June 2025: Some Ones, Somewhere

This was one of my favourite Jane Street puzzles so far; there’s something that’s so satisfying about extracting information from an initially indecipherable collection of objects and shapes, and this was one of the purest instances of that I’ve experienced.

I collaborated on this puzzle with Fred Vu since we first saw the puzzle not long after it was uploaded, and thought we’d have a good chance of placing highly if we worked together.

Stats:

Difficulty: 6/10

Enjoyability: 10/10

Leaderboard Placing: 2/276

The approach:

The only real information that’s immediately accessible comes from the scrabble tiles placed in the top row and left column of pictures. If you read these across the top first and then down the left, you obtain the words “PARTRIDGE TILING”. Neither me nor Fred had heard of partridge tilings before, and there’s not too much content on partridge tilings on the internet, but a quick Google search nevertheless revealed most of what we needed to know. It turns out that for all natural numbers \(n\), the following satisfying identity is true (as can easily be proven by induction):

Now, the left-hand side of the above equation is equal to the area of the disjoint union of one square of side length one, two squares of side length two, and so on, up to \(n\) squares of side length \(n\). Meanwhile, the right-hand side is obviously equal to the area of one square of side length \(1+2+\cdots+n\). It follows that in principle, it may be possible to tile a square of side length \(1+2+\cdots+n\) with one tile of side length one, two tiles of side length two, and so on. Such tilings are indeed possible for some values of \(n\), and are called partridge tilings. Looking at some pictures of such tilings, it becomes clear that the nine images in the puzzle show incomplete partridge tilings, and the natural progression is to try and complete these.

The first order of business is to then determine what \(n\) is. By counting the tiles placed on the tables in each picture, one concludes that these are incomplete partridge tiling of order 9, and moreover one can determine what size tile each colour corresponds to. The incomplete tilings have been chosen in such a way that each one has a unique completion, and some Python code was written to recursively place the remaining tiles and discover these unique solutions (in fact, the most time spent on this problem was in hard-coding the existing tile placements for this part of the problem).

The next order of business is working out how to extract information from the complete tilings. This requires two key insights, one of which was provided by Fred and the other of which was provided by me. The first is that it’s the location of the single tile of size one in each tiling which is important (which the puzzle’s title hints at, as Fred surmised). The second is that each row and each column of the tilings corresponds to a letter. I deduced this by observing the proximity of some of the scrabble tiles which had been placed; for example, the ‘D’ and ‘G’ tiles in the top-right image have just enough room between them for imaginary ‘E’ and ‘F’ tiles to fit. This, and other examples, lead to the theory that the columns cycle through the alphabet as you read left-to-right across the entire collection of images, as do the rows as you read top-to-bottom. If this is true, then every image would confer two characters of information: one from the column and one from the row in which the tile of size one is placed. Together with the words “the” and “a” spelled out in scrabble tiles in the top- and bottom-left photos, this would be enough to form a short sentence. This theory turns out to be correct, and leads to the sentence “the sum of cubes is a square”, which describes the identity which motivated the concept of a partridge tiling in the first place. How satisfying!

Note: we were able to deduce the final sentence before having completed all nine partridge tilings. In fact, we probably could have guessed it a little sooner and perhaps placed first instead of second.