How to play the triangle game

Extract

We are given n sticks of lengths 1, 2, 3, …, n and three are selected at random. Which selections enable a triangle to be formed?

This question can be written in the form of a game: you win if a triangle can be formed from the three numbers interpreted as side lengths, you lose if they do not.

We let ℕ_n = {1, 2, 3, …, n}, and see there are two variants of the game:

• • From ℕ_n draw at random three numbers sequentially. At each stage do not replace the drawn number.

• • From ℕ_n a number is drawn at random and is recorded. In three stages three numbers are drawn. Replace the number at each stage.