Teaser 2689: The right choice
From The Sunday Times, 6th April 2014 [link]
I am organising a tombola for the fete. From a large sheet of card (identical on both sides) I have cut out a lot of triangles of equal area. All of their angles are whole numbers of degrees and no angle exceeds ninety degrees. I have included all possible triangles with those properties and no two of them are identical. At the tombola entrants will pick a triangle at random and they will win if their triangle has a right-angle. The chances of winning turn out to be one in a certain whole number.
What is that whole number?
[teaser2689]
S2T2 
Jim Randell 8:59 am on 1 December 2022 Permalink |
Once we have chosen the three angles of the triangle (each an integer between 1° and 90°, that together sum 180°), we can form a triangle with the required area by scaling it appropriately.
This Python program counts the number of possible triangles and the number of them that are right angled.
It runs in 53ms. (Internal runtime is 1.0ms).
Run: [ @replit ]
from enigma import (decompose, fraction, printf) # generate possible triangles triangles = lambda: decompose(180, 3, increasing=1, sep=0, min_v=1, max_v=90) # count: # t = total number of triangles # r = number with a right angle r = t = 0 for (a, b, c) in triangles(): t += 1 if c == 90: r += 1 # output solution (p, q) = fraction(r, t) printf("P(win) = {r}/{t} = {p}/{q}")Solution: The chance of winning is 1/16.
There are 720 triangles, and 45 of them are right angled.
I think it would be annoying to choose a triangle with an 89° angle.
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