You have 1,000 silver coins and 1 gold coin. Every round you toss all remaining silver coins and the gold coin. All silver coins that land on the same side (heads or tails) as the gold coin are set aside and not tossed anymore. You keep tossing the remaining silver coins (and gold coin) until no silver coins remain.

What’s the distribution of the number of silver coins that are showing heads at the end of the process?

Thanks Asaf Aharoni for this cool riddle!

Spoiler Alert - Hints Ahead

This riddle has a straightforward solution via guessing the answer (which is somewhat surprising!) and showing that it holds via induction. This involves developing a couple of algebraic expressions. The riddle also has a much nicer solution that you can work out in your head. Hint: consider what happens if you keep tossing the coins an infinite number of times and interpret the tosses as a binary expansion of a fraction between 0 and 1.