Problem
Find opposite faces on a cube net
Five squares are joined in a staircase. The top square carries a star. Exactly one more square is added edge to edge. Find every place that makes it a cube net.
Reasoning · Grade 5-2 Nets of a Cube
Your answer
Watch out It is easy to test a few spots, but there are nine to check.
How to solve
Strategy Make a Systematic List — The set of places the sixth square can go is small and completely knowable — walk round the outside of the staircase and every free edge is one candidate. That gives a finite list of 9, which is exactly the situation where listing them all and then eliminating is honest, complete work. Before testing them one by one there is a shortcut worth finding: fold the five squares that are already there and see which faces of the cube they cover. Five of the six faces get covered, they pair up into opposite pairs, and one square is left without a partner — so the sixth square has to become that partner. That single observation explains every answer and every rejection. Rolling a die across each candidate figure, or cutting one out and folding it, checks each verdict physically.
1STEP 1
One square is missing, not more
Exactly one square is missing.
6 - 5 = 1
Counting faces on a real cube and subtracting the squares already drawn is a first-grade subtraction that fixes exactly how much freedom the problem allows.
K.G.B.4Make A Systematic List2STEP 2
List every place the square could go
There are nine possible spots.
9 free edges → 9 candidate positions
Tracing the outline once and marking each free edge guarantees the list is complete, which is the only way to be sure no answer is missed.
1.G.A.2Make A Systematic List3STEP 3
Fold the five squares and find the odd one out
Folding the five reveals the face with no partner.
(1,3) ⇔ (3,2), (2,3) ⇔ (3,1), (2,2) ⇔ the new square
Six faces make three opposite pairs, so once five faces are placed the missing one is completely pinned down — the whole problem becomes 'where does the partner of (2,2) go?'
6.G.A.4Create A Physical Representation4STEP 4
Cross off the three spots that put four squares at one corner
Cross off the three spots that crowd a corner.
squares round one corner = 4 ＞ 3 = faces at a cube corner
Spotting a 2 by 2 block is pure looking — no folding needed — so three of the nine candidates fall in a single glance.
6.G.A.4Eliminate PossibilitiesAny spot that puts four squares around one corner is impossible, so those placements are crossed off.
Why?
Exactly three faces of a cube meet at every corner, so a placement demanding four is refuted straight away.
🧱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?
Once every impossible spot has been ruled out, the survivors are the only placements worth folding.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
5STEP 5
Cross off the two spots that fold onto a square already there
Cross off the two spots that overlap a face.
(0,3) → face of (3,1), (3,0) → face of (1,3)
Rolling the die two or three squares further than feels necessary is what catches an overlap; the collision is invisible in the flat picture but obvious the moment the die repeats a face.
6.G.A.4Visualize Spatial Relationships6STEP 6
Check the four survivors really fold
The remaining four all fold up.
(1,4), (2,4), (4,1), (4,2)
Nine candidates minus three corner failures minus two overlap failures leaves four, and four is exactly the number of blank figures the book printed — the count itself says nothing was missed.
6.G.A.4Create A Physical RepresentationAnswer
4 places
9 − 3 − 2 = 4
The four answers exactly fill the four blank copies printed in the book, and 3 corner failures plus 2 overlap failures plus 4 successes accounts for all 9 candidate positions with none left over. Every answer obeys the pairing rule found in step 3 — the new square always lands opposite (2,2) — and in all four completed nets the face opposite the star is still the square at (3,2), which is right, because the five original squares never move relative to one another. Each answer really has 6 squares, and each folds shut into a cube with no face bare and no face doubled.
Takeaway
Fold the five squares you already have, find the one face with no partner, and the sixth square must go wherever it can become that partner — four places work!
- One square is missing, not more
- List every place the square could go
- Fold the five squares and find the odd one out
- Cross off the three spots that put four squares at one corner
- Cross off the two spots that fold onto a square already there
- Check the four survivors really fold