This week’s puzzle can be thought of as a missing page from the Phantom Tollbooth, a book very near and dear to my heart. Here is the hero of our story, Milo, with his faithful companion, Tock the watchdog:
As our puzzle begins, Milo and Tock (and the illustrious Humbug) are caught in the trap of the Terrible Trivium. The Trivium is a faceless man who aspires to waste the time of those he catches, by smooth talking them into trivial, endless, time-wasting tasks. Like emptying a well with an eyedropper. Anyway, Milo and his friends have been wasting time for a while, and are about to make their getaway, when the Trivium gives them one last task–win a simple game involving dominos and a checkerboard.
I’ll skip the whole story–listen to the linked audio to hear the dialogue! The task that the Trivium sets Milo on is to cover a modified checkerboard with dominos, without any overlap of dominos or any dominos hanging over the side. You see, the dominos are rectangles that each exactly cover two squares of the checkerboard. It is easy to cover a standard 8 by 8 checkerboard with dominos by these rules–simply put 4 dominos end to end in each row. However, the Trivium has changed the checkerboard. He’s gotten out his saw and cut the upper left and lower right corner squares off of the board. His challenge to Milo is to cover this new board by the same rules: each domino covers two squares, no dominos should overlap or hang over the edge. Milo starts in to work…
Milo tries and tries different rearrangements of the dominos. Each time, he finds that it doesn’t quite work out. The Trivium is incredibly pleased with this–he knows that there are millions of arrangements of dominos on a checkerboard like this, and that it will take Milo his entire life to try them all. However, Milo suddenly stops. He realizes that he’s been had. Milo has noticed a very simple fact that makes it clear that he will never be able to cover the board in dominos. Though the Trivium is furious, he can’t argue with Milo’s simple and impeccable logic, and he is forced to let Milo and his friends leave. Of course, they go on to finish their quest and rescue the Princesses Rhyme and Reason. All because of Milo’s amazing observation.
The puzzle–What very simple fact did Milo notice, and how did he know, then, that he could never cover the board?
Send your solution to me at firstname.lastname@example.org! The author of the best solution that I receive by the middle of next week will win an incredible Math Mostly/Somewhat Science T-shirt, courtesy of the Wesleyan University Department of Mathematics and Computer Science. Make sure and tune in to WESU 88.1 FM (in Middletown, or stream live at wesufm.org) next week, from 2:30-3:00 on Friday afternoon (Eastern time) to hear the solution to the puzzle. I’m so excited to read people’s solutions!
[Checkerboard picture source–David Hardman via Wikimedia Commons]