as a square of area 370

Lo and behold, these triangles could be rearranged as below.

The lake is therefore the difference in area of the big triangle (0.5*9*17 = 76.5) and the three smaller pieces (28 + 0.5*5*7 + 20 = 65.5) which is 11.

So there you have it. In my opinion, this proves that the solution works but gives no indication (to me at least) of how one might solve similar situations in the future and does not enlighten the puzzler to some heretofore unknown rule, theorem or hypothesis. Not a very satisfying solution for me.