Usually when I’ve posted puzzles to the blog the’ve had a basis in Probability or Statistics. One exception to this was the mutilated chessboard puzzle, whose inclusion I justified by pointing out that mathematical logic is an important strand in the theory that underpins Statistics. Definitely not the only strand, but important nonetheless.

In this same spirit, here’s another puzzle you might like to look at. I’ll give references to the author and so on when I write a follow-up post with the solution. But, if you think I should only be doing posts that are strictly Probability or Statistics related, please just ignore this post. It’s only related to Statistics in the same way that the mutilated chessboard puzzle was. Having said that, I will use follow-up discussion to this puzzle as a lead-in to some important statistical ideas.

Anyway, here’s the puzzle. Look at this grid of squares…

Actually, you have to imagine the grid extending indefinitely downwards and to the right. In the top left-hand corner of the grid you can see a 2-by-2 section of the grid that’s been marked with red lines, and 3 coins have been placed in that section. Your task is to remove the coins from that section by following these rules:

- Coins are removed one at a time.
- When you remove a coin you must replace it with two coins, one immediately below and one immediately to the right of the one that’s been removed.
- If a coin does not have a free space both immediately below and to the right, it cannot be removed until such space becomes available.

You have to find the sequence of moves that results in the section inside the red square being emptied of coins, or explain why it’s impossible to find such a sequence.

To make things easier, you can try the puzzle using this gif. Again, I’ll give credits and references for this work when I write the follow-up post.

To play just click on the green flag to start and then click successively on the coins you’d like to remove. You will only be allowed to remove coins according to the rules above, and when you do legally remove a coin, two new coins are automatically added, again according to the stated rules. If you just want to start over, press the green flag again.

(I don’t know what the red button is for, but DON’T PRESS IT).

So, can you find the sequence of moves that releases all of the coins from the red square? Or if it can’t be done can you explain why?

Please write to me if you want to discuss the puzzle or share your ideas. I’ll write a post with a solution shortly.