Tanya Khovanova's Math Blog

The puzzle In the Details by Derek Kisman is one of my favorite MIT Mystery Hunt puzzles of all times. I even wrote a blog post advertising it and another post with comments on the solution. This puzzle type became known as fractal word search.

In the standard word search, you have a grid of letters and a list of words. You need to find the words written in the grid in all eight directions. The unused letters provide the answer, or a clue to the answer, of the word search puzzle.

In the fractal word search, you have a grid of letters and a list of words. It looks like a regular word search, but you will not find all the words inside the given grid. The given grid is only a snapshot of the whole grid on some particular level k. To go to level k+1, you have to use replacement rules: each letter is replaced by a 2-by-2 block of letters. This creates a much bigger grid where you might find more words from the given list.

An interesting question is, where do you find the replacement rules? In Derek’s puzzle, the rules were not given, but the initial grid was level 2. So you could notice that this grid can be decomposed into 2-by-2 squares, and there are only 26 different squares, implying that each square corresponds to a letter. Assuming that replacing the 2-by-2 squares with correct letters will allow you to find more words from the list, you can decipher the replacement rules. This will allow you to get to level 1 as well as any other level. Small levels are easy to search, but on large levels, the grid gets so huge that it might not fit in the memory of a computer, or a million computers. That is why this puzzle was presented at the MIT mystery hunt, but not at any other puzzle hunt. It is quite difficult: one of the given words is hiding on level 86.

I liked the puzzle so much, I included it in one of my lectures. After I gave my talk at Brown University, a student, Klára Churá, approached me. She got as fascinated with the puzzle as I was. We ended up collaborating on the paper Fractal Word Search: How Deep to Delve. As the title suggests, we focused on finding the upper bound of the level where we could find a word of a given length. We had two parameters: the size of the alphabet n and the block size b used in the replacement rules.

For different reasons, the most interesting case is words of length 3. I will leave it to the reader to figure out why this is the most interesting case, or the reader can check our paper. We showed that any word of length 3 appears no later than level n³ + n² + 1.

When we posted the paper, I sent the link to Derek. He immediately wrote a program and showed that our bound is fairly tight. His code is available at GitHub. He created a configuration that puts a 3-letter word at depth LCM(a,b,c)+1, where a,b,c ≤ n-3. If n is even, this gives us a lower bound of (n-3)(n-4)(n-5) + 1. If n is odd, this gives us a lower bound of (n-4)(n-5)(n-6) + 1. In any case, asymptotically, it is very close to our upper bound.

I am not as excited about the MIT Mystery Hunt as I used to be. So, for this year’s hunt, I didn’t go through all the puzzles but present here only the puzzles that were recommended to me. I start with math, logic, and CS.

• Scan, a bunch of numbers. The puzzle requires some programming.
• Flamingo, several grids that look like Nikoli puzzles.
• Jigsaw Slitherlink.
• Under Pressure, evaluating equations.

Then we have some word puzzles.

• Deposit at the Word Bank.
• Body Language, a crossword.
• Fren Amis, a crossword in many languages.
• Look Before You Leap, a crossword with seemingly meaningless clues.

Now, the rest.

• Queen Marchesa to g4 starts with a Magic-the-Gathering card showing a chess piece: white queen.
• New Rules, weird sentences about games.
• Itineratum de Urbes Universum, looks like Latin.
• Retro Chess Puzzle.

Every year, I used to blog about math-related puzzles from that year’s MIT mystery hunt. However, I want to skip this year. I participated in the hunt, and a few of the puzzles I saw were not good enough. Moreover, the puzzles had boring steps or were inelegant. I am not motivated to sift through every math puzzle in the hunt and check its quality.

This year, instead of cataloging all the math puzzles, I will present a short list of puzzles from the hunt that were recommended to me by others. Most of these puzzles are unrelated to math, but I checked them out, and they seem cool.

After the hunt, I asked two people about their favorite puzzles. Both of them mentioned Showcase. Looking at it, I know why. I’ll give you a hint: this puzzle will appeal to people doing competitive programming.

Here are some other recommendations.

