Answer: Tuesday Teaser #15
Since we’re making two random cuts, let’s mark the first cut as x and the second cut as y.
It’s clear that we can’t always make a triangle, but what exactly are the conditions we need to have to make a triangle? Well, we simply need that for the longest side to be shorter than the sum of the two shorter sides.
In the case where x < y, we have the three sides x, y-x and 1-y and we can obtain three inequalities for the conditions needed:
- (x) + (y-x) > 1-y => y > 1/2
- (x) + (1-y) > y-x => y-x < 1/2
- (y-x) + (1-y) > x => x < 1/2
In the reverse case, where y > x, we arrive at three likewise inequalities:
- (y) + (x-y) > 1-x => x > 1/2
- (y) + (1-x) > x-y => x-y < 1/2
- (x-y) + (1-x) > y => y < 1/2
By plotting a graph of x against y, we can easily arrive at the answer:
The first set of inequalities relate to the top-left corner, the second set relate to the bottom-right corner. It’s now clear that the probability is 1/4.
Good old algebra.
