These puzzles insist only that the apex angle of the tiles be less than 90 degrees. The illustrated solution achieves 72 degrees for the squattest of the component triangles. How many triangles do we need if the apex angle is constrained to be *less than* 72 degrees? Can these dissections be achieved regardless of how sharp the apex angle constraint is? For example, can an equilateral triangle be dissected into isosceles triangles, all of whose apices are less than 15 degrees? If yes, how many are needed? If no, what is the sharpest constraint that *can* be achieved? On Thu, Feb 23, 2012 at 4:59 AM, Bill Gosper <billgosper@gmail.com> wrote:

Ouch. I never imagined that an "oversized incircle" could isoscelize all seven subtriangles! Had I merely sliced off the largest possible 45-67.5-67.5 and repeated on the remaining 45-22.5-112.5 the process shown here<http://gosper.org/acutefail.png>, I would have found at least -Floor[Tan[69]] solutions instead of Round[Tand[69]].

Many thanks, Hans! --rwg

hans>

...

So, can someone scan in (the picture(s) from) that '61 article? http://chesswanks.com/txt/AcuteIsoscelesDissectionOfAnObtuseTriangle.png

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