• Scicabulary, this is a word puzzle.
• Natural Transformation, you are presented with some very structured images.
• Collage, this is a funny farm.
• Cute Cats, you are presented with many images, some of them with cats.
• Hall of Archaeology, this is a Nikoli-type puzzle.
• Museum Rules, this puzzle looks like a children’s picture book.
• This Puzzle is Just Another Regular Cryptic, this is a cool cryptic puzzle.
• Parsley Garden, you are presented with a grid of words.
• Dispel the Bees, you are presented with crossword clues.

Every year, after the MIT Mystery Hunt was over, I would go through all the puzzles and pick out the ones related to mathematics. This year, I didn’t feel like doing it. Besides, I think it is more important to collect quality puzzles rather than all the puzzles by topic. So my new collection is the puzzles recommended to me, which I might like.

I start with math and logic puzzles.

• Crooked Crossings: A lovely mixture of language and logic.
• Albumistanumerophobia: You are given datasets.
• Lists of Large Integers: You are given integers that are products of two primes. Some of the integers are ridiculously large. I love the puzzle.
• The Colour Out of Space: You are given long division cryptarithms.
• Dancing Triangles: You are given Nikoli-type puzzles on triangular grids.

I continue with computer science.

• The Day You Begin: You are given an interactive input box requesting students’ names.
• Red Herring: You are given a weird image.
• Sorcery for Dummies: You are given an interactive box.

I carry on with some non-math fun.

• Oxford Children’s Dictionary: You are given incomplete dictionary definitions.
• The Times They Had: You are given colorful scored epitaphs.
• La Ronde: You are given a crossword with weird numbering. I quite enjoyed working on it.
• Love, Actually: During the hunt, this puzzle had an intermediate interaction during which an extra clue was sent separately.

I conclude with the plot device round. All the puzzles in this round are relatively easy. But our team got stuck on them until we realized that we already had the answer, which was not a single word. Here are some of the puzzles that were specifically recommended.

• Calculated Whisk: A fun word puzzle.
• A Crying Shamus: You are given crossword clues.
• First Contact Lens: You are given crossword clues and letters.
• Heavenly Bodice: A variation on Star Battle.
• THE NONSENSE SHOW: You are given poems. (This puzzle is number 4 on the Fruit Around page.)

I usually collect puzzles related to math after each MIT mystery hunt. I just discovered that I never reviewed the 2013 hunt, though I started writing the post. Plus, I knew the puzzles of that year better than other year’s puzzles: I was on the writing team. Not to mention, the hunt had some real gems.

I start with mathematical puzzles:

• In the Details by Derek Kisman. This is one of my favorite puzzles ever involving evil fractal word search. I even posted the puzzle and the solution on my blog.
• Integers and Sequence by me. I used my number gossip database to include cool facts about numbers. The numbers form sequences, as hinted by the title. (The title had a typo: sequences should be plural.)
• Permuted by Victoria de Quehen, David Roe, and Andrew Fiori, with illustrations by Kiersten Brown, Turing machine by Adam Hesterberg, and ideas from Daphna Harel and David Farhi. The puzzle has many mini puzzles related to different areas of mathematics.
• Basic Alphametics by David Farhi and Casey McNamara. I love this puzzle and often give it to my students.
• 50/50 by Derek Kisman; server by Ben Buchwald. This is a multi-layered statistics puzzle with a fantastic design and should be included in statistics books. It was the most difficult puzzle of the hunt. Unfortunately, the link is broken, but luckily, I blogged about this puzzle.

Logic puzzles:

• Color Sudoku by Byon Garrabrant and Tanya Khovanova.
• Turnary Reasoning by Timothy Chow, Alan Deckelbaum, and Tanya Khovanova. The puzzle looks like a mixture of chess, checkers, and Magic the Gathering.
• Paint-by-Symbols by R.M. Baur, based on an idea by Eric Wofsey. As the name suggests, this is a paint-by-numbers puzzle.
• Time Conundrum by David Farhi. Many people loved this puzzle.
• Portals by Palmer Mebane. Ten interconnected Nikoli-type logic puzzles. It is a very difficult puzzle with a fantastic design; I immensely enjoyed testing it.
• Random Walk by Jeremy Sawicki. Another Nikoli-type masterpiece that I enjoyed.
• Lineup by Charles Steinhardt and Palmer Mebane. Another paint-by-numbers with a twist.
• Agricultural Operations by Palmer Mebane, based on an idea by Dan Zaharopol. Kenken with a mathematical twist. It might be the most mathematical puzzle in the hunt, a masterpiece that I greatly enjoyed.
• Lojicomix by Robyn Speer and Alex Rozenshteyn; art by DD Liu.

