Day 8: Eight Numbers in a Hat
A teacher organises a game to improve her students' arithmetic.
She puts the numbers \(0,1,2,\dots, 7\) into a hat.
Every turn one student takes out two numbers from the hat and replaces them with the sum of their sum and their product (so the number of numbers in the hat decreases by one).
The game ends when there is one number left in the hat. Incredibly, it turns out that this number is always the same. What is this number?