Solution to 15×15 — MIT Mystery Hunt 2021

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Answer: STOWING

by Brian Chen

Solve the crossword normally. Well, not really. When you switch between across and down clues, all cells’ contents are passed through a consistent, reversible cipher per cell, although some cells’ contents won’t change. (This cipher is just equivalent to an atbash composed with a Caesar shift, the latter of which varies per cell, although it isn’t necessary or even particularly helpful to realize this to solve the puzzle.)

The answers are:

Across

1A Highlight : DRAW ATTENTION TO
16A Like Tim the Beaver : ANTHROPOMORPHIC
17A Doves, collectively : ANTI-WAR MOVEMENT
18A By the way : PARENTHETICALLY
19A One way to slice it : AGAINST THE GRAIN
20A A famous one of these mentions a bear : STAGE DIRECTIONS
21A Disorder affecting about 1/4 of humans : NEARSIGHTEDNESS
22A Time, it's often said : FOURTH DIMENSION
23A P, for example : COMPLEXITY CLASS
24A Legal position whose plural is odd : ATTORNEY GENERAL
25A Be firm : PUT ONE’S FOOT DOWN
26A Early humans : HUNTER-GATHERERS
27A Group terrified of "it" : KNIGHTS WHO SAY NI
28A Act impulsively : SHOOT FROM THE HIP
29A This, for example : CROSSWORD PUZZLE

Down

1D American standard : STARS AND STRIPES
2D "You've proven me wrong" : I STAND CORRECTED
3D Anger : FLY OFF THE HANDLE
4D Generic name : WHATCHAMACALLIT
5D Be self-evident : GO WITHOUT SAYING
6D Two alliterative services, or their provider : BED AND BREAKFAST
7D Moderately unusual pets : PRAYING MANTISES
8D Immediately : AT THE DROP OF A HAT
9D Internet predictor : MARSHALL MCLUHAN
10D Household object that contains mercury : FLUORESCENT LAMP
11D Like some nation-states and doctors : INTERVENTIONIST
12D Start of many games : SICILIAN DEFENSE
13D Normal : PAR FOR THE COURSE
14D "That's my best offer" : TAKE IT OR LEAVE IT
15D Feared place of sailors : DAVY JONES’ LOCKER

The complete across grid:

The complete down grid:

Take the letters in the completed grid that don’t change when you flip directions:

They say: WITH AWAY, HIDING ONESELF ON A SHIP. The seven-letter answer, reinforced by the seven squares on the bottom, is the word STOWING.

Author’s Notes

This puzzle was basically inspired by thinking about the fun of putting in a long answer to a clue without any crosses when solving a crossword and have it turn out to be exactly the right number of letters, and trying to write a puzzle around the distilled essence of that action.
During testsolving, a lot of people guessed STOW.
In testsolving, diffing the two grids proved natural but also not technically easy to execute. Many testsolvers tried to superimpose two images in image editing software and then got bitten when, say, the letters I and T or P and R superimposed too well. One approach in GIMP to obtain a fairly reliable diff is to superimpose the two grids in “Grain extract” layer mode, select the inverse of the exactly-gray pixels, and then color those in on one of the grids. In the end it may have been easier, if temporarily more tedious, to read off the letters manually.
For some amount of time at the start, I wanted to figure out a simple cipher scheme that would give you no information at all about which squares were fixed points. Specifically, I wanted:
1. The cipher in each square should still take a single integer from 0 to 25 as a key.
2. For every pair of letters (A, B), there should exist a key K such that the cipher maps A to B. (By counting, the key also has to be unique for each pair.)
3. For every key K, the cipher has at least one fixed point. (By counting again, this implies each key has exactly one fixed point and every letter is a fixed point for a distinct key.)
4. Ideally, the cipher is an involution.
5. Ideally, the cipher is very simple to describe.
I don’t think it’s possible to satisfy all of these criteria with a single cipher. Getting the first three is equivalent to finding an n × n Latin square with all distinct numbers along one diagonal for n = 26. This task is easy when n is odd, but surprisingly difficult when n is even and I was not able to find any solutions that satisfied d and e, so I just dropped criterion c. (The best I found was “A simple method for constructing doubly diagonalized Latin squares", Gergely 1972.)