Jane Street puzzle writeup: Some Ones, Somewhere

Overview:

This puzzle involved extracting a sentence from a PDF file. The PDF contained images of mosaics arranged in a 3×3 grid, with some Scrabble letters on the sides. (Here’s the link to the PDF: https://www.janestreet.com/static/pdfs/puzzles/june-2025-puzzle.pdf)

Solving:

Starting out, I removed the perspective from these images.

This allowed me to figure out the relative sizes of the squares. Originally, these sizes didn’t seem to suggest anything, but after multiplying them by the factor 45, a pattern began to emerge.

The sizes were now as follows:

• Grey: 9

• Brown: 8

• Yellow: 7

• Light Blue: 6

• Pink-Red: 5

• Dark Blue: 4

• Orange: 3

• Green: 2

It appears that a square with a side length of 1 is missing.

Now the next question: I assumed the original mosaic had to be recreated somehow, so how many of these squares am I allowed to use?

To figure that out I simply counted the squares on the board as well as the table next to it. This resulted in the same distribution as the sizes above.

To verify I multiplied all the sizes squared by their count to figure out if I could actually cover the grid.

The grid is 45×45 so in total it contains 2025 spaces. Calculating the area of all squares yields 2024 spaces. So we are one space short.

I assumed that this 1×1 square would have to be in line with some of the scrabble letters, so I could extract text. But this turned out to be false.

Nevertheless I created a small tool to aid with solving these grids.

An hour or so of trying and failing to solve the board to satisfy the constraints of this assumption. I gave up and just tried to solve the board.

This turned out to be more straightforward than expected, after less than half an hour I managed to solve all of the grids.

So, what’s next?

I had the solved boards but no clue on how to extract a sentence.

This was by far the most difficult part of the puzzle.

I tried to solve the boards differently, but was unsuccessful.

I tried drawing lines between the letters to see if the lines and spaces would intersect.

One out of these desperate attempts actually was successful.

I wrote down the alphabet and placed it next to the board (I was using Excalidraw) with the intention that crossing out the Scrabble letters would leave me with something interesting. That didn’t really happen but I found something else.

I noticed that the Scrabble letters and their spacing were perfectly aligned with the alphabet.

Below is an image showing the finished grids with the repeating alphabet next to the sides (this was created in Excel which would have made this discovery trivial).

Now I just guessed that the letter from the left came first and the one from the top second.

This results in:

SU MO FC
UB ES IS
SQ UA RE

SUMOFCUBESISSQUARE
Solution: SUM OF CUBES IS SQUARE

Tool: https://static.korff.dev/blog/some-ones-somewhere (AI used for translating and some UI tweaks)