Welcome to The Riddler. Every week, I offer up a problem related to the things we hold dear around here: math, logic and probability. These problems, puzzles and riddles come from lots of top-notch puzzle folks around the world — including you! You’ll find this week’s puzzle below.
Mull it over on your commute, dissect it on your lunch break, and argue about it with your friends and lovers. When you’re ready, submit your answer using the link below. I’ll reveal the solution next week, and a correct submission (chosen at random) will earn a shoutout in this column. Important small print: To be eligible, I need to receive your correct answer before 11:59 p.m. EDT on Sunday — have a great weekend!
Before we get to the new puzzle, let’s return to last week’s. Congratulations to 👏 Brian Skinner 👏 of Boston, our big winner. You can find a solution to the previous Riddler at the bottom of this post.
Now here’s this week’s Riddler, a picky eater’s lunchtime puzzle, inspired by @hatathi.
Every morning, before heading to work, you make a sandwich for lunch using perfectly square bread. But you hate the crust. You hate the crust so much that you’ll only eat the portion of the sandwich that is closer to its center than to its edges so that you don’t run the risk of accidentally biting down on that charred, stiff perimeter. How much of the sandwich will you eat?
Extra credit: What if the bread were another shape — triangular, hexagonal, octagonal, etc.? What’s the most efficient bread shape for a crust-hater like you?
And here’s the answer to last week’s Riddler, concerning some mathematicians who resign the next Friday after they realize they’ve made a mistake. The super-mathematician, who shows up and informs the department that someone has made a mistake in a paper, really does a number on the department: All 10 mathematicians resign, simultaneously, on the 10th Friday. Why? Here’s the explanation adapted from our winner this week, Brian Skinner, a physics postdoc at MIT:
We can do a recursion on the number of mathematicians, working our way up from a simpler case, in which there is only one mathematician. Let’s call this lone person A. She would resign on the first Friday, knowing that the “someone” who the super-mathematician said had made a mistake must be her because she is the only possibility.
If there are two mathematicians, then mathematician A knows that the other, we’ll call her B, will retire on Friday if A has never made a mistake (remember that the mathematicians don’t know whether they themselves have erred, but they know the others have) because the only possibility would be that B was the mistake-maker. So neither A nor B will resign on the first Friday, and both will use that information to infer that they have each made a mistake. So both A and B will resign on the second Friday.
If there are three mathematicians, A, B and C, then A will assume that if she has never made a mistake, B and C will immediately eliminate her from consideration and will go on to play the two-person game that is described above. The expected outcome is that both B and C will resign on the second Friday. If neither B nor C resigns on the second Friday, then A’s assumption about the two-person game is wrong, and she knows that she has made a mistake. So A resigns on the third Friday, and B and C do as well. Continue this recursive logic, and you’ll reach the conclusion that all 10 must resign on the 10th Friday. Ruthless play by that super-mathematician!
Brian suggests that the super-mathematician goes on to write a best-selling management book entitled “I, Super-Mathematician: How I managed to get an entire math department to resign simultaneously after only 10 weeks.” I’d read it.
Paul Van Metre gives us a bit more intuition: The super-mathematician has given the members of the math department something they hadn’t had before: common knowledge. Now that all the mathematicians know that everybody knows that at least one of them has made a mistake, they will observe how everybody else reacts to this information. Because each one knows that, of the k mathematicians in the department, there is either k-1 or k among them that have made a mistake, they will wait until the k-1 Friday before deducing that they themselves have made a mistake and will resign on the kth Friday.
Elsewhere in the mathematical and puzzling worlds:
- “Super Mario Bros.” is NP-hard. [Gizmodo]
- This company makes $8,000 jigsaw puzzles that are “maddeningly difficult.” [The Fader]
- Puzzles about stamps and spirals. [Wall Street Journal]
- Puzzles on paths. [Expii]
Have a great weekend! Enjoy your sandwiches.