Tanya Khovanova's Math Blog

I was on the writing team for the 2018 MIT Mystery Hunt. I am pleased that the hunt got very positive reviews from the participants. I spent tons of hours working on the hunt and it is good that folks liked it. I edited and tested a lot of puzzles. Here is my review of these year’s puzzles that are math-related.

I already posted an essay about the puzzles I wrote myself. Four of my five puzzles are math-related, so I am including them below for completeness. I will mention the topic of each puzzle unless it is a spoiler.

I start with Nikoli-type puzzles. Four elegant Nikoli-type puzzles were written or cowritten by Denis Auroux. In all of them the rules of the logic are stated at the beginning. That means the logic part doesn’t contain a mystery and can be solved directly.

• Good Fences Make Sad and Disgusted Neighbors (by Denis Auroux). You can guess by the title that this puzzle was in the emotions round corresponding to sadness and disgust. This is an interesting variation on the hexagonal Slitherlink. This is a relatively easy puzzle.
• Shoal Patrol (by Denis Auroux and James Douberley). Each grid is a combination of Battleship, Minesweeper, and a loop puzzle. These are difficult, but satisfying puzzles. The extraction step is not mathematical and not completely trivial.
• Submarine Patrol (by Denis Auroux and James Douberley). This is a 3D version of Shoal Patrol.
• Hashiwokakuro (Count your bridges) (by Denis Auroux). This is a mixture of Hashi and Kakuro. I enjoyed the puzzle while I tested it. The extraction is trivial.
• A Learning Path (by Tanya Khovanova and Xavid). This is a path logic puzzle that was targeted for new hunters. It contains self-referencing hints and solving techniques.

There were several puzzles that were very mathematical.

• X Marks the Spot (by Zachary Chroman). A very non-trivial puzzle about triangle centers.
• Tournament Organization (by Scott Harvey-Arnold ). A tournament reconstruction puzzle.
• Cash Cab (by Erica Newman and Justin Melvin).
• Voter Fraud (By Ben Weissmann). A voting puzzle.
• The 10,000 Puzzle Tesseract (by Charles Steinhardt, Zachary Chroman, and Scott Harvey-Arnold ). The most difficult puzzle of the hunt, by far.
• Games Club (by Tanya Khovanova and Sergei Bernstein). This puzzle is about combinatorial two-player games.
• Murder at the Asylum (by Tanya Khovanova). This is a difficult puzzle about liars and truth-tellers.
• Studies in Two-Factor Authentication (by Brandon Avila). A very elegant puzzle, that is one of my favorites.

There were also some math-related or computer-sciency puzzles.

• The Next Generation (by Colin Liotta). I enjoyed being an editor of this puzzle.
• Disorientation (by Alex Churchill). This puzzle has a beautiful visual component.
• Message in a Bottle (by Nathan Fung). The puzzle doesn’t look like it has something to do with mathematics, but my testing of it was very satisfying. I guessed from the start what it was about.
• Self-Referential Mania (by Justin Melvin). Self-referential logic puzzle, which I enjoyed editing.
• Bark Ode (by Elizabeth French, Justin Melvin, and Erica Newman). The pictures are so cute.
• Executive Relationship Commandments (by Robin Deits, John Toomey, and Michele Pratusevich). I didn’t see this puzzle until after the hunt. I wish I could have tested this puzzle with my son Alexey, who is a computer scientist.

There were also several decryption puzzles:

• Word Search (by Tanya Khovanova). A crypto word search.
• Texts From Mom (by Elizabeth French and Justin Melvin ). A text enciphered with emojis.
• Marked Deck (by Colin Liotta and Leland Aldridge). One of my favorite puzzles. Hunters received a physucal deck of cards that was laser cut. You can buy the deck at Etsy. The art in this puzzle is beautiful, but the puzzle also has a non-trivial decryption step.

I was on the writing team of this year’s hunt, which was based on the movie “Inside Out.” One of our goals was to create an easy first round to allow small teams to have a full hunt experience. Our first round consisted of 34 puzzles related to five basic emotions: joy, sadness, disgust, fear, and anger. Each emotion had its own meta puzzle. And the round had a meta-meta puzzle and a runaround. As I tend to write easy puzzles, I contributed three puzzles to this emotions round. The puzzles had references to corresponding emotions that were not needed for the solve path. They were inserted there for flavor.