Computer science puzzles:

• This Page Intentionally Left Blank by Dan Gulotta, with some help from Robyn Speer. Another puzzle worthy of textbooks: It covers different ways how information can be hidden.
• Halting Problem by Dan Gulotta. You have to analyze programs in different languages: the programs are designed to run for a VERY LONG time.
• Evolution by Karen Rustad; idea and some screenshots by Asheesh Laroia. A puzzle about email clients.
• Git Hub by Robyn Speer. A git repository puzzle.
• Call and Response by Asheesh Laroia, Glenn Willen, and Charles Steinhardt. You are given only an IP address.

Crypto puzzles:

• Infinite Cryptogram by Anders Kaseorg.
• Security Theater by Sean Lip. I enjoyed this puzzle that uses various ways to encrypt information.
• Open Secrets by Tanya Khovanova and Robyn Speer. A puzzle with many famous ciphers.
• Caesar’s Palace by Jason Alonso, with casino-formatting by Robyn Speer, clue phrase by Adam Hesterberg. An encrypted crossword which I greatly enjoyed.
• Famous Last Letters by David Farhi. Another encrypted crossword.

Word puzzles:

• Split the Difference by R.M. Baur and Eli Bogart. A puzzle with cryptic clues, which I enjoyed.
• Mind the Gaps by R.M. Baur and Eli Bogart. You are given an empty rectangular grid without black cells and cryptic crossword clues.
• Changing States by Charles Steinhardt, Robyn Speer, and David Roe. A lovely easy puzzle which I enjoyed tremendously.
• Wordplay by Ken Fan, Matt Jordan, Tanya Khovanova, Derek Kisman, and Ali Lloyd. You are given grouped cryptic clues.
• Plead the Fifth by Eli Bogart and R.M. Baur, based on an idea by Eric Wofsey. An elegant puzzle with several AHA moments.
• CrossWord Complex by Robin Baur, Eli Bogart, and Jeff Manning. The puzzle consists of six crosswords that turn into serious mathematics.
• Ex Post Facto by Derek Kisman; idea by Tom Yue. An inventive multi-layered crossword, which I greatly enjoyed.
• The Alphabet Book by Halimeda Glickman-Hoch and Robyn Speer, based on an idea by Bryce Herdt. Loved it.
• Funny Story by Lilly Chin, Eric Mannes, and Jenny Nitishinskaya.
• Loss By Compression by Charles Steinhardt and Robyn Speer. A cool puzzle.
• A Regular Crossword by Dan Gulotta, based on an idea by Palmer Mebane. A hexagonal crossword with regular expressions as clues.
• A Set of Words by David Farhi. You are given several boggle grids. I loved this puzzle very much.
• Czar Cycle by Yasha Berchenko-Kogan. This is a weird puzzle that uses English, Russian, and Greek alphabets.

Misc puzzles:

• Substance Abuse by David Reiley and Timothy Chow, with assistance from Jill Sazama, Shrenik Shah, and Glenn Willen.
• Puzzle-Regarding Vehicle by Katie Steckles, Paul Taylor, and Ali Lloyd. An interesting and difficult puzzle.
• A Walk Around Town by Sean Lip, Ludwig Schmidt, and Freddie Manners, with help from Mary Fortune and Jonathan Lee. The puzzle looks like a walk around town but has more to it. The idea is awesome.
• Something in Common by Dan Gulotta and Tanya Khovanova.
• Too Many Seacrest by Robyn Speer. A cool and funny puzzle.
• Analogy Farm by Robyn Speer, Adam Hesterberg, and Alan Deckelbaum; additional code by Anthony Lu and John Stumpo. A superb analogy game. I loved it so much that after we solved it, I came back and resolved it myself. I am not sure it works anymore, though.
• I Can See For Miles by David Roe, based on an idea by Charles Steinhardt. You are given aerial views of buildings. I usually do not like puzzles that involve googling, but using google maps in this puzzle made me feel like I am flying around the world.

