In our first Math Circle of the year, we ushered in Gardner’s centennial with a look at some problems from his classic book My Best Mathematical and Logic Puzzles. The problems seem almost whimsical because of how compelling they are. They are, in fact, quite serious; they touch on deep mathematical issues. I had 5 problems prepared to look at today, but the 2 we had time for were so interesting that they took up all of our time.
First, we considered “The Mutilated Chessboard.” “The props for this problem are a chessboard and 32 dominoes. Each domino is of such size that it exactly covers two adjacent squares on the board. The 32 dominoes therefore can cover all 64 of the chessboard squares. But now suppose we cut off two squares at diagonally opposite corners of the board and discard one of the dominoes. Is it possible to place the 31 dominoes on the board so that all the remaining 62 squares are covered? If so, show how it can be done. If not, prove it impossible.”1
As is our custom in math circle, the students posed questions, posited conjectures, and tried a bunch of ideas.
- Can we use more than 31 dominoes? (…no)
- Are 31 enough to cover all the squares?
- Why are we using paper, instead of real, dominoes? (…because Gardner uses the phrase “covers exactly”)
- Can we switch the position of the removed squares? (…that’s up to you. Will it still be the same problem?)
- Let them overlap
- Tilt them
- 31 dominoes seems like enough to cover 62 squares since 31+31=62, so it should work
- It might not be working b/c there are not really 31 dominoes here (let’s count them!)
- It doesn’t seem to work – this may be because of the location of the removed squares
There were many more than listed here. The students spent half an hour testing various conjectures: 5 students, 1 chessboard, 31 tiles, total cooperation. Wow, that Gardner really knew how to write a great question. (Even the students pointed this out.) Interestingly, if you look up this question online, many versions do not require you to give a reason if it doesn’t work. Our students working conjecture was that it doesn’t work with diagonal corner squares removed, but it does with adjacent corners. They do not know why, but had enough for one day.
They saw that I had brought my fox puppet Waggy, and wanted to know which Gardner problem involved him. So I told them:
“Waggy has a secret life. On Saturday afternoons, he goes to New York City. He has two activities he likes to do there – He likes to go to Central Park with his boombox to rollerskate to the song What Does the Fox Say with a group of like-minded skaters, and he also likes to go to Fox Beach on Staten Island to chase wildlife. To visit Staten Island, he takes a train on the downtown side of the platform (and then takes the ferry); to visit Central Park he takes a train on the uptown side of the same platform. Since he likes roller skating and wildlife chasing equally well, he simply takes the first train that comes along. In this way, he lets chance determine whether he rides to Central Park or Staten Island. The young fox reaches the subway platform at a random moment each Saturday afternoon. Uptown and downtown trains arrive at the station equally often – every 10 minutes. Yet for some obscure reason he finds himself spending most of his time on Staten Island; in fact on the average he goes there nine times out of ten. Can you think of a good reason why the odds so heavily favor Staten Island?”2
This is based upon Gardner’s problem “Bronx vs. Brooklyn.” The math and most of the wording are the same. But I updated it to make it appropriate for our group of kids. (Email me for the original version if you’re curious.) We spent a few minutes discussing the saga of Fox Beach, a real neighborhood in NYC that may be soon reverting to its natural state. Then began the questions and conjectures. After half an hour, the kids had come up with the same explanation that Gardner did. It was a truly collaborative solution that rose up from conjecture built upon conjecture built upon conjecture. The solution became clear during the final 2 minutes of class, so we may revisit this one again to make sure everyone is on board with it. Everyone seemed to be today, but sometimes deep insights can be fleeting.
Everyone was sad that time was up. “Are there more problems like those?” Fortunately, the answer is yes.