• A Learning Path (jointly with Xavid) is a Nikoli-type logic puzzle that was targeted for new hunters. It contained self-referencing hints and solving techniques. It was in both disgust and fear.
• Word Search is a word search with a twist. The words are related to fear and sadness.
• Games Club (jointly with Sergei Bernstein) is a puzzle on the topic of combinatorial two-player games. It is pure sadness.

I also wrote another easy puzzle called A Tribute: 2010-2017 (jointly with Justin Melvin, Wesley Graybill, and Robin Diets ). Though the puzzle is easy, it is useful in solving it to be familiar with the MIT mystery hunt. This is why the puzzle didn’t fit the first emotions round.

I also wrote a very difficult puzzle called Murder at the Asylum. This is a monstrosity about liars and truth-tellers.

The puzzle Family Ties was written for the 2013 MIT Mystery Hunt, but it never made it to the hunt. Here’s your chance to solve a puzzle no one has seen before. I wrote the puzzle jointly with Adam Hesterberg. The puzzle is below:

Mathematics professor S. Lee studies genealogy and is interested in the origins of life.

1. Alexei Mikhailovich Ivanov
2. Alexei Petrovich Ivanov
3. Amminadab
4. Anna of Moscow
5. Arador
6. Arahad II
7. Arassuil
8. Arathorn I
9. Arathorn II
10. Aravorn
11. Argonui
12. Asger Thomsen
13. Caecilia Metella Dalmatica
14. Egmont
15. Eldarion
16. Ellesar
17. Endeavour
18. Faustus Cornelius Sulla
19. Henry Frederick
20. Hezron
21. Isaac
22. Ivan the Great
23. Ivan the Terrible
24. Jacob
25. James I and VI
26. James V
27. Jens Knudsen
28. John Francis
29. Joseph Patrick
30. Joseph Patrick
31. Jørgen Jensen
32. Judah
33. Knud Nielsen
34. Margaret Stuart
35. Maria Donata
36. Mary Stuart
37. Matthew Rauch
38. Mikhail Ivanovich Ivanov
39. Niels Møller
40. Ole Pedersen
41. Peder Petersen
42. Peter Jørgensen
43. Petr Alexeyevich Ivanov
44. Pharez
45. Ram
46. Robert Francis
47. Rose Elizabeth
48. Søren Thomsen
49. Thomas Olsen
50. Ursula Gertrud
51. Vasily I of Moscow
52. Vasily II of Moscow
53. Vasili III of Russia
54. Yuri of Uglich
55. Zerah

The last MIT Mystery Hunt was well-organized. It went smoothly—unlike the hunt that my team designed the year before. Sigh. As I do every year, here is the list of 2014 puzzles related to math.

• Build Your Own Sudoku. As it says: Sudoku.
• Oyster Card. Nikoli puzzles related to the London tube.
• A Rose by Any Other. Paint by numbers. I enjoyed this puzzle during the hunt.
• Please Remain Seated. A roller-coaster puzzle, where the extraction needs some math.
• Marking Territory. A Shikaku variation.
• SAT III. Several optimization problems.
• Growth and Fixed Costs. Some financial mathematics.
• Magic Mushrooms. A very cool puzzle with variations of Skyscrapers, Nurikabe, Hashi, Heyawake, Corral, Filomino, Hitori, and Slitherink. I discovered it after the hunt, and enjoyed solving individual puzzles. The extraction is difficult, though.
• Glazing Over. A broken Sudoku puzzle.

There were also several puzzles requiring decoding or having a CS flavor.

• The Spy Who Read Me. Several cipher texts.
• A Puzzle with the Answer CLEMENS. Some encrypted words.
• A Puzzle with the Answer CRONIN. A stream of data.
• A Puzzle with the Answer RICE. Unraveling a huge list of instructions.
• Callooh Callay, World! Programs in a fictional programming language.
• Some Nutty Lines. My favorite decoding puzzle.
• WAT? Some crazy looking cryptograms.
• SKI Trees. Computer sciencey ski slopes.

I want to mention one non-mathematical puzzle.

• Operator Test. It is based on puzzles from the previous years and one of them was Wordplay, co-written by me.

The MIT Mystery Hunt starts on Friday. My old team—Manic Sages— fell apart after last years’ hunt. My new team—Death and Mayhem—started sending us daily practice puzzle to prepare for the hunt.