Other puzzles I worked on:

• Cambridge Waldo by Tanya Khovanova starring Ben Buchwald, Adam Hesterberg, Yuri Lin, Eric Mannes, and Casey McNamara. The goal of this puzzle was to allow a chance for people to go for a walk around Cambridge. I am not sure we were successful: it could be that people used Google street-views to solve this puzzle.
• Lost in Translation Gaetano Schinco, Natalie Baca, and Tanya Khovanova. You are given tangrams.
• Linked Pairs by Herman Chau and Rahul Sridhar, with crosswords by Adam Hesterberg, Dan Gulotta, and Jenny Nitishinskaya, Manya Tyutyunik, and Tanya Khovanova. You are given three crossword grids, each with two sets of clues.
• Magic: The Tappening by John Wiesemann, with help from Dan Gulotta and Tanya Khovanova.

Each year I look at the MIT Mystery hunt puzzles and pick ones related to mathematics, logic, and computer science. I usually give additional comments about the puzzles, but this year’s titles are quite descriptive. Let’s start with mathematics.

• Yes or No?
• Exactly.
• Numbers.
• Triangles.
• The Greatest Jigsaw.
• Clueless.
• Ignorance.
• The IMO Shortlist.

Now Nikoli-type logic puzzles. I really enjoyed “Fun with Sudoku” during the hunt.

• Better Bridges.
• ★.
• Bounce House.
• Slitheɹlᴉuʞs.
• Fun With Sudoku.

And computer science.

• Illiterate Programming.
• A Bit of Light.
• Complexity Evaluation.
• Analog Circuitry.
• Escape From Life.
• Extensive.
• In C.
• Befuddled.

• 4, 6, X, 9, 10, 12, 14, 15, 16
• 1, 2, 6, 24, 120, X, 5040, 40320, 362880
• 2, X, 3, 4, 7
• 1, 2, 3, 4, 5, 6, X, 8, 9, 153, 370, 371
• X, 2, 3, 4, 5, 6, 7
• 6, 28, 496, 8128, X, 8589869056, 137438691328
• 0, 1, 1, X, 4, 7, 13, 24, 44, 81

This is the puzzle I designed for yesterday’s event at the Museum of Mathematics. This puzzle is without instructions — figuring out what needs to be done is part of the fun. Solvers are allowed to use the Internet and any available tools. The answer to this puzzle is a word.

2020 MIT Mystery Hunt

Every year I write about latest MIT Mystery Hunt puzzles that might be appealing to mathematicians. Before diving into mathy puzzles, I would like to mention two special ones:

• Fortune Cookies—Our team laughed at this one.
• No Clue Crossword—Our team was puzzled by this one.

Unfortunately math wasn’t prominent this year:

• Food Court—This is a probability puzzle that is surprisingly uninspiring. There is no mystery: the puzzle page contains a list of probability problems of several famous types. But this puzzles can find great use in probability classes.
• Torsion Twirl—Mixture of dancing and equations. I love it.
• People Mover—Logical deduction at the first stage.

On the other hand, Nikoli-type puzzles were represented very well:

• The Ferris of Them All—Several different Nikoli puzzles on a wheel.
• Toddler Tilt—Not exactly a Nicoli puzzle, but some weird logic on a grid, some music too.
• The Dollhouse Tour—Not exactly a Nicoli puzzle, but some weird logic on a grid, some pictures too.
• The Nauseator—The first part of the puzzle is a huge nonogram.
• Domino Maze—A non-trivial Thinkfun puzzle.
• Backlot—Finding a path on a grid with a fractal structure.
• Whale—Variation on Rush Hour.

Some computer sciency puzzles:

• Hackin’ the Beanstalk—Hidden algorithms.
• Turtle—LogoWriter.
• Bear—Origami.

Cryptography:

• The Scottish Display—You are given trigrams.
• Pig—The pigpen cypher.
• ANDARAC—You are given gibberish looking telegrams.

A couple of puzzles with the mathy side hidden:

• Tunnel of Love.
• Pied Piper—A puzzle type that I like and promoted is hidden here.

Every year I review MIT mystery hunt from a mathematician’s point of view. I am way behind. The year is 2020, but I still didn’t post my review of 2019 hunt. Here we go.

Many puzzles in 2019 used two data sets. Here is the recipe for constructing such a puzzle. Pick two of your favorite topics: Star Trek and ice cream flavors. Remember that Deanna Troi loves chocolate sundae. Incorporate Deanna Troi into your puzzle to justify the use of two data sets.

