Time For Some Abstract Math. Drink Up.

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.

Last weekend, hundreds of expert puzzle solvers descended on Cambridge, Massachusetts, for the 2019 MIT Mystery Hunt, and hundreds more solved remotely across the country. I attended and wrote about the Hunt last year; it’s one of the oldest and most complex puzzle competitions in the world — and basically a Riddler column come to life. I loved it so much that I’ve decided to make it an annual tradition to reprint a couple of the Hunt’s most Riddlery puzzles. This year’s were written by members of a puzzling superteam called Setec Astronomy, and they are set within the theme of this year’s Hunt: holidays. (There’s always a theme.)

Fair warning: These are very hard! And their setups are more arcane than those of the typical Riddler — in some cases you may go down a few dead ends before you even know what it is you’re trying to solve. When I finally solved a couple of Hunt puzzles last weekend, I felt like I was levitating out of my chair. The answer to each puzzle is an English word or phrase — you’ll know it when you’ve found it, I promise. The language preceding the puzzles is a very lightly edited version of what Hunt teams saw. It’s known in Hunt-speak as “flavor text” — it both describes the puzzles and offers clues as to how to begin solving. You can also find some solving resources on the Hunt’s website. If you need another hint, find me on Twitter.

## Puzzle 1: Bubbly

Puzzle by Sami Casanova, art by Jesse Gelles

Ray and Blanche sit down with their jug of champagne to play a game of Bubbly. Blanche, as she always does, goes first. Blanche surveys the opening state with a smile …

… and pops bubble S.

Now it’s Ray’s turn, and he chooses to pop bubble H. Since it’s impossible to pop H without popping both A and E (which surround H), A and E are popped as a consequence.

It is Blanche’s turn again, and her move is to pop bubble T.

“You’ve got me — I resign,” says Ray. “If I pop I, you win with N as the last possible move. And if I pop N, you win with I.”

“Don’t worry,” says Blanche. “You never had a chance! I was guaranteed to win, as long as I started with bubbles E, N or S.”

“Congratulations!” says Ray. “Shall we play again?”

“I would, but it looks like we’re running low on champagne,” Blanche replies. “How about our new friend here sits in for me, and you two start while I get us a remedy for our champagne situation?”

What are your winning starting moves?

## Puzzle 2: State Machine

By Matt Gruskin

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.