Today’s puzzle was written by Paul Hlebowitsh. As usual, the answer is a word or a phrase.

Puzzle 3: Humans

“After a long day taking care of the animals, it’s good to unwind by letting the taps flow.”

Tap 1: Sierra Nevada Pale Ale
Tap 2: Lagunitas Pale Ale
Tap 3: 21st Amendment “Brew Free! or Die IPA”
Tap 4: Wachusett Blueberry
Tap 5: Harpoon UFO

Alice, Bob, Carol, Danny, Erica, Fred, and Gregario, the seven children of Noah, went to a local bar before the flood to get drinks. Accounts of night vary, and no one remembers exactly what happened, but some facts have become clear:

1. Everyone had two drinks, in some order.
2. Danny liked his first drink so much he had it again. Everyone else had drinks from different breweries.
3. Erica was the only person to drink an IPA.
4. Four people had the Wachusett Blueberry as their first drink.
5. One person had the Sierra Nevada Pale Ale as their first drink.
6. Alice only drank beers with headquarters in California, in order to spite Bob and Danny.
7. Two Harpoon UFOs were ordered, as well as two Sierra Nevada Pale Ales. 8
8. Alice’s second drink was the same as Fred’s first drink.
9. Bob and Danny hate California and refuse to drink any beer from a company that’s headquartered there.
10. Alice and Erica had the same first drink.
11. Fred had a Harpoon UFO.

Who ordered which drinks and in what order?

This is the promised solution to the puzzle Integers and Sequences that I posted earlier. The puzzle is attached below.

Today I do not want to discuss the underlying math; I just want to discuss the puzzle structure. I’ll assume that you solved all the individual clues and got the following lists of numbers.

• 12 42 18 40 30 24 20
• 2 1 132 42 429 14
• 7 9 1 8 5 3 10 4
• 92 117 70 145 35 1 22 12 5
• 137 1 37 13 107 1013 113
• 30 12 2 42 6
• 70 4030 836 7192

Since the title mentions sequences, it is a good idea to plug the numbers into the Online Encyclopedia of Integer Sequences. Here is what you will get:

• not clear
• Catalan numbers with 5 missing: 1, 1, 2, 5, 14, 42, 132, 429
• not clear
• Pentagonal numbers with 51 missing: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145
• Primeval numbers with 2 missing: 1, 2, 13, 37, 107, 113, 137, 1013
• not clear
• Weird numbers with 5830 missing: 70, 836, 4030, 5830, 7192

Your first “aha moment” happens when you notice that the sequences are in alphabetical order and each has exactly one number missing. The alphabetical order is a good sign that you are on the right track; it can also narrow down the possible names of the sequences that you haven’t yet identified. Alphabetical order means that you have to figure out the correct order for producing the answer.

Did you notice that some groups above are as long as nine integers and some are as short as four? In puzzles, there is nothing random, so the lengths of the groups should mean something. Your second “aha moment” will come when you realize that, together with the missing number, the number of the integers in each group is the same as the number of letters in the name of the sequence. This means you can get a letter by indexing the index of the missing number into the name of the sequence.

So each group of numbers provides a letter. Now we need to identify the remaining sequences and figure out in which order the groups will produce the word that is the answer.

Let’s go back and try to identify the remaining sequences. We already know the number of letters in the name of each sequence, as well as the range within the alphabet. The third sequence might represent a challenge as its numbers are small and there might be many sequences that fit the pattern, but let’s try. The results are below with the capitalized letter being the one that is needed for the answer.

• abundAnt
• caTalan
• dEficient or iMperfect
• pentaGonal
• pRimeval
• proNic or proMic
• weiRd

What is going on? There are two sequences that fit the pattern of the third group and the sequence for the sixth group has many names, two of which fit the profile but produce different letters. Now we get to your third “aha moment”: you have already seen some of the sequence names before, because they are in the puzzle. This will allow you to disambiguate the names.

Now that we have all the letters, we need the order. Sequences are mentioned inside the puzzle. You were forced to notice that because you needed the names for disambiguation. Maybe there is something else there. On closer examination, all but one of the sequence names are mentioned. Moreover, with one exception the clues for one sequence mention exactly one other sequence. Once you connect the dots, you’ll have your last “aha moment:” the way the sequences are mentioned can provide the order. The first letter G will be from the pentagonal sequence, which was not mentioned. The clues for the pentagonal sequence mention the primeval sequence, which will give the second letter R, and so on.

