Adjacency and the Opposite-Face Rule

A shared edge is a hinge

Two squares that share an edge on the pattern become two faces that share an edge on the cube. This is the one relationship folding cannot break, and the reason is mechanical: the shared edge is the hinge the fold turns about, so the two squares are still joined along it when the cube closes.

The black square and the gray square touch along one edge of the pattern, and they touch along one edge of the cube. Nothing you do to the pattern — reading it upside down, folding it in a different order — can separate them. So an option that shows those two faces not touching is wrong, however plausible the rest of it looks.

Two apart in a line means opposite

The second relationship you can read straight off the page: two squares in the same row or column with exactly one square between them end up on opposite sides of the cube. In a run of four that gives you two pairs at once — the first square with the third, the second with the fourth.

Here is a 1-4-1 pattern numbered like a die, with the counts chosen so that the pairs come out 1 with 6, 2 with 5, and 3 with 4:

The cube shows 1, 2 and 4. Turn it around and you get the three faces it was hiding:

6, 3 and 5 — the partner of every face in the first drawing, and not one repeat. The two lone squares are a pair as well: they sit two apart in a column, with the square of the run between them. That holds for every 1-4-1 layout, including the ones whose caps are offset to different columns — there the caps still end up opposite, but the straight-line reading no longer sees them.

The reading finds some pairs, not all

This is where the rule gets oversold, so be exact about it. Two apart in a line always means opposite. The reverse does not hold: plenty of opposite pairs never sit two apart in a line.

Same six numbers, same three pairs, on a 3-3 layout. The reading still finds 1 with 6 and 3 with 4, both of them two apart in a row. It cannot find 2 with 5, because those two squares sit diagonally from each other, and diagonal neighbours tell you nothing.

On the 2-2-2 staircase it finds nothing at all. The cube still has its three pairs — every cube does — but not one of them appears as two-apart-in-a-line anywhere on this pattern.

Treat the reading as a free result rather than a method: take the pairs it hands you, and fold for the rest. On a centred 1-4-1 it hands you all three. On a staircase it hands you none.

Opposite faces never share a drawing

Here is what makes all of this worth the trouble. An option draws exactly three faces of the cube — a top, a left and a right, meeting at one corner. Three faces that meet at a corner are pairwise neighbours, so no drawing can ever contain both members of a pair.

That turns the opposite-face reading into an eliminator. Adjacent faces may or may not appear together, depending on how the figure is turned, and their absence proves nothing. But a figure that shows you two faces the pattern says are opposite is impossible, and one look settles it.

The black square and the gray square on this pattern are the two lone squares of a centred 1-4-1 — one column, two apart, with a square of the run between them. They are a pair. So no cube folded from this pattern can show black and gray at the same time, and figures A and D, which show both, are out on that single reading. Of the two survivors, the marked figure is the answer.

One shared edge that an option contradicts, or one pair it puts on the same drawing, is enough on its own. You never have to finish folding the pattern to convict an option.

One thing to carry forward: reflecting a cube keeps every adjacency intact but reverses the arrangement of the faces around each corner, which is what makes the mirror image a possible wrong answer — the Trap Anatomy lesson takes it from there.

The takeaway

Squares that share an edge on the pattern share an edge on the cube, always. Squares two apart in a straight line, with one square between them, end up opposite — but that reading finds only some of the pairs, and on a staircase layout it finds none. Since every drawing shows three faces meeting at a corner, an option displaying a known pair together is wrong on sight.