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
From Tom Hanrahan, Riddler Nation’s Master Mazist, a jumbled journey:
Walking through an enchanted land, you are informed that you are coming up to a maze. You reach a very odd set of words arranged in hexagonal patterns. Accompanying them are instructions for how to navigate through the hexes to arrive at the maze’s end. You see the words:
IF BLUE Z ASKS AMY EE DANCES QUEEN Z O O
“OK, what,” you ask yourself. Luckily, you find further instructions, involved though they may be:
- These letters form the pattern of how to navigate this maze. Once you enter, you must follow the options provided.
- Your goal is to exit the maze (“Win!”).
- When moving from one hex to another, you must obey the basic two rules:
- If you come to a hex with a consonant, you must turn left: either a mild left (60 degrees, or one hex to the left), or a sharp left (120 degrees).
- If you come to a hex with a vowel (“Y” is a vowel), you must turn right: either a mild right (60 degrees, or one hex to the right), or a sharp right (120 degrees).
- You may never proceed straight or back directly up.
- If you travel outside of the pictured hexes, or enter the dreaded gray hex, you must return to the start.
- You must pass through the letter “M” before you are allowed to finish.
Can you navigate your way to the end of the maze?
Riddler Classic
From Benjamin Danard, a navigational challenge of a different kind:
There are 48 contiguous states in the United States. Each of these states shares at least one border with another state. You are planning a road trip in order to visit as many states as possible. The only restriction is that you can only cross a border from one state to another one time. You can visit a state any number of times, as long as you do not enter or leave it from a border you have already crossed.
How many states can you visit without crossing the same border twice?
Solution to last week’s Riddler Express
Congratulations to 👏 Jonathan Chow 👏 of Ottawa, Ontario, winner of last week’s Riddler Express!
Last week, you were given an empty 4-by-4 grid and a marker. You could color in any of the individual squares you liked and leave any of them untouched. After you did that, I would then secretly cut out a 2-by-2 piece of the square and show it to you, without rotating it. You then had to tell me where the piece was (e.g., “top middle” or “bottom right,” etc.) in the original 4-by-4 square. The challenge: Could you design a square for which you’d always know where the piece came from?
Indeed you could. In fact, there were 6,188 different ways to do so. This puzzle’s submitter, Tyler Barron, illustrated 150 of them:
Perhaps the most elegant solution is to simply color in a small 2-by-2 square smack in the middle of the big 4-by-4 square. One might call it the donut. It’s quick to check whether that one, for example, works as a solution to challenge. No matter what piece I cut, the pattern you’ve colored will be different — a colored piece in the lower right, or along the bottom, and so on — and so you’ll be able to tell me where I cut in the original square.
Mmm, donuts.
Solution to last week’s Riddler Classic
Congratulations to 👏 Sjoerd De Vries 👏 of Nancy, France, winner of last week’s Riddler Classic!
Last week you were charged with constructing an optimal tournament which consisted of some predetermined number of players and number of games. Specifically, you wanted to devise a tournament in which the best player won most often. All the players were ordered by a skill level, though you didn’t know the order, and the better player won each individual game two-thirds of the time. Your challenges were to devise these tournaments for four players and four games, and for five players and five games.
For four players and four games: First, name the teams A, B, C and D. Have A play B, and then have the winner play both C and D, and finally have the winners of those two games play for the championship. (If the winners are the same, that player wins the championship). The best player wins this tournament 38 out of 81 times, or about 46.9 percent of the time. (Note that this is a somewhat higher chance than the standard three-game bracket-style tournament in which the best player wins about 44.4 percent of the time.)
For five players and five games, it’s a bit more involved: As before, name the teams A, B, C, D and E. First, pit A versus B, and have the winner (say it’s A) play C. If A wins, have A play D and E, and have the winners of those games play for the title. If A loses, have A play D. If A wins, A plays E and the winner plays C. If A loses, have C play E and that winner play D. The best player wins this one — which I would love to hear explained on SportsCenter — about 1,496 out of 3,645 times, or about 41 percent of the time.
This puzzle’s submitter, Erich Friedman, has put together a handy table — frankly, one of my all-time favorite tables — with the optimal tournaments and their relevant probabilities for various combinations of numbers of players and games. Not too surprisingly, the chance that a tournament produces the best player as its winner increases with the number of games, and decreases with the number of players.
May the best player win … you know, some of the time.
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.