The answer is GRANTER.

Many old-timers criticized the 2013 MIT Mystery Hunt. They are convinced that a puzzle shouldn’t have more than one “aha moment.” I like my “aha moments.”

*****

• (the largest integer n such that there exists a Platonic solid with n vertices, a Platonic solid with n edges, and a Platonic solid with n faces)
• (the largest two-digit tetrahedral number)/(the smallest value the second smallest angle of a convex hexagon with all integer degrees can have)
• (the number of positive integers less than 2013 that are divisible by 100, but not divisible by 70)
• (the number of two-digit numbers that produce a square when summed up with their reverse) ⋅ (the smallest number of weighings on a balance scale that guarantees to find the only fake coin out of 100 identical coins, where the fake coin is lighter than other coins)
• (the only two digit number n such that 2ⁿ ends with n) − (the second smallest, and conjectured to be the largest, triangular number such that its square is also triangular)
• (the smallest non-trivial compositorial number that is also a factorial)
• (the sum of the smallest three positive pronic numbers)

*****

• (the digit you get when you sum up the digits of 2013²⁰¹³ repeatedly until you get a single digit) − (the greatest common factor of the indices of the Fibonacci numbers divisible by 13)
• (the largest common divisor of numbers of the form p² − 1 for primes p greater than three) − (the largest sum of digits that can appear on a 12-hour digital clock starting from 1:00 up to 12:59)
• (the largest Fibonacci number, such that it and all positive Fibonacci numbers less than it are deficient) + (the difference between the sum of all even numbers up to 100 and the sum of all odd numbers up to 100) − (the first digit of a four-digit square that has the first two digits the same and the last two digits the same)
• (the smallest composite Jacobsthal number) ⋅ (the only digit needed to express the number of diagonals of a convex hendecagon)/(the smallest prime divisor of 13²⁰¹³ + 1)
• (the smallest integer the fate of whose aliquot sequence is unknown) + (the largest amount of money in cents you can have in American coins without having change for 2 dollars) − (the repeated number in the aliquot cycle of 95) ⋅ (the second-smallest integer n such that the Russian word for n has n letters)
• (the smallest positive even integer that’s not a totient)

*****

• (the number of letters in the last name of a famous Russian writer whose year of birth many Russians use to help them memorize the digits of e)
• (the number of pluses you need to insert in a row of 20 fives so that the sum is 1000)
• (the number of positive integers less than 2013 such that not all their digits are distinct) − (the number of four-digit numbers with only odd digits) − (the largest Fibonacci square)
• (the number of positive integers n for which the sum of the n smallest positive integers evenly divides 18n)
• (the number of trailing zeroes of 2013!) − (the number of sets in the game of Set such that every feature is different on all three cards) − (an average speed in miles per hour of a person who drives somewhere with a speed of 420 miles per hour, then drives back using the same route with a speed of 210 miles per hour)
• (the smallest fortunate triangular number)
• (the smallest weird number)/(the only prime one less than a cube)
• (the third most probable product of the numbers showing when two standard six-sided dice are rolled)

*****

• (the largest integer number of dollars you can’t pay if you have an unlimited supply of 9-dollar bills and 13-dollar bills) − (the positive difference between the two prime numbers that do not share a unit digit with any other prime number)
• (the largest three-digit primeval number) − (the largest number of distinct SET cards without a set)
• (the number conjectured to be the second-largest number such that two to its power has no zeroes) − (the largest number whose cube has at most two distinct digits and no zeroes)
• (the number of 5-digit palindromic integers in base 5) + (the only positive integer that is five times the sum of its digits)
• (the only Fibonacci number that is a double of a prime) + (the only prime p such that p! has p digits) − (the only fixed point of look-and-say operation)
• (the only number whose concatenation with itself is prime)
• (the only positive integer that that differs by 1 from a square and a nonsquare cube) − (the largest number such that its divisors are each 1 less than a prime)
• (the smallest admirable number)
• (the smallest evil untouchable number)

