A table's four columns are a triangular and a quadrilateral pyramid and a triangular and a quadrilateral prism. There are rows for vertices, faces and edges. The bottom row is vertices plus faces minus edges. Fill the table and find the rule.
Your answer
Watch outAdding each face's sides double-counts, because an edge belongs to two faces.
How to solve
StrategyMake a Systematic List — Counting the parts of a solid from a flat drawing is where mistakes happen, so for each solid I count in a fixed order — base first, then top or apex, then the sides — instead of trying to sweep my eye round the picture. That is what stops me forgetting the dashed edges at the back. Once the table is full the bottom row makes the pattern jump out. Then, rather than counting a pentagonal solid all over again from a picture I do not have, I use the same base-then-sides recipe with 5 in place of 3 or 4, which is the easier related problem I have already solved four times.
1STEP 1
Count the triangular pyramid and confirm the given numbers
The triangular pyramid gives 4, 4, 6.
v = 3 + 1 = 4, f = 1 + 3 = 4, e = 3 + 3 = 6
The two printed numbers act as a free check on the counting recipe. If base-plus-apex had not given 4 and 4, the recipe would be wrong and every other column would be wrong too.
6.G.A.4Make A Systematic List
2STEP 2
Count the quadrilateral pyramid
The quadrilateral pyramid gives 5, 5, 8.
v = 4 + 1 = 5, f = 1 + 4 = 5, e = 4 + 4 = 8
In any pyramid the faces and the vertices come out the same, because each side of the base contributes one slanted face while each corner of the base contributes one vertex, and a polygon has as many sides as corners.
6.G.A.4Make A Systematic List
3STEP 3
Count the triangular prism
The triangular prism gives 6, 5, 9.
v = 3 + 3 = 6, f = 2 + 3 = 5, e = 3 + 3 + 3 = 9
Picture the prism as a triangle that has been slid straight upward: every corner traces one upright edge, every side sweeps out one rectangular face, and you end up with a second copy of the triangle on top.
6.G.A.4Visualize Spatial Relationships
4STEP 4
Count the quadrilateral prism
The box gives 8, 6, 12.
v = 4 + 4 = 8, f = 2 + 4 = 6, e = 4 + 4 + 4 = 12
Everyone has handled a box, so this column is the one you can verify with your hands: hold a cereal packet and count its corners, its flat sides and its folded edges.
6.G.A.4Visualize Spatial Relationships
5STEP 5
Fill in the bottom row
The bottom row is 2 every time.
4+4-6 = 2, 5+5-8 = 2, 6+5-9 = 2, 8+6-12 = 2
The three counts change a lot from column to column — v runs 4, 5, 6, 8 and e runs 6, 8, 9, 12 — so getting the same 2 four times in a row is far too tidy to be an accident.
4.OA.A.3Make A Systematic List
6STEP 6
State the rule found in part (2)
The rule is vertices plus faces minus edges is 2.
v + f - e = 2
A rule found from four examples is a conjecture, and stating it with letters is what makes it testable: now any new solid either fits v + f - e = 2 or it does not.
6.EE.A.2Look For A Pattern
7STEP 7
Test the rule on a pentagonal pyramid
A pentagonal pyramid also gives 2.
v = 5 + 1 = 6, f = 1 + 5 = 6, e = 5 + 5 = 10, 6 + 6 - 10 = 2
You never have to see a pentagonal pyramid to count it: the same base-then-apex recipe that worked for the triangle and the quadrilateral works with 5, because nothing in the recipe depended on the number.
6.G.A.4Solve An Easier Related Problem
8STEP 8
Test the rule on a pentagonal prism
A pentagonal prism also gives 2.
v = 5 + 5 = 10, f = 2 + 5 = 7, e = 5 + 5 + 5 = 15, 10 + 7 - 15 = 2
Two fresh solids that were not used to find the rule both obey it, which is much stronger evidence than four more examples of the same two shapes would have been.
Using a letter for the number of base sides turns four separate checks into one line of arithmetic, and watching the n terms cancel shows exactly why the leftover is always the plain number 2.
6.EE.A.2Look For A Pattern
The rule has to keep working, because vertices minus edges plus faces comes to 2 for every solid with flat faces.
Why?
That one little sum lands on 2 whatever the solid, so it links the three counts however many faces there happen to be.
🧱Euler's formula for solidsFor any solid with flat faces, vertices minus edges plus faces always comes out to 2, so a count you cannot see can be recovered.
Why?
A single solid where the sum missed 2 would end the rule, so testing new shapes is a real check and not a formality.
🧱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.
Answer
2
4 + 4 − 6 = 2
Every count is a whole number and each one grows sensibly as the base gains a side: from the triangular to the quadrilateral pyramid, v goes 4 to 5, f goes 4 to 5 and e goes 6 to 8, exactly the plus 1, plus 1, plus 2 that one more base side should give; from the triangular to the quadrilateral prism, v goes 6 to 8, f goes 5 to 6 and e goes 9 to 12, again the expected plus 2, plus 1, plus 3. The quadrilateral prism column can be checked against a real box: 8 corners, 6 faces, 12 edges. A second independent check on the edge counts is to add up the sides of all the faces and halve, since every edge is shared by exactly two faces: for the triangular prism that is 3 + 3 + 4 + 4 + 4 = 18, and 18 halved is 9, matching. And the algebra in the last step explains, rather than merely observes, why the bottom row is all 2s.
Takeaway
Count corners, add flat faces, take away edges — for every one of these solids the answer is 2, every single time.
Count the triangular pyramid and confirm the given numbers