Today, we are going to play a game. It takes place on a grid with an infinite number of rows and columns, and it starts with three coins in the top left corner of the grid, as illustrated here.
The game has one move. At any moment you can remove any coin, and replace it with two coins, one in the cell immediately below the removed coin, and one in the cell immediately to the right of the removed coin. For example, just say we remove the coin two along on the top row, the grid will now look like this:
One more thing: you are only allowed to remove a coin in the way described above if the cell below that coin and the cell to the right of that coin are empty. In other words, if a coin has an empty cell below it and an empty cell to its right, then you can remove that coin and replace it with two coins, one in the cell below it and one in the cell to its right. That’s it. There are no more rules. Let’s redraw the initial grid with a red line separating the four cells in the upper left corner. These four cells are called the ‘bank’.
The challenge: design a strategy that gets all the coins out of the bank. If you can’t, prove that it is impossible.
Get some coins from your pocket or purse. Draw a grid. Play around with the coins on the grid. The challenge should be quite easy, right?
Warning: this is a tricky puzzle. But it’s worth the effort. The solution is spectacular, involving a simple insight. Well, simple, once you know it.
The problem was devised by the Argentine mathematician Carlos Sarraute. As the writer of a column about puzzles, I am constantly looking for new problems. Most of the time, what I come across are variations of puzzles I have seen before. This one, however, was completely new to me. It blew me away.
PLEASE NO SPOILERS and I’ll be back with the solution and a discussion at 5pm UK time today.
In other puzzle-related news, I have a new book out! So You Think You’ve Got Problems? is a compendium of 200 of my favourite brainteasers, together with historical context and discussion of the mathematical ideas involved. I’ve chosen puzzles I find especially ingenious, and that provoke feelings of wonder and surprise. It’s out next month, but you can pre-order here.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
