# As short as possible

What shape should replace the question mark?

• For the solution, are you looking for an image file specifically, or if we can give a verbal description of what replaces the ?, would that be sufficient? Commented Nov 16, 2023 at 17:05
• @Vaekor Yeah, preferably an image file and then explaining with words Commented Nov 16, 2023 at 17:19

The solution would look like:

First,

We can write the numbers from 1-36 in the squares of the grid in order. Then we can overlay the red lines from the bottom onto that grid.

Then,

We can look at the numbers in the cells traversed by the line. For the second figure, a cube, its line traces the numbers 1, 8, 27. For the third figure, a hypercube, its line traces the numbers 1, 16. These are the sets of cubic and fourth power numbers less than or equal to 36. The relationship is that if a figure depicts an n-dimensional figure, then the line will trace all integer powers of n less than or equal to 36 in ascending order in the given 6x6 grid.

Therefore,

The line for the square, a 2 dimensional figure, must trace over all square numbers up to 36 in ascending order. These are 1, 4, 9, 16, 25, and 36, and the result is the following:

• This is indeed the intended solution. Good job! Commented Nov 18, 2023 at 8:49

My best visual representation would look like this:

My verbal description (bear with me) would be something like:

In middle pair, the cube is paired with the shortest distance around the surface area of the cube. We'll reword this as "the shortest path through the 2-D faces around the 3-D cube". With that perspective, the third pair is a hypercube (4-D cube) paired with "the shortest path through the 3-D cubes around the 4-D cube", since each "face" of the hypercube is in fact a 3-D cube. Because of our limited perception of the hypercube, this visually appears as a straight line passing through the cubes. For the first pair, we therefore need "the shortest path through the 1-D edges around the square", which could take us either up and left, or right and up.

• But where does "Ordered" fit in? Commented Nov 17, 2023 at 16:44
• I took the title to suggest rot13(jr ner svaqvat gur fubegrfg cngu trbzrgevpnyyl), while I took "ordered" to suggest rot13(jr ner neenatrq va beqre ol qvzrafvbanyvgl (vr, gung gur qvzrafvba vf vzcbegnag)). As for the 6 x 6 arrangement, I'm not sure -- a wild guess might be rot13(fvapr n phor unf 6 snprf, n ulcrephor unf 6 "snprf" bs phorf, rnpu bs juvpu unf fvk fvqrf, juvpu jbhyq znxr 36 fdhnerf ba gur bhgfvqr bs gur ulcrephor) Commented Nov 17, 2023 at 17:03
• Sry but this is not the intended answer. The answer I'm looking for is based on something simpler. Commented Nov 17, 2023 at 18:00