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, a new kind of maze:

Bad news: the enemies of Riddler Nation have forced you into a maze. And this maze is weird. The rules are as follows.

- You move between boxes in a grid: up, down, left or right, but never diagonally.
- Your goal is to arrive in the finish square, designated here by a “☺.”
- Your movement is dictated by the symbol inside the square you have just moved to, and each direction is relative to where you’d be facing if you were physically walking the maze. “S” means you continue straight, “R” means you turn right, “L” means you turn left, “U” means you make a U-turn, and “?” gives you the option of any of those four directions.
- An “X” ends your game in failure — think hot lava. (But hey, you can always start over!)
- If you are forced to exit the maze’s grid, you lose — more hot lava.

Your maze is below. You may enter the maze anywhere along the perimeter, giving you 32 options. Some of these, however, immediately fail. If you enter at a “U” on the top of the maze, for example, you must immediately U-turn out of the maze, so you lose.

Can you reach the smiley face? If so, how many moves does it take?

## Riddler Classic

From Jordan Miller and William Rucklidge, three-deck monte:

You meet someone on a street corner who’s standing at a table on which there are three decks of playing cards. He tells you his name is “Three Deck Monte.” Knowing this will surely end well, you inspect the decks. Each deck contains 12 cards …

- Red Deck: four aces, four 9s, four 7s
- Blue Deck: four kings, four jacks, four 6s
- Black Deck: four queens, four 10s, four 8s

The man offers you a bet: You pick one of the decks, he then picks a different one. You both shuffle your decks, and you compete in a short game similar to War. You each turn over cards one at a time, the one with a higher card wins that turn (aces are high), and the first to win five turns wins the bet. (There can’t be ties, as no deck contains any of the same cards as any other deck.)

Should you take the bet? After all, you can pick any of the decks, which seems like it should give you an advantage against the dealer. If you take the bet, and the dealer picks the best possible counter deck each time, how often will you win?

## Solution to the previous Puzzle 1: Bubbly

Congratulations to 👏 Brian Leebrick 👏 of Lynn Haven, Florida, winner of last week’s Puzzle 1!

Last week brought two puzzles from the 2019 MIT Mystery Hunt. They were constructed by members of a puzzling superteam called Setec Astronomy. The answer to each puzzle was an English word or phrase.

In the first puzzle — by Sami Casanova with art by Jesse Gelles — you were presented with a series of images of arrangements of bubbles, some of which were inside of others. You and another player took turns popping the bubbles, and the player who popped the last bubble won. Your job was to identify the winning first moves in each image, and from there extract the word or phrase that was the answer.

This game is a variant of Nim, and for each game, the first player can guarantee themselves a win — the goal of the puzzle is to find all possible winning moves for the starting player.

In the images below, the winning first moves (sometimes there are multiple options) for each game are colored in (using a different color for each game). Proper strategies in Nim can be found using a mathematical idea called the Nim sum.

In the final image, all winning first moves for all games are shown superimposed together.

If you read the letters assigned to each bubble from left to right, top to bottom, they spell **ANSWER RELIEF PITCHER**.

You can also find a Python program that solves this puzzle on the official Hunt solution page.

## Solution to the previous Puzzle 2: State Machine

There were … no winners of last week’s Puzzle 2! So congratulations to my editor, I guess? He predicted that this puzzle was too hard. (He says the same thing at least every other week, so he was bound to be right one of these times.)

The second Hunt puzzle, by Matt Gruskin, began with the following cryptic introduction, known in Hunt-speak as “flavor text.” It’s meant to provide hints as to how to begin solving.

“In Presidents Day Town, you come across an inscription commemorating our nation’s history:

“Our Founding Fathers started this nation from nothing. Our values change over time, and after a number of generations we have finally eliminated all negativity.”

What followed were a series of mathematical expressions, beginning with -9 – ◐ + ◀ – ◳ + ▥ + ◉ and ending with +9 – ◩ – ◍ + ◫ + ◑ + ◱ – ▨. And that was it. So what gives? Here’s a lightly edited version of the solution from from the official Mystery Hunt page:

This puzzle consists of a set of 48 algebraic expressions, containing 48 unique symbols, as well as one numerical constant per expression. Each of the 48 continental states has its own symbol and its own expression. Each state’s expression uses exactly the symbols of the states geographically adjacent to that state, and the solver may eventually realize that within each expression, the states are referenced in alphabetical order — using these facts, we can uniquely match symbols and expressions to states.

States are considered adjacent when they share a land border. The puzzle’s creators considered the Four Corners states to be adjacent to one another. The algebraic expressions will provide confirmation that you’ve picked the correct bordering states.

The next step is to assign an initial value to each state. The flavor text, “Our Founding Fathers started this nation from nothing” is intended as a clue to solvers that the initial value assigned to each state should be zero.

We can now treat the algebraic expressions as transition functions from one “state” of state values to the next. During each transition, we replace a state’s value with a value computed from its geographically adjacent states, according to that state’s algebraic expression.

After repeating this process several times (it turns out that three iterations are needed to solve the puzzle), we reach a state where all values are either zero or positive. Indexing into state names using the positive values — so that a 2 for Idaho gives its second letter, D, or a 5 for Maine gives its fifth letter, E — and writing these letters in the locations of their states on the map, we can read a word from each cluster of states on the map and extract the answer phrase **KATIE BAR THE DOOR**.

## 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.