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Reading a Flat Pattern
Cube Nets and Their Layout Families
Summary
The run and the caps
Most cube patterns are built the same way: four squares in a straight run, one square above the run, and one square below it. That is the 1-4-1 layout, and twelve of the sixteen cube layouts the practice generator can draw belong to it.
The run of four does the real work. Each square hinges onto the next, and the fourth closes back onto the first, so the run wraps into a closed band around the cube. A band of four faces leaves exactly two sides of the cube open, and the two lone squares are the caps that close them.
Read the run first, always. Once you have found four squares in a line you know where the band is, and the two squares left over are the caps — no folding required.
The caps can sit at any column
Nothing says the lone squares must sit above and below the same square of the run. They can be offset from each other, and they can hang off either end. Here is the most extreme version, with one cap above the first square of the run and the other below the fourth:
Same band, same caps, same cube — the folded drawing is identical to the centred cross above it. The engine agrees: it compares patterns by the set of drawings a cube can produce across all twenty-four orientations, and for these two patterns that set is exactly the same. Offsetting the caps changes your bookkeeping, not the solid. It costs you the convenience of a symmetric picture, which is why the same cube feels harder to fold from an offset pattern.
Three layouts that hide the band
Not every cube pattern gives you a run of four. Three other layouts appear, and all three are drawn from the generator's hard tier: the 1-3-2, the 2-2-2 staircase, and the 3-3.
Each of these still folds into a cube with a band and two caps — that structure belongs to the cube, not to the drawing. What changes is that you can no longer see it. Across all sixteen certified layouts, the run of four is the only band that appears on the page as a connected straight run, and it appears only in the 1-4-1. In the other three layouts the four band squares are scattered, and the two caps are ordinary-looking squares with nothing to mark them out.
So the layout family is worth naming before anything else, for two different reasons. A 1-4-1 tells you where the band is. A 1-3-2, a staircase or a 3-3 tells you that nothing can be read off the page and you will have to fold square by square instead — which is the next lesson's method.
What the generator holds
Sixteen layouts, each one certified by the fold engine rather than assumed: a candidate is only admitted if folding it puts all six squares on six different sides of the cube, each face centred on its own side. Four layouts are rated easy, six medium and six hard. A cube item at the easy setting picks from the four easy layouts, a medium item from ten, and a hard item from twelve — the hard tier drops the friendly centred crosses entirely.
Six squares is not a certificate
A six-square shape is not automatically a cube net. Both of these are hexominoes, and neither one folds:
The two rows of three fail because there is no run of four anywhere in them: fold them and three squares end up facing the same way, leaving three sides of the cube bare. The run of five fails for the opposite reason — the band only has room for four, so the fifth square comes all the way around and lands back on the first.
That is the run-and-caps reading working as a certificate. Four in a line, and two left over to cap the ends: get that reading and the shape folds. Fail to get it and either you have one of the three awkward layouts, or you do not have a cube net at all.
Read the layout on a real pattern
Run of four along the middle row; one lone square above the third, one below the second. It is a 1-4-1 with offset caps — the band is the middle row, and the two lone squares close its ends. What the shading on those squares implies about the cube is the next topic's business.
The takeaway
Find four squares in a line: that run is a band wrapping the cube, and the two leftover squares cap it. The caps may sit at any columns without changing the solid. Three layouts — the 1-3-2, the staircase and the 3-3 — hide the band completely, and any six-square shape that yields no run-and-caps reading is not a cube net.
Practice Questions
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