Wrong Face, Wrong Neighbour

The markings are right; their addresses are wrong

Once you have settled whether a mirror image can exist, the remaining wrong answers are almost all built the same way: the engine takes the correct solid and moves a marking to a face it does not belong on. Nothing is added, nothing is redrawn, nothing changes size. Every shade you expect is still a shade you expect. Only the addresses are wrong.

That is what makes this family the one people lose points to. It survives a glance, it survives a shade-by-shade inventory, and it is beaten by exactly one question: which face is next to which?

Start with a pattern where every face is marked

A die is the clearest case, because there are no blanks to hide behind. Fold the cross below and each of the six numbers lands on a definite face, with a definite neighbour on every side.

The pattern

The correct die

The fold also decides which numbers you will never see together. On a strip of four, the first cell ends up opposite the third and the second opposite the fourth, and the two flaps end up opposite each other. So on this pattern the 1 is opposite the 3, the 2 is opposite the 4, and the 5 is opposite the 6.

Note what those pairs are not. On a real die opposite faces sum to seven; on this one they sum to 4, 6 and 11. The generator drops the six numbers onto the pattern at random and lets the fold decide what faces what. Read the opposite pairs off the pattern in front of you — a habit imported from real dice will convict the right answer and acquit a trap.

Four ways to move a marking

Each of these produces a solid that no folding of the pattern can produce, so each is a legal wrong answer:

• A marking moved to another face. One shade leaves its face and lands on a face that was blank, or on a face that belonged to something else.
• Two markings traded. Both faces are still marked and both markings still exist, but they have exchanged places — so the neighbours are wrong on both.
• A marking duplicated. One face's artwork is copied onto a second face. Now one marking appears twice and another has disappeared entirely.
• A face carrying its opposite's marking. The strongest version: a face is drawn with the marking that the pattern puts on the face directly behind it.

The check that convicts them

Here is the last of those four, built on the die above. The top face has been given the marking from the face opposite it:

Trap — a 3 where the 1 belongs

Fold the pattern and the 1 is the face between the 2 and the 6 at this corner. The option shows a 3 there instead — and the 3 is the face directly behind the 1, so a 3 can never occupy the 1's position. One check, one conviction, and you never had to look at the rest of the solid.

The trade version can be even quicker, because it puts the evidence on two panels at once:

Trap — two faces that are back to back

A 1 and a 3 are on show together. The pattern makes those two opposite faces, and opposite faces are back to back — no viewpoint shows both. The option is dead before you have counted a single pip on the third panel.

That is the opposite-face check, and it is decisive whenever every face carries a distinct marking. When a pattern marks only two or three faces and leaves the rest blank, it loses its teeth: two blank panels are consistent with almost anything, so a blank tells you nothing. There you fall back on adjacency — which shade sits beside which, and in what order they run around the corner you are looking at.

Counting shaded faces proves nothing

The most common way to lose one of these is to check the wrong quantity. Here is the correct solid for a three-shade pattern, and beside it the duplicate trap built from it:

The pattern

The correct cube

Trap — one shade twice, one missing

The trap still has three shaded faces, exactly as the pattern demands. It still uses shades the pattern uses. What it does not have is one of each: the gray has been copied onto the face that should carry the stipple, so a gray appears twice and the stipple has vanished. An inventory that stops at "three shaded faces, black and gray present" passes this option straight through.

So make the inventory about position, not about quantity. Name the marking on each visible face, then name what the pattern says should be beside it. Every member of this family fails that sentence somewhere, and the failure is usually on the first or second face you check.

A word on what this family is not

Do not confuse a moved marking with a turned one. In everything above, each marking keeps its own orientation and simply sits on the wrong face; an option whose markings are all on the correct faces but one of them is turned a quarter is a different trap with a different cure, and it is the next topic. Do not confuse it with the mirror image either — there, every marking is on the face it belongs on, and only the handedness is wrong. Practice the checks on the pattern folding practice generator, naming the opposite pairs before you look at the options.

The takeaway

Most wrong options move a marking rather than change it: relocated, traded, duplicated, or lifted from the face opposite. All four are convicted by position — which face is next to which, and which two faces the fold puts back to back so they can never be seen together. Read the opposite pairs off the pattern rather than from any habit about real dice, and never let a count of shaded faces stand in for a check of where they sit.