*****

• (the alphanumeric value of MANIC SAGES) + (the sum of all three-digit numbers you can get by permuting digits 1, 2, and 3) + (the number of two-digit integers divisible by 9) − (the number of rectangles whose sides are composed of edges of squares of a chess board)
• (the integer whose standard Roman numeral representation is alphabetically later than all others) − (the number you get if you divide a three digit number with identical digits by the sum of the digits)
• (the largest even integer that is not a sum of two abundant numbers) − (the digit in the first position where e and π have the same digit)
• (the number formed by the last two digits of the sum: 1! + 2! + 3! + 4! + . . . + 2013!)
• (the only positive integer such that if you sum the digits and the squares of the digits, you get the original number back) + (the largest prime factor of the smallest Carmichael number)
• (the smallest multi-digit hyperperfect number such that more than half of its digits are the same) − (the sum of digits that cannot be the last digits of squares) ⋅ (the largest base n in which 8n is not written like 80) ⋅ (the smallest positive integer that leaves a remainder of 2 when divided by 3, 4, and 5)
• (the smallest three-digit brilliant number) − (the first decimal digit of the number that in hexadecimal gives the house number of Sherlock Holmes)

*****

• (the number of evil minutes in an hour)
• (the number of fingers on ten hands) − (the smallest number such that its square has a digit repeated three times)
• (the number of ways you can rearrange letters of MANIC)/(the number of ways you can rearrange letters of SAGES)
• (the only multi-digit Catalan number with digits in strictly decreasing order)
• (the smallest perfect number)

*****

• (the largest product of positive integers that sum up to 10) + (the smallest perimeter of a rectangle with integral sides of area 120) − (the day of the month of the second Thursday in a January that has exactly 4 Mondays and 4 Fridays)
• (the second-largest number with all distinct digits, such that all the words in its American English representation start with the same letter) + (the largest square-free composite number that contains each of the digits 1, 2, 3, 4 exactly once in its prime factorization) + (the number of ways you can flip a coin 10 times so that the number of heads is the same as the number of tails) + (the smallest positive integer such that 2 to its power contains 2013 as a substring) + (the sum of five prime numbers formed from the digits 2, 3, 5, 7, 8, 9 where each digit is used exactly once) + (the number of days in a year where the day of the month is odious) + (the sum of the digits each of which spelled out has an alphanumeric value equal to the meaning of life, the universe, and everything) ⋅ (the sum of all prime numbers p such that p + 20 and p + 40 are also prime) + (the first digit of the total number of legal moves of the Black king in chess)
• (the second-largest three-letter palindrome in Roman numerals)/((the smallest composite number not divisible by any of its digits)/(the last digit of 2013²⁰¹³) − (the digit in position 2013 of the string formed by concatenation of all integers into one stream: 123456789101112…)) − (the number of days in a year such that the month and the day of the month are simultaneously composite)
• (the second-smallest cube with only prime digits) ⋅ (the smallest perimeter of a Pythagorean triangle)/(the last digit to appear in the units place of a Fibonacci number) + (the greatest common divisor of the sums in degrees of the interior angles of convex polygons with an even number of sides) + (the number of subsets that you can form from the set {1,2,3,4,5,6,7,8,9} that do not contain two consecutive numbers) − (the only common digit of 2013 base 8 and base 9)

The most personal puzzle that I wrote for the 2013 MIT Mystery Hunt was Integers and Sequences based on my Number Gossip database. I named it after the first lecture that I prepared after I decided to return to mathematics. It is still my most popular lecture.

Many of the clues in this puzzle are standard math problems that are very good for math competition training. Other clues are related to sequences and integer properties.

You might wonder why I often ask for the second largest integer with some property. Isn’t the largest one more interesting than the second largest? I do think that the largest number is more interesting, but exactly for this reason the largest number is available on my Number Gossip website and therefore is googleable. For example, my Number Gossip properties for 3000 contain the fact that 3000 is the largest palindrome in Roman numerals. This is why in the puzzle I used a slightly different clue, i.e. “the second largest three-letter palindrome in Roman numerals.”

It took me many hours to find non-googleable variations of interesting properties for this puzzle. Unfortunately, its non-googleability evaporated as soon as my solution was posted, right after the hunt. In any case some clues in this puzzle are useful for math competition training, and I plan to use them myself in my classes. The puzzle is attached below. I will post the solution in a couple of weeks.

