Edge and Count Bookkeeping for Shape-Only Nets

When there are no markings, count

Plenty of patterns in this section carry no shading at all. Nothing is black, nothing is hatched, no square has a diagonal across it — the pattern is a bare outline and the four figures are bare solids. There is still plenty to read, but it is all arithmetic: how many panels, how wide each one is, and which panels have to match each other.

On a shape-only pattern, start with the panel count. It is the fastest measurement on the page, it needs no folding, and it usually rules out at least one figure on its own.

A profiled prism: two caps plus one wall per edge

Most shape-only items are a flat outline extruded to a depth — an L, a T, a channel, a staircase. The pattern of one of those is always built the same way: a strip of rectangles that wraps once around the outline, plus the two end panels that close it.

Three facts come out of that construction, and all three are checkable at a glance:

• The panel count is fixed. An outline with six edges gives six wall rectangles plus two caps — eight panels, always.
• Each rectangle's width is the length of one edge of the outline, and the rectangles appear in the outline's own order, going around it. Walk the outline edge by edge and walk the strip rectangle by rectangle; they keep step.
• Every rectangle in the strip has the same height, because that height is the depth of the solid. A strip whose rectangles are not all the same height is not a prism's wall.

A slope announces itself in the strip

A ✓

B

C

D

Count first: three rectangles and two triangles, so the outline has three edges — a triangular end. Now compare the three rectangles. Two of them match the triangle's two legs. The third is wider than either, and it is not the width of any upright wall the wedge has.

That is what a sloped edge looks like on a flat pattern. A rectangle whose width matches no edge of the caps is the slope, and its width is the length of the slanted face, which is longer than either side it spans. On a rectilinear outline every strip width matches an edge you can see on the cap; the moment one does not, there is a slanted face on the solid, and any figure without one is finished.

Tapered figures: one slant height, and a frame for the top

A

B

C ✓

D

A taper is drawn as a base panel with four flaps splayed out from it and the top panel attached beyond one of those flaps. Two checks make this family quick.

First, all four flaps are drawn to the same height on the page. They are the four slanted faces of one solid, and on a figure that tapers evenly they share a single slant height, so a pattern whose flaps disagree in height is not this solid at all.

Second, and more useful, the short edges of the flaps must exactly frame the attached top panel. Look at the flap the top panel hangs off: the top panel is precisely as wide as that flap's short edge. Follow the other flaps around and their short edges give the other side of the top panel the same way. That is the taper check — the amount the flaps narrow by is the amount the solid narrows by, and it fixes both the top face and the height in one reading.

Assemblies: six squares per cube, minus two per joint

A ✓

B

C

D

A cube assembly's pattern is the whole outside surface of the solid, laid flat: one square per exposed face. Every cube would contribute six squares on its own, and every place two cubes are glued together buries one face on each of them — two squares gone per joint.

So count the squares in the pattern and solve. The pattern above has eighteen. Four cubes joined at three seams give 6 × 4 − 2 × 3 = 18, which is the count on the page; five cubes would need six seams to come down to eighteen, and five cubes can never be glued at more than five. Get the cube count from the square count before you try to reconstruct anything, and figures with the wrong number of cubes are gone for free.

Boxes: three dimensions, each appearing twice

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

A

B

C

D

A rectangular box's pattern is six rectangles in three matching pairs, because opposite faces of a box are identical. Read the three shapes off the pattern and you have the box: here the panels come in 2 × 3, 2 × 4 and 3 × 4, and the only box with those three faces is 2 × 3 × 4. A rectangle that has no partner in the pattern means you have misread a panel, not that the box is irregular.

The marked panel is the 2 × 4 one. That is two claims to test, and between them they clear the board:

• One figure is a 3 × 3 × 4 block — a dimension the pattern never offers. Out on proportions alone.
• One is the right 2 × 3 × 4 block, but its marked face is the 3 × 4 one. Right box, wrong face.
• One is the right block with the marking drawn as hatching rather than solid gray. Right box, wrong marking.
• One is the right block with the 2 × 4 face marked in solid gray.

A

B

C

D ✓

The pattern folds into a 2×3×4 box whose marked face measures 2×4. Check each option's proportions edge by edge, then check that the marked face has those exact dimensions and the same shading style — the wrong options shade a face of the wrong size, stretch the box, or lose the marked face entirely.

Every one of those judgments came from measuring panels, and none of it required folding the pattern up. Practice the counting on shape-only items in the pattern folding practice section.

The takeaway

Shape-only patterns are settled with arithmetic. Count the panels: a profiled prism is two caps plus one wall rectangle per edge of its outline, in the outline's own order, and every rectangle in the strip is the same height. A width that matches no edge of the caps is a slanted face. A tapered figure's flaps share one slant height and their short edges frame the attached top panel exactly. An assembly's squares number six per cube less two per joint. A box's six panels are three shapes, each drawn twice.