Problem of the Month
These problems are my attempt to give high school students, like you, a glimpse into "real mathematics" by creating a math puzzle which will challenge you to think in ways that you have never thought before. Typically, in your high school math classes, you are given a problem to solve and a fixed step-by-step procedure for solving that type of problem — follow the steps, and you are certain to get an answer. However, "real mathematics" is not like that. When mathematicians work on problems, they often start off with no idea about how to solve the problem. There are lots of math questions that no one yet knows how to solve. I invite you to stretch your mind and take my mathematics challenge. I look forward to seeing what you come up with.
— Joseph DiMuro, Associate Professor for the Department of Math and Computer Science
The Last Jellybean
It's Easter time, and siblings Alice and Bob have managed to get an impressive haul of jellybeans; they have 38 jellybeans between them. They decide to play a game with the jellybeans before they start eating them.
Alice and Bob put their jellybeans in a pile on a table, with a bowl nearby. Here's how the game works: Alice splits the jellybeans into two piles. Bob chooses one of the piles, and puts all the jellybeans from that pile into the bowl; those jellybeans are now out of the game. Bob then splits the remaining jellybeans into a pile, and Alice puts one of those two piles into the bowl. This process continues until one player can't split the remaining jellybeans into two piles, because there's only one jellybean left. That player loses.
Here's a sample game: Alice divides the 38 jellybeans into piles of size 14 and 24. Bob puts the pile of size 24 into the bowl, and splits the remaining 14 jellybeans into piles of size 5 and 9. Alice puts the pile of size 5 into the bowl, and splits the remaining 9 jellybeans into piles of size 2 and 7. Bob puts the 7 jellybeans into the bowl, and splits the remaining 2 into two piles of size 1. Alice puts either jellybean into the bowl, and is stuck with the last jellybean. So Bob wins.Remember, there are 38 jellybeans, and Alice will split the pile first. If both siblings play optimally, who will win? And what is the winning strategy?
Send your answers to email@example.com by April 30, 2018.