MIT’s magazine Technology Review has a neat section in the back called “Puzzle Corner.” The section always has clever problems, though I’ve never thought to share one with the kids until seeing the most recent issue. It was problem N/D #3 that caught my attention this week:
Link to the November / December Puzzle Corner
The problem is referred to as a “famous problem” though I have not seen it before. Here’s the statement:
“Some men sat in a circle, so that each had two neighbors.
Each had a certain number of coins. The first had one coin more
than the second, who had one coin more than the third, and so
on. The first gave one coin to the second, who gave two coins to
the third, and so on, each giving one coin more than he received,
for as long as possible. There were then two neighbors, one of
whom had four times as much as the other.”
The two questions are (1) how many people were in the circle, and (2) how much money did they have?
Although putting together a full solution to this problem on their own is almost certainly too advanced for my kids, the neat thing about this problem is that it is still accessible to them. They are able to play around with some of the simple situations and notice the key ideas. Turing those ideas into algebraic equations was difficult for them, though hopefully instructive nonetheless. At the beginning of this project I wasn’t sure if we’d make it all the way to the end, but I was pleasantly surprised to find that we did. Here’s how our morning went:
We began with an introduction to the problem and then worked through a few simple examples. Right from the start my younger son started noticing some of the patterns in the problem that lead you down the path to the solution. Having the Lego figures and snap cubes for visual cues seemed to really help the kids notice these patterns:
Next we moved on to a slightly more complicated example. We noticed a pattern about one of the players in the game running out of money. It is interesting to hear kids talk about these patterns when they don’t quite have the mathematical language in their vocabulary to describe them. The cool thing for me in this part is that my younger son began to describe the patterns in terms of “n”.
Continuing with our exploration, we now looked at a situation where the person with the lowest amount of money at the start had more than one coin. The specific case we look at here starts with the person with the least amount of money having two coins. The three things that I was pleased about here were:
(1) the boys continued to describe the patterns with “n”‘s,
(2) they noticed what the pattern of money would be at the end of the game, and
(3) they formed a theory about who would run out of money.
I was happy to hear all of these conversations.
The next step – translating the patters that we’ve seen into math – was probably the most difficult. This discussion went a little long, but I thought it was important for them to try to figure out the equations on their own. There was one little stumbling block coming from the fact that there are two variables, but overall they made good progress translating their ideas to math:
So, having written down some expressions for the number of coins at the beginning and the end of the game in the last video, we now use the fact that the total number of coins doesn’t change during the game to write down an equation that will help us solve the puzzle. The main math involved in this stage is evaluating the sum 0 + 1 + 2 + 3 + . . . . + (n-1). My older son was able to remember the clever idea that makes adding up this set numbers really easy!
Looking at this piece now, I wish I had done a better job explaining the process. The first expression that we write down is the number of coins at the beginning of the game. The second expression is the number of coins at the end of the game. Since those expressions have to be the same, we can solve for the complicated expression that gives the “large” number of coins that the one player has when the game stops. I need to go back in time and add that explanation in to the video!
Next up is solving the equation we wrote down in the last video. This part is just a little bit of algebra, though the expressions involved are complicated enough that we have to be extra careful not to make any mistakes. Luckily (and maybe miraculously!) we work through the algebra without a mistake. Amazingly, solving our complicated equation just boils down to a problem about prime numbers!
Now my favorite part – after slogging through all the algebra in the last three videos we got an answer. The problem was engaging enough for the boys that their reaction to getting that answer was “should we try it?”! Super happy to see that level of engagement, and glad that they found this problem to be as fun as I did. The last step here was just checking the result:
So, I think this problem in Technology Review is a great problem to work through with kids. There are so many great opportunities to notice and speculate about patterns, and that process is such an important part of learning math. Also, even if turning the ideas in the patterns into algebraic equations is near the edge of their mathematical ability – as is the case with my kids – they should still be able to learn something from looking at the equations. At least in this experiment today, the kids were able to stay engaged with the problem through the 10 minutes discussion of that algebra. At the end they were really excited to see that the solution we found actually worked! Thanks to “Puzzle Corner” for this super fun problem.