Solution: Pyramid Schemes

Answer: DELTA

Written by Jonah Ostroff

Solvers are given a standard-looking criss-cross grid, a list of words to fit in it, and a triangular diagram of dots pointing to more dots. One odd feature is that at the bottom corners of the grid appear to be connected to two more copies of the grid, one to the right and one to the left. We’ll figure out what’s up with that later.

The grid has one given letter (a C, indicated with an arrow), and it’s possible to solve the grid uniquely starting from that C. While solving the grid, solvers might notice that it’s impossible to make progress without using information derived from the C. Without the C, every single entry can be filled in multiple ways, and “locally” this always works. For example, the horizontal word in the bottom row could be TREMBLED (bordered by SET then LID) or DAYLIGHT (bordered by LID then SET). This suggests that with different givens, there might be a different answer.

Indeed, that’s what’s going on with the additional copies: the letters from each grid feed into the two below it, and those grids can be solved uniquely from those givens. The grids then form an infinite pyramid, like in the diagram: each copy’s bottom two answers feed into the grids below it, and each grid solves uniquely from the givens it gets from its parents.

The numbered squares in the solution to the original grid read FIFTH LAYER. So the next thing to do is work downward to produce the fifth layer of the pyramid. Of the fifteen grids in the first five layers, eight are repeats, so solvers must complete seven versions of the grid. Some parts of these seven solutions can be skipped, as described below. (One could continue past the fifth layer, and indeed the pyramid can be extended infinitely, but no new versions appear after row five.)

Here's a chart of the seven grids’ input and output letters, along with the letter pointed at by the arrow:

Using this table, it’s possible to construct as much of the pyramid as desired. In the fifth layer, read the marked squares (the ones indicated by the arrows) to get an appropriately triangular answer: DELTA.

The first five layers of the pyramid are shown in the Appendix.

Logical process

There are seven solutions to the criss-cross, but not every word can be placed in every space of its length. Here’s an annotated version of the grid and word bank, and some commentary on the logic.

Author’s Notes

I’ve always found “infinite” puzzles cool, and this is my first time attempt at writing one. This particular construction was inspired by The World's Tallest Cryptic and Curious Customs, both by Kevin Wald. This puzzle was originally intended for the math department, under the title “Pascal’s Triangle”, with a nine-letter answer. It turns out you can extend this idea to a 9-letter answer, but it’s a mess—the word bank needed 40 words just for the meta-logic (that is, getting the correct inputs to propagate to the next layer), and that was before I added words for extraction and connectivity. By contrast, the final version of the puzzle uses just 16 words for meta-logic, and 8 more words to fit in the extraction. The remaining 12 are only there to connect the various pieces of the grid.

Appendix: 1st five grid layers