Eliminate on One Decisive Relationship

Read the options before you fold

A pattern fixes an enormous number of facts at once: which squares finish up opposite each other, which edge of a face every marking touches, how each mark is turned when it lands. An item asks you about almost none of them. The four figures offered agree about most of the solid and disagree about one or two specific things, and those one or two things are the only facts you are being scored on.

So read the four figures first — not to judge them, but to find out what they are arguing about. The options tell you which relationship is decisive; the pattern tells you which side of the argument is right. Building the whole solid in your head before you know what you are hunting for is how a fifteen-second item costs you a minute.

Pick one relationship the pattern fixes

The pattern

Two marks here, on squares that share a fold edge: one square has its lower half shaded black, and the square directly below it is shaded solid gray. Because those squares are neighbors on the sheet, the black face and the gray face meet along that same edge on the folded cube — and the black half is the half lying against it.

That last sentence is the whole test. The black band must sit against the seam where the black face meets the gray face. One pair of faces, one edge contact, one thing to look for. Everything else the pattern fixes — which face is opposite which, where the blank squares land — stays unexamined until it has to be.

Test it against all four

Now run that single check across the options, in order, before you look at anything else about them.

Option A

Option B

Option C

Option D

• A. The black band lies along the far edge of its face, hard against the cube's right-hand outline. The seam with the gray face is the front vertical edge, on the other side of the white half. Out.
• B. The black band runs down the front vertical edge — the seam where the black face meets the gray one. It survives.
• C. The gray face has moved to the top, and the black band still hugs the front vertical edge, which here is the seam with a blank face. The seam with gray is the upper edge of the black face, and nothing is shaded along it. Out.
• D. The black band lies along the bottom of its face. It touches the lower half of the seam with gray instead of lying along the seam. Out.

One relationship, three kills, no folding of blank faces and no bookkeeping about opposites. That is the usual yield: a well-chosen pair of marked faces normally takes out two or three options at once, because the wrong figures are built by disturbing exactly the kind of fact you just tested.

What makes a relationship decisive

Not every true statement about the pattern does work. "The black face touches the gray face" is perfectly true, and it is also true of all four cubes above — in every one of them the two marked faces meet along an edge. A fact all four options agree on eliminates nothing, no matter how carefully you verified it.

A relationship earns its keep only when the options disagree about it, which is why you read them first. Between two candidate relationships, prefer the one that is finer: which edge of a face a marking touches discriminates far more often than which faces carry markings at all, because the second is easy for a wrong figure to get right by accident.

If more than one option survives your first test, do not restart. Keep the survivors and pick a second relationship — ideally about a different thing, such as a mark on a third face, or the turn of a marking rather than its position. Two cheap tests in sequence beat one exhaustive reconstruction, and you should never end up verifying the entire solid. Verify only what the options disagree about, and stop the moment one is left standing.

The same method on a live item

Which cube results when the pattern is folded into the page?

A

B

C

D

Read the four cubes: they disagree about which face carries the gray half and about which edge each shaded half lies against. Then read the pattern. The gray half occupies the side of its square furthest from the black square next to it, so on the folded cube the gray band has to sit on the far side of its face from the black face. Run that one check and see how many cubes it removes before you look at anything else.

A

B ✓

C

D

In the key, the gray band lies along the top face's back-left edge and the black face is the panel on the right — the gray band is as far from the black face as it can be, exactly as the pattern draws it. You can generate as many of these as you like in the pattern folding practice section; the discipline worth practicing is choosing the relationship before you start folding.

The takeaway

Read the options first and find the one or two things they disagree about. Turn that into a single relationship the pattern fixes — one pair of adjacent marked faces, or one marking's contact with one edge — and test that relationship against all four figures before you examine anything else. It will usually leave one option standing; when it leaves two, add a second relationship rather than starting over. You are never asked to build the whole solid, only to settle the argument the options are having.