Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. There are two types: Riddler Express for those of you who want something bite-size and Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,1 and you may get a shoutout in next week’s column. If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter.
Riddler Express
Paul Schafer, a very long walk:
You find yourself on the eastern shore of Lake Tahoe and you start a long journey on foot by walking southeast. After you cross the border into Utah, you keep walking. When you reach the Four Corners Monument, you step diagonally across it into New Mexico. There, you replenish your water supply and walk east for many miles until you cross into Oklahoma. You keep walking east until you reach Tulsa. You rest up for a few days and then head east to Fayetteville, Arkansas. Here, you turn southward and hike the 300 miles to Shreveport, Louisiana. You enjoy a po’boy sandwich and turn back toward the north. You cross back into Arkansas, step into Missouri close to Branson, and then, in Iowa, you pass just a few miles east of Des Moines as you head north. Finally, you decide you’re too tired to go on, so you end your trek at the SPAM Museum in Austin, Minnesota.
You’ve just walked through parts of nine states. What do these nine contiguous states have in common that none of the other 41 states share?
Riddler Classic
From Michael Sarkis, a puzzle that his friend was asked to answer during an interview at an investment bank:
There is a square table with a quarter on each corner. The table is behind a curtain and thus out of your view. Your goal is to get all of the quarters to be heads up — if at any time all of the quarters are heads up, you will immediately be told and win.
The only way you can affect the quarters is to tell the person behind the curtain to flip over as many quarters as you would like and in the corners you specify. (For example, “Flip over the top left quarter and bottom right quarter,” or, “Flip over all of the quarters.”) Flipping over a quarter will always change it from heads to tails or tails to heads. However, after each command, the table is spun randomly to a new orientation (that you don’t know), and you must give another instruction before it is spun again.
Can you find a series of steps that guarantees you will have all of the quarters heads up in a finite number of moves?
Solution to last week’s Riddler Express
Congratulations to 👏 Tori Courtney 👏 of Phoenixville, Pennsylvania, winner of last week’s Riddler Express!
Last week, you were faced with a series of number pairs corresponding to some coincidences in American history, and your job was to fill in the missing pair.
2, 6
?, ?
9, 23
17, 36
26, 32
The missing pair is 41, 43.
The pairs are the orders of the U.S. presidents who have shared last names. The second president was John Adams and the sixth president was John Quincy Adams. The ninth president was William Henry Harrison and the 23rd president was Benjamin Harrison. And so on. The only such pair missing was the elder and younger George Bush, who were the 41st and 43rd presidents, respectively. That pair comes second because the list is in alphabetical order: Adams, Bush, Harrison, Johnson, Roosevelt.
Solution to last week’s Riddler Classic
Congratulations to 👏 Patrick Nevins 👏 of Cincinnati, winner of last week’s Riddler Classic!
Last week’s Riddler Classic cast you as a university professor who was walking from one campus building to another through a square courtyard, 50 feet by 50 feet. You entered the courtyard from the center of its west wall and exited from the center of its south wall. You didn’t like sharp turns during your walk, so you made sure any turn you made had at least a 5-foot radius. You also don’t like to cross your own path when walking. Here’s an example of how you might’ve walked across the courtyard:
The puzzle: Given these constraints, what was the longest walk you could’ve taken?
The longest possible walk — assuming your feet are small enough — is infinite. So your next lecture might start a little bit late. The key to plotting this endlessly lengthy constitutional lies in spirals.
Our winner Patrick showed the basic setup — a type of spiral to which you can add as many loops as you want, while never crossing yourself or making any sharp turns:
Solver Daniel Wilcox’s solution was inspired by Fermat’s spiral. And solver Mike Seifert surmised that you must be a professor of medieval art and architecture, and therefore familiar with the idea of a unicursal labyrinth — a type of lengthy meditative path which your walk through the courtyard will come to resemble. Mike provided the following example of a path, which can be made arbitrarily long by adding more and more rings, closer and closer together:
Finally, solver Laurent Lessard provided the following “proof-by-GIF” of the existence of an infinite path, showing that more courtyard loops can be packed in endlessly:
Enjoy your walk. See you never.
Want more riddles?
Well, aren’t you lucky? There’s a whole book full of the best puzzles from this column and some never-before-seen head-scratchers. It’s called “The Riddler,” and it’s in stores now!
Want to submit a riddle?
Email me at oliver.roeder@fivethirtyeight.com.