Cube-net fold check
Test whether six connected squares fold into six different cube faces.
Adult planning suggestion: about6–9 · Shape & spatial thinking
- Players
- 2–3 per group
- Time
- 15–25 minutes
- Materials
- Two six-square nets, pencil and adult-assisted scissors; tape optional.
Prepare
Use equal squares with integer coordinates. Net 1 has squares(1,0),(0,1),(1,1),(2,1),(1,2),(1,3). Net 2 is the 2×3 rectangle(0,0),(1,0),(0,1),(1,1),(0,2),(1,2). Label each square so an overlap remains detectable after folding.
Play
Predict valid or invalid before cutting. Cut only the outside outline, then fold ninety degrees along every shared grid edge while keeping all six squares connected. A valid cube has six distinct faces, no overlapping face, no gap and no extra cut separating a square. Do not stretch or trim a square to force a fit.
Worked example
Net 1 is valid: treat(1,1) as a base; the four neighbouring squares become four sides and the remaining square(1,3) closes the opposite face. Net 2 is invalid: a 2×2 block within it creates a closed cycle of fold constraints that cannot become four different faces without a conflict.
Finish and check
Check the physical folds or a labelled face model rather than just count six squares. Record which labelled faces meet and whether an overlap occurs. A failed net is a useful completed classification, not an invitation to erase a face. Finish after both predictions have been tested and explained.
Adaptation and limits
Thirty learners may use ten trios; an adult can pre-cut the outlines or demonstrate with a large copy. Our diagrams and labels are original. The NRICH source discusses cube views and nets; it does not validate our exact invalid-net proof or guarantee suitability for every learner.
Purpose, observation and provenance
Within this activity: Test whether six connected squares fold into six different cube faces.
Ask which face would overlap and whether all six labels remain distinct after folding.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
NRICH — A Puzzling Cube — Background on relating nets and cube faces; our two diagrams and folding audit are separately written.
Age bands are planning suggestions. Goals describe observable actions, not measured long-term gains or institutional endorsement.
Free alternative: Copy the supplied grids, labels and cards onto reused paper. No specialist product is needed. Adults may prepare pieces or act as a scribe.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Player materials · Cube-net fold check
Net 1:(1,0)(0,1)(1,1)(2,1)(1,2)(1,3).
Net 2:2×3 rectangle. Predict, fold, record distinct face count.
Additional pieces / private cards
Player record · keep host key separate
| Net | Prediction | Overlap? | Conclusion |
|---|---|---|---|
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
Host key · separate from players
Net 1 valid, six unique outward faces; Net 2 invalid. Six equal squares alone are insufficient.
Run the round
Use equal squares with integer coordinates. Net 1 has squares(1,0),(0,1),(1,1),(2,1),(1,2),(1,3). Net 2 is the 2×3 rectangle(0,0),(1,0),(0,1),(1,1),(0,2),(1,2). Label each square so an overlap remains detectable after folding.
Predict valid or invalid before cutting. Cut only the outside outline, then fold ninety degrees along every shared grid edge while keeping all six squares connected. A valid cube has six distinct faces, no overlapping face, no gap and no extra cut separating a square. Do not stretch or trim a square to force a fit.
Net 1 is valid: treat(1,1) as a base; the four neighbouring squares become four sides and the remaining square(1,3) closes the opposite face. Net 2 is invalid: a 2×2 block within it creates a closed cycle of fold constraints that cannot become four different faces without a conflict.
Check the physical folds or a labelled face model rather than just count six squares. Record which labelled faces meet and whether an overlap occurs. A failed net is a useful completed classification, not an invitation to erase a face. Finish after both predictions have been tested and explained.
Thirty learners may use ten trios; an adult can pre-cut the outlines or demonstrate with a large copy. Our diagrams and labels are original. The NRICH source discusses cube views and nets; it does not validate our exact invalid-net proof or guarantee suitability for every learner.