On one hand, two data sets guarantee that the puzzle is new and fresh. On the other hand, often the connection between two topics was forced. Not to mention that puzzle solving dynamic is suboptimal. For example, you start working on a puzzle because you recognize Star Trek. But then you have to deal with ice cream which you hate. Nonetheless, you are already invested in the puzzle so you finish it, enjoying only one half of it.

Overall, it was a great hunt. But the reason I love the MIT mystery hunt is because there are a lot of advanced sciency puzzles that can only appear there. For example, there was a puzzle on Feynman diagrams, or on characters of representations. This year only one puzzle, Deeply Confused, felt like AHA, this is the MIT Mystery hunt.

Before discussing mathy puzzles I have to mention that my team laughed at Uncommon Bonds.

I will group the puzzles into categories, where the categories are obvious.

Mathy puzzles.

• Bubbly—Combinatorial game theory.
• Rules of Order—Fractals mixed with movies.
• Cubic—A long list of cubic polynomials with integer coefficients.
• Twisty Passages—Famous number properties that remind my of my Number Gossip website.

Here are some logic puzzles, in a sense that Sudoku is a logic puzzle.

• Lantern festival—A cool mixture of Slitherlinks and Galaxies.
• Invisible Walls.
• Place Settings.
• Middle School of Mines—Minesweeper.
• Moral Ambiguity—Nonograms with a twist.
• Connect Four—Mastermind. There was a strong hint that the extraction step was also mastermind. My team spent some time trying to mastermind the ending, until we backsolved. The extraction step was not mastermind. The final grid in the puzzle had the word CODE written in red. It corresponded to letters CDEO found at that location. Given that the letters were not in alphabetical order, it gave the ordering, which didn’t exist in the puzzle. Anyway, you can see that I have a grudge against this puzzle. This could have been a great puzzle. But it wasn’t.
• Schematics—Tons of Nikoli puzzles of different types.

Now we have logic puzzles or another type, where you need to draw a grid. These are puzzles of the type: Who lives in the White House?

• Compromised.

Now we have logic puzzles or yet another type, where you need to figure out which statements are true and which are false.

• True and False².

Now some cryptography.

• Tales From The Crypt.
• You’re Gonna Need a Bigger Gravy Boat.
• The Bill.
• Helvetica Is Only an Okay Font.
• A Good Walk Spoiled—Some decyphering and Vim.
• Have You Seen Me?
• Chain of Commands.
• Picture Book.

And some programming.

• Turtle Power!

Miscellaneous.

• Hexed Adventure II: Hexed Again!—A text adventure game with a lot of logic.
• Loaded—Dice of different shapes.
• The Lives of Π—Large numbers.
• Battle of the Network Stars—A lot of actors in a grid.

This is a math blog, but from time to time, I write about other things. Today I have something to say about puns, which I adore.

I also like gym, but rarely go there: it doesn’t work out. I stopped using stairs, because they are up to something. I wanted to learn how to juggle, but I don’t have the balls to do it.

I work at MIT, the work place with the best dam mascot: Tim the Beaver. My salary is not big, and I stopped saving money after I lost interest. I’m no photographer, but I have pictured myself outside of MIT too. I am a mathematician, which is the most spiritual profession: I am very comfortable with higher powers. I praise myself on great ability to think outside the box: it is mostly due to my claustrophobia. I am also a bit of a philosopher: I can go on talking about infinity forever.

I would love to tell you a joke. I recently heard a good one about amnesia, but I forgot how it goes.

My biggest problem is with English. So what if I don’t know what apocalypse means? It’s not the end of the world!

I never get tired of puns and here is my list of pun puzzles from the MIT Mystery Hunt:

• We Are All Afraid to Die. I edited this 2018 puzzle and enjoyed it very much.
• Mass Aid. I didn’t work on this 2018 puzzle, but it looks like a lot of fun.
• Capital Punishment. 2017 puzzle that looks tempting.
• Typecasting. A 2017 puzzle with pictures of actors.
• Losers. A 2016 puzzle that I would be very bad at.
• Be a Star!. A 2015 puzzle that I enjoyed very much.
• Feeling Bluefin. A 2015 puzzle that looks delicious.
• Timbales. A 2011 puzzle that might be the most famous pun puzzle.

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