Turning, folding and stacking in your head

You look at a drawing on paper and picture a solid. You turn a solid over and imagine its far side. This ability is sometimes called spatial sense. The four works here are practice for it.

Nets of a cube (the site’s “Cube Nets”)

Cut a cube open along its edges and lay it flat: that is a net. Shapes made of six joined squares number 35 when rotations and flips count as the same. Of these, 11 fold into a cube. The program tried all 35 shapes and checked that exactly 11 fold. A real 3D folding check gives the same result (cube-net/engine.test.ts).

The 11 fall into four groups by layout: six of the “1-4-1” type, three of “2-3-1”, one of “2-2-2” and one of “3-3” (6 + 3 + 1 + 1). There is a quick way to rule shapes out. No shape containing a 2×2 square folds into a cube, which the program also checked. A 2×3 block and a straight line of six do not fold either. When a face lands on another, one face gets used twice and another is left open.

In history, Dürer printed nets of the regular solids in a book in 1525. Whether every convex polyhedron has a net is said to be still unsolved.

How many blocks? (the site’s “Block Count”)

From a picture of stacked blocks seen from above and to one side, you answer how many blocks there are in all. Counting only the visible faces comes up short, because some blocks hide behind others. In this work a block is hidden only if another block sits exactly in front of it along the line of sight. From the picture you reason about the hidden ones.

The program checked that for each puzzle, trying every stacking that matches the picture gives only one possible total. Even in the harder puzzles, where several stackings look the same, the total is the same (block-count/engine.test.ts). Some pictures do not fix the total, so this needs checking. The puzzles use only those where the answer is unique.

Fold, punch, unfold (the site’s “Paper Fold”)

Fold a sheet a few times and punch one hole. When you unfold it, how many holes are there, and where? One fold puts two layers of paper together. Punch and you make a hole in each layer. Unfolded, the hole has a mirror-image partner across the crease. With n folds, the number of holes is 2 to the power n times as many: 4 times for two folds, 8 times for three.

The program checked that two methods give the same answer. One goes forward, asking where each cell of the paper lands after folding. The other mirrors the holes back, fold by fold. It also checked that every cell of the folded paper receives 2 to the power n cells of the original (paper-fold/engine.test.ts). A side with an odd number of cells cannot be halved, so it cannot be folded.

A tower of four cubes (the site’s “Four Cubes”)

Stack four cubes with coloured faces so that all four colours show on each of the four sides. Each cube has 24 orientations, so trying everything means 24 to the power 4 = 331,776 stackings. This kind of puzzle is said to have been hugely popular in the 1960s, but similar ones date back much further, one even to a patent in 1900. There is a way to solve it with graphs. Draw a graph joining the colours on opposite faces of each cube, then find a front-back set and a left-right set that share no edge.

The program checked that a cube has exactly 24 orientations, that its turns move like a real cube, and the number of solutions for each set by trying every combination of 24 to the power 4. The work’s three sets have 16, 8 and 3 solutions (four-cubes/engine.test.ts). Solutions that become the same by turning the whole tower or flipping it over count as one.

What they share

• Move it to check: when turning in your head is hard, move things step by step and check.
• Count what you cannot see: hidden blocks, stacked paper, the opposite face.
• Count every possibility: 11 of 35, a few of 24 to the power 4. Counting lets you state it with confidence.

All four are puzzles made for this site. No commercial puzzles or pictures were used.