Planar face ledger
Count the exterior face and audit how legal planar edits preserve V−E+F.
Adult planning suggestion: about 9–12 · Shape & spatial thinking
- Players
- 1–4 with adult support
- Time
- 15–25 minutes
- Materials
- Square boundary and one diagonal
Prepare
Draw a connected square graph with four vertices and four boundary edges. Edges meet only at drawn vertices. Faces include the unbounded exterior region, so the starting face count is 2, not 1.
Play
Add one diagonal between opposite vertices without crossing an existing edge. It splits one bounded face into two: V stays 4, E becomes 5, F becomes 3. Compute 4−5+3=2.
Worked example
Next put a new vertex halfway along one existing edge, replacing that edge with two. V and E each increase by 1 and F stays 3; the expression remains 2. The two resulting segments count as two edges.
Finish and check
A dangling edge to a new vertex outside the square raises V and E by 1 but creates no new face. A crossing without a vertex is not permitted in this drawing; do not quietly treat it as either an overpass or a junction.
Adaptation and limits
Finish after a partner traces every edge and region for the three stages. The formula V−E+F=2 here assumes a connected plane graph; disconnected drawings require a different count. No claim about walking every edge is involved.
Purpose, observation and provenance
Within this activity: Count the exterior face and audit how legal planar edits preserve V−E+F.
Ask the counter to point to the unbounded exterior face and account separately for each graph edit.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
Discrete Mathematics: Planar Graphs — Reference supports the stated concept; our fictional numbers, instructions and paper diagrams are independently prepared, not reproduced teaching pages.
Secondary practice: Cooperation and explaining a check.
For a class of 30, use ten groups of three. Prepare one player-material set per group and keep the host key with the adult. Rotate operator, recorder and checker after each worked example; a reader or scribe may help without choosing answers. Avoid public ranking or diagnostic labels.
Age bands are planning suggestions. Background references may address older learners and do not validate suitability for this age band. 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.
Rules to keep beside the materials
- Draw a connected square graph with four vertices and four boundary edges. Edges meet only at drawn vertices. Faces include the unbounded exterior region, so the starting face count is 2, not 1.
- Add one diagonal between opposite vertices without crossing an existing edge. It splits one bounded face into two: V stays 4, E becomes 5, F becomes 3. Compute 4−5+3=2.
- Next put a new vertex halfway along one existing edge, replacing that edge with two. V and E each increase by 1 and F stays 3; the expression remains 2. The two resulting segments count as two edges.
- A dangling edge to a new vertex outside the square raises V and E by 1 but creates no new face. A crossing without a vertex is not permitted in this drawing; do not quietly treat it as either an overpass or a junction.
- Finish after a partner traces every edge and region for the three stages. The formula V−E+F=2 here assumes a connected plane graph; disconnected drawings require a different count. No claim about walking every edge is involved.
Planar face ledger — player materials
| Stage | V | E | F incl outside | V−E+F |
|---|---|---|---|---|
| Square | ||||
| Add diagonal AC | ||||
| Subdivide AB |
Host check — keep separate
Square 4−4+2=2; diagonal 4−5+3=2; subdivided 5−6+3=2.
Examples are stipulated model cases, not measured experimental or learning outcomes.
Related activities
Activity library · One-stroke house: rules and printable materials