## Po-Shen Loh’s coin problem part 2

Yesterday we began looking at a problem that Po-Shen Loh presented during his fantastic public lecture at the Museum of Math. Here’s his lecture:

and here’s the conversation we had yesterday about the problem in his talk:

Po-Shen Loh’s coin problem

It may help to look at the first part of our project before going through this one since we sort of dive in where we left off and don’t really explain the problem in great detail.

As a quick overview, though, yesterday’s problem was about a row of 6 boxes each containing one coin. You play a game in which the only rule is that you can take a coin out of one box and replace it with two coins in the box immediately to the right (so you cannot apply this move to the right most box). You can stop playing the game at any time and take all of the money in the boxes. The question is what is the most amount of money you could win by playing this game?

Today’s project is looking at a twist on that game (explained in the video below). That twist has a surprising impact on the value of the game. Finding the maximum value of the new game is a problem that is too difficult for kids, but playing this new game is a fantastic math activity for kids.

Here’s the introduction to the new game and some thoughts from the boys:

Next I had my older son play the new game to see what would happen. He just finished 6th grade hopefully this video shows how a 12 year old might think about this game:

After my older son’s walk through the game I had my younger son play. He just finished 4th grade, so hopefully this video shows how a 4th grader can think about this game (and also how a kid can learn a lot from seeing it played just once):

Finally, we talked for a bit about strategy for the game. Then I revealed the actual problem from the International Mathematics Olympiad. The boys were a little skeptical that you could ever find a number as big as the number from the problem (!!):

After we finished the boys played the game one more time off camera and found a value of $7*2^{27}$. Not bad for their third try!

This is such a fun problem. I’m excited to see the math camp kids play around with this problem today and I also can’t wait to use a version of this project for 4th and 5th grade Family Math night next year.

I’m so happy to have seen Loh’s MoMath talk – I never would have thought that you could transform IMO problems into great project for kids!