Solution to Cross-stitched Words

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Answer: MANIAC

by Herman Chau

The first step is to solve the crossword clues. While doing so, we might notice that the answers are in alphabetical order and are all six letters long. This helps us confirm most of our answers and once we have enough clues solved, we may notice that in fact the second and fifth letters of each answer are the same. Below are the answers to the crossword clues.

Clue Answer
Witty remark BONMOT
Alternative to Greek salad CAESAR
French pianist Alfred CORTOT
Tyson show COSMOS
Double dagger DIESIS
____ bubble DOTCOM
Non-competitive race FUNRUN
They're often all scratched up? LOTTOS
Title character of an Ozzy Osbourne album MADMAN
Units of length in the UK METRES
Performed wrongly MISDID
Maxims MOTTOS
Star jelly genus NOSTOC
Competitor to OneNote NOTION
Nag PESTER
Wicker material RATTAN
Parting place? REDSEA
Rue REGRET
Reposed RESTED
Sports with a lot of horseplay? RODEOS
In short supply SCARCE
Kurdistan Province's capital SENNEH
Snow White's dwarves e.g. SEPTET
Member of the 2021 writing team, perhaps SETTER
In a way SORTOF
Japanese poems TANKAS
Haiku e.g. TERCET
Ring on a saddle TERRET
Like the next clue, compared with this one TERSER
Sluggish inactivity TORPOR
Midsections TORSOS
Command to delete files UNLINK

The heading for the crossword clues is "Individual Cross-stitches" and at the bottom of the page is a diagram that seems to indicate a cross-stitch. Although cross-stitches are often done in rows, it is possible to make an individual cross-stitch. When an individual cross-stitch is made, it follows a cross shape as in the diagram with the thread crossing in the middle. Since the second and fifth letters are the same in each crossword answer, this suggests that we can "cross-stitch" each word such that the second and fifth letters are the crossing in the middle. For example, the crossword answer BONMOT can be cross-stitched as follows.

N T
 O
M B

We can then attempt to fill in these cross-stitched words into the grid given. Some uncommon letters are given to us to start with. We can fill in a few words using the given letters. Note that we make an assumption in the case of the F and K that both occurrences in our crossword answers use the given letter. This is in order to try filling in the grid as compactly and efficiently as possible.

N T
O
M B
N N
U
R F
O
T S
N S
A
L K T
N
I U G T
E
R R

There are no further strong constraints on how to fill the remaining crossword answers into the grid, but playing around with some possible fills we find that the words intersect well, suggesting that we are doing the right thing. Some experimentation also reveals that the words must be quite tightly packed into the grid and that a greedy fill should get close to a complete solution. Indeed, just filling in the grid greedily by placing words wherever it can intersect two letters from another word gets us to a filled grid with only a couple possible ambiguities that are quick to eliminate via trial and error. The final filled grid is shown below.

D A E S D N T
E C I I A O
S R S D M M B
O E
D S T R R
O E E
E R T S T N N
A E O A U
S C T L T R F
O O
T N H R T S
O E E O
I N S R R T M
A E O O O
L K T P T C D
N E
I U G T S S
E E O
R R M C

Now that we've filled in all the cross-stitched words, we might notice that we haven't used the shaded red squares in the diagram at the bottom of the puzzle page. The diagram suggests that there is a recursive step where we wish to cross-stitch the filled grid somehow to extract the answer. If we look at the cross stitches present in the shaded red squares, it turns out that each of the shaded red regions corresponds to a 3×2 rectangle with some cells filled in by cross-stitches. This and the tactile nature of cross-stitching suggest that we should extract letters from the shaded red regions via Braille and cross-stitch one more time to get the final answer. Below is the filled in grid with appropriate shading to help determine the Braille letters.

D A E S D N T
E C I I A O
S R S D M M B
O E
D S T R R
O E E
E R T S T N N
A E O A U
S C T L T R F
O O
T N H R T S
O E E O
I N S R R T M
A E O O O
L K T P T C D
N E
I U G T S S
E E O
R R M C

Converting the Braille to letters and reading in the cross-stitch order gives the answer MANIAC.

Author’s Notes

The solution path to filling the grid ends up being more a like a Numberlink logic puzzle than a traditional logic puzzle. Rather than local reasoning about where words must go, following our noses and trying to pack the words efficiently ends up being the best strategy. Uniqueness was a concern and an earlier version of the fill had a small uniqueness issue that required changing some words around. Yannick Yao helped write a uniqueness verifier and the final version of the grid is unique up to UNLINK and COSMOS being able to fit in the bottom left and bottom right corners in either order. However, those extraneous solutions don't have a single connected component, unlike the solution presented above.

At first glance, uniqueness of the fill seems quite surprising since only 4 letters are given and the shape of the grid is entirely unconstrained. When constructing the puzzle, however, I noticed that it was actually very difficult to get a dense fill. We're restricted to 6 letter words that need to cross themselves and furthermore many of our letters are crossed 3 or 4 times, compared to traditional crosswords in which each letter is crossed exactly twice. As a result, the crossings are actually fairly constrained and it's perhaps not so surprising that the fill ends up being unique.