-9 – ◐ + ◀ – ◳ + ▥ + ◉

-9 – ◪ + ◰

-8 – ▦ – ◔

-7 + ▲ + ▣ – ◫ + ◲ – △ – ⌘

-6 – ◭ + ◩ + ▩ + ◈ + ◆ – ◑ – ◲ – △

-6 + ◵ – ◇

-6 – ◴ + ◆ – ▼ – ◒ + ▶

-5 – ◵ – ◇ + ■ + ◒ – ◷

-5 – ◵ + ◁ – ● + ○ – ◷

-5 + ▩ + ◍ + ◶ – ◑ + ◓ – ▧

-4 + ◭ + ▣ – ⌘

-3 + ◕ + ▽ – ◔ + ◮ + △

-3 + ◀ + ◫ – ◑ – ◲

-3 + ▤ – ◒ + □ + ▶

-3 + ◁

-1 – ◭ + ▲ + ▥ + ◲

-1 + ◩ – ◆ + ▼ – ▷

-1 – ◆ + ◨ – ▷ + ◒ + □

+0 + ◕ + ◭ + ▦ – ◆ + ▣ + ◫ + ◔ + □

+0 + ◧ + ◀ – ◪ – ◳ – ▥ – ▧

+0 + ▽ + ▦ + ▣ + △

+0 + ▦ + ◮ – △ – □

+0 – ◩ + ◴ – ◫ – ▷ + △ + □ – ▶

+0 – ◴ – ▷ – ▨

+0 + ◶ – ◳ – ◰ – ◉ + ✮ + ▧

+1 + ◕ – ◭ – ▲ + △

+1 – ◀ – ▩ + ◈ – ◫ + ◱ + ▧

+1 – ◩ + ▩ + ▼ – ◍

+2 + ◴ – ▩ – ◆ – ◫ – ▨

+2 – ◇ – ◁ + ●

+3 – ◧ – ◐ + ◪ – ◰ + ◉

+3 + ◧ + ◀ + ◲ + ⌘ + ◉

+3 + ◪ – ◓ – ◱ – ▧

+3 – ◆ – ◨ + ◔ – △ + ▶

+3 + ◇ + ● – ○

+3 – ◍ + ◶ + ◱

+4 + ◧ – ◳ – ◰

+4 – ▩ + ◓ – ◱ – ▨

+4 + ◬ – ◇ + ◷

+4 + ◨ – ■ – ◒

+5 + ▤ + ◨ – ■ + ● – ▷ – ▶

+6 + ▤ + ● – ◒

+8 – ◭ – ◀ + ◈ – ◫ – ▥ + ⌘

+8 + ◐ – ◪ – ◳ + ✮

+8 – ◀ – ◪ – ◶ + ◑ + ◱ + ◉

+9 – ◕ + ▦

+9 – ◧ + ◈ + ◑ – ▥ – ◲ – ◉ – ▧

+9 – ◩ – ◍ + ◫ + ◑ + ◱ – ▨

## Solution to the previous Riddler Express

Congratulations to 👏 Chuck Culman 👏 of Audubon, Pennsylvania, winner of last week’s Riddler Express!

Last week, you were staring at a two-character, seven-segment display, like the kind you might find on a microwave. You wondered, as one does: How many numbers can this screen show in a way that is not ambiguous if the display happens to be upside down? The number 81, for example, would not fit this criterion — if the display were upside down, the number would look like 18. The number 71, however, would be fine. It’d look something like 1L — not a number.

There are 100 total numbers the display could show — 00 through 99. Of those, 58 can be safely turned upside down without being ambiguous.

We could go through the list of two-digit numbers one by one — 100 isn’t that many. But we could also look for patterns. For starters, any two-digit number with a 3, 4 or 7 in it is safe — those digits don’t look like a digit anymore when they’re turned over. There are 51 numbers containing these digits, which solver Angelos Tzelepis helpfully illustrated.

Further, some digits look like themselves when turned upside down on such a display: 0, 1, 2, 5 and 8. So we can add “00,” “11,” “22,” “55” and “88” to our total, bringing us to 56. Finally, there are two digits that do look like other digits when turned upside down, but they look like each other: 6 and 9. Thus “69” and “96” still look like “69” and “96” when flipped. Adding those two to our total brings us to 58, our answer.

This puzzle’s submitter, Tyler Barron, plotted the share of numbers that were potentially confusing, both for our two-digit case and for larger displays.

This chart will prove very helpful in my daily life: For some reason, my microwave keeps flipping upside down, and there seems to be nothing I can do to stop it.

## Solution to the previous Riddler Classic

Congratulations to 👏 Jim Ferry 👏 of Leesburg, Virginia, winner of last week’s Riddler Classic!

Last week took us to the world of crossword puzzles and their grids’ rules and conventions. Crossword grids are typically rotationally symmetric, meaning they look exactly the same if turned upside down. All words in the grid must be at least three letters long. All letters must appear in a “down” word and an “across” word. And there can be no “islands” of white squares — the grid must be completely interconnected. Given that a weekday puzzle is usually 15-by-15, how many possible crossword puzzle grids are there?

For starters, solver Laurent Lessard drew the 7-by-7 grids — all 397 of them.

Beautiful! But the numbers get very big very quickly from there as we approach a standard 15-by-15 puzzle. Tyler Barron shared his approach and code, which he used to generate about a thousand 10-by-10 grids. Our winner this week, Jim Ferry, found 40,575,832,476 valid 13-by-13 grids using a dynamic programming approach.

But the programmatic approaches ground to a halt after that and no one — myself included — was able to pin down the exact number of valid 15-by-15 crossword puzzle grids. “This turned out to be very difficult,” Tyler wrote. So it remains an open question! If you make progress, let me know, and we’ll be sure to feature your solution in a future column.

The good news for constructors — and for solvers like me — is that there are still many, many, many puzzles left to go. The XWord Info database, which collects New York Times crosswords and data about them, contains only about 25,000 puzzles. We have at least, oh, tens of trillions to go. Happy solving.

## 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.

## Footnotes

1. Important small print: For you to be eligible, I need to receive your correct answer before 11:59 p.m. Eastern time on Sunday. Have a great weekend!

Oliver Roeder was a senior writer for FiveThirtyEight. He holds a Ph.D. in economics from the University of Texas at Austin, where he studied game theory and political competition.