*****

• (the largest integer n such that there exists a Platonic solid with n vertices, a Platonic solid with n edges, and a Platonic solid with n faces)
• (the largest two-digit tetrahedral number)/(the smallest value the second smallest angle of a convex hexagon with all integer degrees can have)
• (the number of positive integers less than 2013 that are divisible by 100, but not divisible by 70)
• (the number of two-digit numbers that produce a square when summed up with their reverse) ⋅ (the smallest number of weighings on a balance scale that guarantees to find the only fake coin out of 100 identical coins, where the fake coin is lighter than other coins)
• (the only two digit number n such that 2ⁿ ends with n) − (the second smallest, and conjectured to be the largest, triangular number such that its square is also triangular)
• (the smallest non-trivial compositorial number that is also a factorial)
• (the sum of the smallest three positive pronic numbers)

*****

• (the digit you get when you sum up the digits of 2013²⁰¹³ repeatedly until you get a single digit) − (the greatest common factor of the indices of the Fibonacci numbers divisible by 13)
• (the largest common divisor of numbers of the form p² − 1 for primes p greater than three) − (the largest sum of digits that can appear on a 12-hour digital clock starting from 1:00 up to 12:59)
• (the largest Fibonacci number, such that it and all positive Fibonacci numbers less than it are deficient) + (the difference between the sum of all even numbers up to 100 and the sum of all odd numbers up to 100) − (the first digit of a four-digit square that has the first two digits the same and the last two digits the same)
• (the smallest composite Jacobsthal number) ⋅ (the only digit needed to express the number of diagonals of a convex hendecagon)/(the smallest prime divisor of 13²⁰¹³ + 1)
• (the smallest integer the fate of whose aliquot sequence is unknown) + (the largest amount of money in cents you can have in American coins without having change for 2 dollars) − (the repeated number in the aliquot cycle of 95) ⋅ (the second-smallest integer n such that the Russian word for n has n letters)
• (the smallest positive even integer that’s not a totient)

*****

• (the number of letters in the last name of a famous Russian writer whose year of birth many Russians use to help them memorize the digits of e)
• (the number of pluses you need to insert in a row of 20 fives so that the sum is 1000)
• (the number of positive integers less than 2013 such that not all their digits are distinct) − (the number of four-digit numbers with only odd digits) − (the largest Fibonacci square)
• (the number of positive integers n for which the sum of the n smallest positive integers evenly divides 18n)
• (the number of trailing zeroes of 2013!) − (the number of sets in the game of Set such that every feature is different on all three cards) − (an average speed in miles per hour of a person who drives somewhere with a speed of 420 miles per hour, then drives back using the same route with a speed of 210 miles per hour)
• (the smallest fortunate triangular number)
• (the smallest weird number)/(the only prime one less than a cube)
• (the third most probable product of the numbers showing when two standard six-sided dice are rolled)

*****

• (the largest integer number of dollars you can’t pay if you have an unlimited supply of 9-dollar bills and 13-dollar bills) − (the positive difference between the two prime numbers that do not share a unit digit with any other prime number)
• (the largest three-digit primeval number) − (the largest number of distinct SET cards without a set)
• (the number conjectured to be the second-largest number such that two to its power has no zeroes) − (the largest number whose cube has at most two distinct digits and no zeroes)
• (the number of 5-digit palindromic integers in base 5) + (the only positive integer that is five times the sum of its digits)
• (the only Fibonacci number that is a double of a prime) + (the only prime p such that p! has p digits) − (the only fixed point of look-and-say operation)
• (the only number whose concatenation with itself is prime)
• (the only positive integer that that differs by 1 from a square and a nonsquare cube) − (the largest number such that its divisors are each 1 less than a prime)
• (the smallest admirable number)
• (the smallest evil untouchable number)

*****

• (the alphanumeric value of MANIC SAGES) + (the sum of all three-digit numbers you can get by permuting digits 1, 2, and 3) + (the number of two-digit integers divisible by 9) − (the number of rectangles whose sides are composed of edges of squares of a chess board)
• (the integer whose standard Roman numeral representation is alphabetically later than all others) − (the number you get if you divide a three digit number with identical digits by the sum of the digits)
• (the largest even integer that is not a sum of two abundant numbers) − (the digit in the first position where e and π have the same digit)
• (the number formed by the last two digits of the sum: 1! + 2! + 3! + 4! + . . . + 2013!)
• (the only positive integer such that if you sum the digits and the squares of the digits, you get the original number back) + (the largest prime factor of the smallest Carmichael number)
• (the smallest multi-digit hyperperfect number such that more than half of its digits are the same) − (the sum of digits that cannot be the last digits of squares) ⋅ (the largest base n in which 8n is not written like 80) ⋅ (the smallest positive integer that leaves a remainder of 2 when divided by 3, 4, and 5)
• (the smallest three-digit brilliant number) − (the first decimal digit of the number that in hexadecimal gives the house number of Sherlock Holmes)

