Problem
How many different cube nets exist
Six flat figures of joined squares are drawn. A net must fold up into a closed cube. No square may be left over and none may overlap. Pick out every figure that is not a net.
Reasoning · Grade 5-2 Nets of a Cube
Your answer
Watch out Six squares seems enough, but a 2 by 2 block never folds.
How to solve
Strategy Create a Physical Representation — Staring at a flat figure and trying to picture the fold is where mistakes happen, so I use a test I can actually carry out with my hands: set a die on one square of the figure and roll it, one square at a time, all the way around the figure. Whatever face of the die is touching the paper gets stamped onto that square. A figure is a net exactly when the six squares receive six different faces of the die — because that is what 'each face covered once' means. Rolling is physical and repeatable, and it never needs me to imagine the finished cube. Two quick screens run first and knock out figures without any rolling at all: count the squares (a net needs exactly 6), and look for a 2 by 2 block (four squares round one corner can never fold). I then organise the six figures into a short list of verdicts so none is skipped.
1STEP 1
Screen 1 — a net must have exactly 6 squares
Figure 2 has only five squares, so out.
faces of a cube = 6, squares in (2) = 5, 5 ≠ 6
Counting the squares and comparing with the 6 faces of a cube is a matching job a young child can do by pointing, and it settles one whole figure before any folding is imagined.
K.G.B.4Make A Systematic List2STEP 2
Screen 2 — no 2 by 2 block is allowed
Figure 4 has a 2 by 2 block, so out.
faces meeting at a cube corner = 3, squares round the shared corner in (4) = 4
Counting how many faces meet at a corner of a real cube (hold one and look) turns an imagining task into a counting task, and the 2 by 2 block is easy to spot by eye.
6.G.A.4Visualize Spatial RelationshipsA figure containing a 2 by 2 block can never be a net, because those four squares would pile onto one corner.
Why?
Only three faces of a cube meet at any corner, so a block demanding four is refuted before any folding is tried.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
Why?
Folding must send each square to exactly one face, so two squares landing on the same face means the figure cannot close.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Set up the rolling test
For the rest, roll a die across them.
Rolling a die across the drawing reproduces the fold exactly, one edge at a time, so a solver who cannot picture the finished cube can still get the right verdict by moving a real object.
6.G.A.4Create A Physical Representation4STEP 4
Roll across figure (5) — two squares collide
In figure 5 two squares land on the same face.
(1,1) and (3,3) → the same face
A strip of 4 behaves like a paper belt round a box: after four squares you are back where you started, so a two-square arm hanging off the belt has nowhere new to land.
6.G.A.4Create A Physical Representation5STEP 5
Roll across (1), (3) and (6) — all six faces come out different
Figures 1, 3 and 6 give six different faces.
6 squares → 6 different faces = 3 opposite pairs
Reading off the three opposite pairs is a second, independent way to see the fold worked: six faces sorted into three pairs is exactly how a cube is built.
6.G.A.4Create A Physical Representation6STEP 6
Collect the verdicts
The non-nets are 2, 4 and 5.
not nets = (2), (4), (5)
Writing one short verdict per figure, in order, is how a fourth grader keeps a six-case check honest — the list itself shows that every figure was judged.
1.G.A.2Draw A DiagramAnswer
2, 4, 5
Three of six figures fail, which is a believable mix for a page that is testing whether you can tell nets from look-alikes — and each failure fails for a different, checkable reason, not for a vague feeling. The counts are right: (2) really does have 5 squares, not 6; (4) really does contain a solid 2 by 2 block; and (5) really does send two squares to one face, since a strip of 4 closes into a belt after four steps. The three survivors each split their six squares into exactly three opposite pairs, which is what a cube demands. As a further check, the three survivors are all shapes that appear among the eleven different nets of a cube taught in this unit.
Takeaway
Count the squares, hunt for a 2 by 2 block, then roll a die across the figure — if any two squares get the same face of the die, it is not a net!
- Screen 1 — a net must have exactly 6 squares
- Screen 2 — no 2 by 2 block is allowed
- Set up the rolling test
- Roll across figure (5) — two squares collide
- Roll across (1), (3) and (6) — all six faces come out different
- Collect the verdicts