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Hard Items and Pacing
What Makes a Hard Item Hard
Summary
A hard pattern-folding item is not a different kind of question. It is the same question with more on the page: a layout that hides where the band is, more marks to hold at once, solids with more edges to account for. Nothing you learned stops working. There is simply more of it to track before you can use it.
The layout goes first
The practice engine keeps sixteen cube layouts and grades each one by how hard the layout is to read. Four of them are the friendly ones: a run of four squares with a cap above it and a cap below it, both caps sitting near the middle. Those four are the entire easy pool.
None of those four appears at the hard tier. Hard draws from twelve layouts, and the four easy ones are not among them. In their place: 1-4-1 runs with the two caps pushed to opposite ends, the 1-3-2, the 2-2-2 staircase, and the 3-3.
The staircase is the clearest example of what you lose. Its longest straight run is two squares. There is no run of four to read the wrapping band off, so the shortcut you use on a cross — band here, caps there — has nothing to attach to. You get the same six squares and the same cube; you just have to walk the shared edges to find out which square is which.
Three marks, and two kinds of mark at once
Part-face shading items carry two marked faces at the medium tier and three at the hard tier. That is the single biggest change in bookkeeping load in the section: a third mark adds not one relationship but two, because every mark has to be placed against every other one.
The hard tier also mixes the vocabularies. About two times in five it makes the third mark a solid whole-face shade sitting alongside the part-face ones. A whole-face shade fixes only which face it is on; a half or a diagonal fixes which face and which way round. On a hard cube you can be tracking both kinds in the same item, and only some of your marks will punish you for losing their direction.
Shape-only solids get more edges
The shape-only prisms are graded by their end profile. Easy draws L-shapes, triangles and trapezoids — profiles with six, three and four edges. Hard draws U-channels, three-step staircases, V-notches, plus shapes and Z shapes — seven to twelve edges.
That count is not decoration. Every edge of the profile becomes one rectangle in the strip that wraps around the block, so a plus-profile pattern hands you twelve wall panels to account for where an L-shape hands you six. The arithmetic is the same arithmetic. There is just twice as much of it.
Bigger assemblies, and a second band
Two things appear at the hard tier that you will not have met below it. Glued cube assemblies are one: the hard pool holds eight of them, and two are built from five cubes rather than four. A five-cube assembly unfolds to twenty-two squares — six per cube, less two for every glued joint — against eighteen for a four-cube one.
Assemblies are also the only place in the section where an assembly's reflection is a genuinely different solid, and then only for three of the eight: the other five are their own mirror image turned around. That is what the handedness audit in the previous lesson is aimed at.
Rolled cylinders are the other. They exist at the medium tier too, but their share of the mix nearly doubles at hard, and only at hard can a cylinder carry two bands instead of one. Two bands means a second spacing to check on top of slant, width and direction.
What does not change
The wrong options are not drawn from a cleverer pool at the hard tier. In the tapered-figure family the list of edits that builds them is identical at medium and hard — one unit taller, one unit shorter, half a unit more taper, half a unit less, a wider base, a deeper base, no taper at all. What changes is the solid those edits land on: hard frustums start taller, and their taper can be half a unit where medium's is always a full one. The same one-unit edit is simply a smaller share of a bigger drawing.
The cylinders shift the same way. When the pattern carries two bands, the wrong-count option is a cylinder with one band missing rather than one added — a subtraction is easier to overlook than an addition, but it is the same check.
So the honest summary of hard items is unglamorous. Same rules, same fold, same relationships. More layout to decode before the relationships are visible, more marks to hold while you decode it, and more panels to count when there are no marks at all.
The takeaway
Hard items change the bookkeeping, not the method. Expect a layout with no readable run of four, three marks instead of two, whole-face and part-face shading in the same item, profiles with up to twelve edges, five-cube assemblies and two-band cylinders. The wrong options are built by the same edits as ever — they only look closer because the solids they sit beside got bigger. If a hard item feels like it needs a new technique, what it actually needs is one more line of bookkeeping.
Practice Questions
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