*****

• (the number of evil minutes in an hour)
• (the number of fingers on ten hands) − (the smallest number such that its square has a digit repeated three times)
• (the number of ways you can rearrange letters of MANIC)/(the number of ways you can rearrange letters of SAGES)
• (the only multi-digit Catalan number with digits in strictly decreasing order)
• (the smallest perfect number)

*****

• (the largest product of positive integers that sum up to 10) + (the smallest perimeter of a rectangle with integral sides of area 120) − (the day of the month of the second Thursday in a January that has exactly 4 Mondays and 4 Fridays)
• (the second-largest number with all distinct digits, such that all the words in its American English representation start with the same letter) + (the largest square-free composite number that contains each of the digits 1, 2, 3, 4 exactly once in its prime factorization) + (the number of ways you can flip a coin 10 times so that the number of heads is the same as the number of tails) + (the smallest positive integer such that 2 to its power contains 2013 as a substring) + (the sum of five prime numbers formed from the digits 2, 3, 5, 7, 8, 9 where each digit is used exactly once) + (the number of days in a year where the day of the month is odious) + (the sum of the digits each of which spelled out has an alphanumeric value equal to the meaning of life, the universe, and everything) ⋅ (the sum of all prime numbers p such that p + 20 and p + 40 are also prime) + (the first digit of the total number of legal moves of the Black king in chess)
• (the second-largest three-letter palindrome in Roman numerals)/((the smallest composite number not divisible by any of its digits)/(the last digit of 2013²⁰¹³) − (the digit in position 2013 of the string formed by concatenation of all integers into one stream: 123456789101112…)) − (the number of days in a year such that the month and the day of the month are simultaneously composite)
• (the second-smallest cube with only prime digits) ⋅ (the smallest perimeter of a Pythagorean triangle)/(the last digit to appear in the units place of a Fibonacci number) + (the greatest common divisor of the sums in degrees of the interior angles of convex polygons with an even number of sides) + (the number of subsets that you can form from the set {1,2,3,4,5,6,7,8,9} that do not contain two consecutive numbers) − (the only common digit of 2013 base 8 and base 9)

The ultimate goal of each MIT Mystery Hunt is to find a hidden coin. So it was highly appropriate that our 2013 team created a coin-weighing puzzle (written by Ben Buchwald, Darby Kimball, and Glenn Willen) as a final obstacle to finding the winning coin:

There are nine coins, one real and eight fake. Four of the fake coins weigh the same and are lighter than the real coin. The other four fake coins weigh the same and are heavier than the real coin. Find the real coin in seven weighings on the balance scale.

Actually, it is possible to find the real coin in six weighings. Can you do that?

I posted the puzzle In the Details two weeks ago. This is the most talked-about puzzle of the 2013 MIT Mystery Hunt. The author Derek Kisman invented this new type of puzzle and it is now called a Fractal Word Search. I anticipate that people will start inventing more puzzles of this type.

Let’s discuss the solution. The words in the given list are very non-random: they are related to fractals. How do fractals work in this puzzle? The grid shows many repeating two-by-two blocks. There are exactly 26 different blocks. This suggests that we can replace them by letters and get a grid that is smaller, for it contains one-fourth of the number of letters. How do we choose which letters represent which blocks? We expect to see LEVEL ONE in the first row as well as many other words from the list. This consideration should guide us into the matching between letters and the two-by-two blocks.

The level one grid contains 18 more words from the list. But where are the remaining words? So we have level one, and the initial grid is level two. The substitution rule allows us to replace letters by blocks and move from level one to level two. When we do this again, replacing letters in level two by blocks, we get the level three grid. From there we can continue on to further levels. There are three words from the list on level three and one word on level four. But this is quickly getting out of hand as the size of the grid grows.

Let’s step back and think about the next step in the puzzle. Usually in word search puzzles, after you cross out the letters in all the words you find, the remaining letters spell out a message. What would be the analogous procedure in the new setting of the fractal word search? In which of our grids should we cross out letters? I vote for grid number one. First, it is number one, and, second, we can assume that the author is not cruel and put the message into the simplest grid. We can cross-out the letters from words that we find in level one grid. But we also find words in other levels. Which letters in the level one grid should we cross out for the words that we find in other levels? There is a natural way to do this: each letter in a grid came from a letter in the previous level. So we can trace any letter on any level to its parent letter in the level one grid.

