Five claims: only five kinds exist; all faces are congruent regular polygons; the same number of faces meets at each vertex; the angles at one vertex total under 360 degrees; the faces are only triangles, squares and regular hexagons. Exactly one claim is wrong. Pick it.
Your answer
Watch outA hexagon-faced solid sounds plausible, but three already lie flat.
How to solve
StrategyMake a Systematic List — The statements that matter (1 and 5) both claim something about a complete list, so seeing five solids in a picture proves nothing — a picture cannot show that a sixth one is missing. Instead I build the list myself. A corner of a regular polyhedron is described by just two numbers: the shape of the face (how many sides it has) and how many faces meet there. So I make a systematic list running through face shapes — triangle, square, pentagon, hexagon, and beyond — and for each one test every possible number of faces per vertex against the 'less than 360 degrees' rule. That is a short list, because the angles grow and the test starts failing quickly. Whatever survives is the whole family; then I read all five statements off the finished list.
1STEP 1
What has to happen at one corner
A corner needs at least three faces totalling under 360.
3 ≤ (faces at a vertex) and (faces at a vertex) × (one face angle) ＜ 360°
Angles around a point add up, and a full turn is 360 degrees — that is the whole of Grade 4 angle sense. Anything short of a full turn leaves a wedge of gap, and taping that gap shut is what makes the flat paper pop up into a corner.
The only fact being borrowed is that a triangle's angles total 180 degrees. Chopping a polygon into triangles turns one hard question into several copies of that one easy fact.
8.G.A.5Solve An Easier Related Problem
3STEP 3
Face = equilateral triangle (60 degrees): three counts work
Once you know one angle is 60 degrees, testing a count is a single multiplication and a comparison with 360 — Grade 4 arithmetic decides a question about solids.
4.NBT.B.5Make A Systematic List
4STEP 4
Face = square (90 degrees): one count works
Squares allow only 3.
3 × 90 = 270 ✓ 4 × 90 = 360 (lies flat)
Four squares round a point is the pattern on every sheet of graph paper, so it is easy to believe that this is the case that lies flat and therefore fails.
4.NBT.B.5Make A Systematic List
5STEP 5
Face = regular pentagon (108 degrees): one count works
Pentagons allow only 3.
3 × 108 = 324 ✓ 4 × 108 = 432 (overlaps)
The gap here is only 36 degrees, which is why the dodecahedron looks almost round: less gap to close means a shallower corner and a rounder solid.
4.NBT.B.5Make A Systematic List
6STEP 6
Face = regular hexagon or anything bigger: nothing works
You do not have to test infinitely many polygons one by one. The angle only ever grows with the number of sides, so once the hexagon already fills a full turn with the minimum of 3 faces, every polygon after it fails too.
4.MD.C.7Make A Systematic List
A regular hexagon or anything larger can never make a solid corner, because three of them already fill a flat turn.
Why?
The angles at a solid corner must fall short of a full turn, or the faces lie flat instead of folding up.
🧱Angles around a point make 360 degreesAngles that fit side by side all the way around one point fill exactly one full turn, 360 degrees.
Why?
Every polygon from the hexagon upwards is ruled out by that one test, so the search stops there for good.
🧱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.
7STEP 7
Collect the survivors — exactly five
So there are exactly five regular polyhedra.
3_triangle + 1_square + 1_pentagon = 5
The list was built by testing every case and never skipping one, so counting the survivors is a complete answer, not a guess — this is why a systematic list can prove that nothing is missing.
4.OA.A.3Make A Systematic List
8STEP 8
Judge the five statements against the finished list
The statement naming hexagons, number 5, is wrong.
Every statement is now checked against one finished table rather than against a feeling, and the false one fails on a specific, checkable point: 'hexagon' where the table says 'pentagon'.
1.G.A.1Visualize Spatial Relationships
Answer
5
3 + 1 + 1 = 5
The five corner types found (3, 4, 5 triangles; 3 squares; 3 pentagons) match the five named solids in the picture one for one, with none left over and none missing, and the face shapes seen in the picture are triangles, a square and a pentagon — exactly the three shapes the argument allows. The three rejected cases are all things you can see on a flat surface: 6 triangles, 4 squares and 3 hexagons around a point are the three ways to tile a floor with one regular shape, which is a good sign the cut-off at 360 degrees is in the right place. Each angle total is also a sensible size: they run 180, 240, 270, 300, 324 degrees, all under a full turn of 360, and the biggest gap (180 degrees, for the tetrahedron) belongs to the pointiest solid while the smallest gap (36 degrees, for the dodecahedron) belongs to the roundest, which is what the picture shows.
Takeaway
A corner needs a gap: add the angles meeting at a point, and if they reach 360 degrees the shape lies flat — that one rule leaves exactly five regular polyhedra, with triangle, square and pentagon faces, never hexagons.
What has to happen at one corner
The angle of each regular polygon
Face = equilateral triangle (60 degrees): three counts work
Face = square (90 degrees): one count works
Face = regular pentagon (108 degrees): one count works
Face = regular hexagon or anything bigger: nothing works
Collect the survivors — exactly five
Judge the five statements against the finished list