Spatial: Nets, Cube Stacks and Symmetry

A five-square arrangement completes into a cube only when it already contains a band of four in a line, because that band is what wraps round the cube. If it does, the missing square is the second cap and it goes on the opposite side of the band from the first. A strip of five will not work, because adding a sixth makes a strip of six that wraps round and collides with itself, and two columns of squares fold over each other whatever you add. Find the band first.

A painted cube splits by where each small cube sat. The eight corners have three painted faces; the cubes in the middle of an edge have two; the ones in the middle of a face have one; and the buried centre has none. For a three by three by three that is eight, twelve, six and one, adding to twenty-seven, and that total is the check. In a four by four by four there are still eight corners, but each of the twelve edges now has two middle cubes, so twenty-four have exactly two painted faces.

To finish a symmetrical row, pair the positions in from each end: the first with the last, the second with the second-last, and so on. With seven tiles the middle one sits on the fold and has no partner, so positions five, six and seven must copy three, two and one — three tiles. With an even number of tiles there is no middle one and every tile has a partner, which changes the count, so always start by asking whether the row is odd or even.

Written plainly, the digits that survive a fold across the middle are nought, one, three and eight, so the largest two-digit number with a horizontal line of symmetry is eighty-eight. A vertical line is a different test: each digit must survive being mirrored and the digits also swap places, so the number has to read the same backwards — which leaves things like eleven, eighty-eight and one hundred and one. Across and down give different answers, so read which line the question means.

A regular shape's order of rotational symmetry is its number of sides, and one step of the turn is three hundred and sixty divided by that order. An equilateral triangle has order three, so one step is a hundred and twenty degrees and two steps are two hundred and forty. A square steps ninety at a time, a hexagon sixty. Read whether the question wants degrees or the order, because those are two different numbers. Quiz and exercises now.