We didn’t find all the words on the list, but the missing words are buried deep in the fractal and each can have at most three parent letters. I leave to the reader to explain why this is so. Because there are so few extra letters, it is possible to figure out the secret message. This is what my son Sergei and his team Death from Above did. They uncovered the message before finding all the words. The message says: “SUM EACH WORD’S LEVEL. X MARKS SPOT.” Oh no! We do need to know each word’s level. Or do we? At this point, the extra letters provide locations of the missing words. In addition, if a word on a deep level has three parents, then it has to be a diagonal word passing through a corner of one of the child’s squares. So our knowledge of extra letters can help us locate the missing words faster.

Also, the message says that the answer to this puzzle will be on some level in the part of the grid that is a child of X. Luckily, but not surprisingly, there is only one letter X on level one. The child of X might be huge. But we could start looking in the center. Plus, we know from the number of blanks at the end of the puzzle, that the answer is a word of length 8. So Sergei and his team started looking for missing words and the answer in parallel. Then Sergei realized that the answer might be in the shape of X, so they started looking at different levels and found the answer before finding the last word on the list. The answer was hiding in the X shape in the center of the child of X on level 167: HUMPHREY.

H..Y
.UE.
.RM.
H..P

Cambridge Waldo puzzle from the 2013 MIT Mystery Hunt was supposed to be easy. Its goal was to get people out of the building for some fresh air. I made this puzzle jointly with Ben Buchwald, Adam (Pesto) Hesterberg, Yuri Lin, Eric Mannes and Casey McNamara. The puzzle consists of 50 pictures of different locations in Cambridge; one of the above individuals was hiding in each picture. Let me use this opportunity to thank my friends for starring in my puzzle and being inventive while doing it.

The puzzle starts with a group picture of my stars. The caption to the picture gives their names. The fact that they are standing in alphabetical order is a clue.

Out of the 50 pictures, each person appears in exactly ten pictures. If you mark the locations of one person on a map, they look like a letter. For example, below are Ben’s locations that form a letter “S”. When you put the letters in the alphabetical order by people names you get the answer to the puzzle: SCAMP.

As you can see, you do not need all ten locations to recognize the letter. You might be able to recognize the letter with five locations, or at least significantly reduce possibilities for the letter. Besides, you do not need all the letters to recognize the answer. We thought that this was an easy puzzle.

And, to make it even easier, the order in which we posted the pictures was not completely random. The pictures of one person were in the order one might walk from one location to another. This played two important roles. If you recognize the person but do not recognize the location on the picture, you can make an approximate estimation of the location because it must be on the path between the previous and next locations. If you recognize the location but not the person, you can guess the person by checking whose path it fits better.

It was difficult to hide people, especially when there were no other people around. So we sometimes used props. We only used one prop per person. Here you can see Pesto with his sarongs in plain view. In the other picture (below) he is hiding under a white sarong. Yuri had a bicycle helmet. In one of the pictures, she hid so well that you couldn’t see her — but you could see her helmet. Ben had a bear hat. In one of the pictures you can only see a shadow of a person, but this person was clearly wearing a hat with bear ears. Eric didn’t have a prop, but my car was eager to make a cameo appearance at the Mystery Hunt, so I hid him in my car in one of the pictures.

As you might guess I made the pictures of different people at different times. So Ben Buchwald was the one who realized that solvers might differentiate people by looking at the data of the picture files. He carefully removed the original time stamps.

Despite our best intentions, our test-solvers decided not to leave their comfy chairs, but rather to use Google-StreetView. We strategically made some of the pictures not Google-StreetViewable, but our test-solvers still didn’t leave the comfort of their chairs. They just became more inventive. I do not know all the things they did to solve the puzzle, but I heard about the following methods:

• Finding a website with parking meter locations by meter number.
• Looking for one-way streets in the proper configuration to find likely places for One-Way/Do-Not-Enter signs.
• Looking up the street cleaning signs in the street cleaning schedule.
• Differentiating the streets by their separator line: double, single, or no line.

I have yet to understand why this puzzle was difficult for the Mystery Hunt teams.