Sunday Times Teaser 2689 – The Right Choice

by Des MacHale

I have cut many triangles of equal area from large sheet of card that is the same on both sides. The angles in all these triangles are whole numbers of degrees and all are 90 degrees or less. I have included all possible, different triangles of this form.

Entrants in a competition pick a triangle at random (replacing it afterwards) and they win if their triangle contains a right angle.

It turns out that their chances of winning are “1 in a whole number”. What is this number?