Question

Alice has 10 balls (all different, imagine each of them have a different color). First, she splits them into two piles; then she picks one of the piles with at least two elements, and splits it into two; she repeats this until each pile has only one element. How many steps does this take?

Answer 1

Alice takes a total of 9 steps to split the 10 different balls into individual piles, where each step is splitting one pile of multiple balls into two smaller piles.

Step-by-step explanation:

Let's solve the problem of how many steps it takes to split 10 different balls into individual piles where each step involves splitting a pile into two. Starting with 10 balls, Alice splits them into two piles. She can pick any two numbers that sum up to 10 (for example, 1 and 9, 2 and 8, etc.), but to minimize the number of splits, she should create the most uneven piles possible. If she chooses the pile with more balls each time, she will minimize the number of steps since every split divides one pile into single balls more quickly.

Here's a step-by-step breakdown:

1. Split the 10 balls into a pile of 1 and a pile of 9 (1 step).
2. Split the pile of 9 into a pile of 1 and a pile of 8 (2 steps).
3. Continue in this manner until all piles have 1 ball. The next splits will be into piles of 7, 6, 5, etc. (3-8 steps).

By continuing in this way, each step reduces the number of remaining multiple-ball piles by one. It takes a total of 9 steps to end up with 10 piles of a single ball each.