Test it, fold it, count it

Fifteen statement cards and three bins.

Always true: a square is a rectangle; an equilateral triangle is isosceles; the diagonals of a rhombus cross at right angles; a quadrilateral's angles add to 360°; a pyramid has as many faces as vertices.

Sometimes true: a rectangle is a square; a rhombus has four right angles; an isosceles triangle is right-angled; a parallelogram's diagonals are equal; a trapezium has two pairs of parallel sides (under the inclusive definition); a kite has four equal sides; a polygon has more diagonals than sides.

Never true: a triangle has two obtuse angles; a prism has an odd number of vertices (it has 2n); a circle has exactly 4 lines of symmetry (it has infinitely many).

For every sometimes card, find one example where it is true and one where it is false.

In count, a solid appears each round and you type its faces, edges and vertices. In explore: Drag the solid, or use the Turn and Tilt sliders, to rotate it; hidden edges show as dashed lines. You can highlight its faces, edges or vertices, and for a polyhedron the Net button unfolds it flat. The readout shows F, E, V and the Euler check.

Polyhedra: cube 6 faces, 12 edges, 8 vertices; cuboid 6, 12, 8; triangular prism 5, 9, 6; pentagonal prism 7, 15, 10; hexagonal prism 8, 18, 12; triangular pyramid 4, 6, 4; square pyramid 5, 8, 5. For every one, F + V − E = 2; for example 8 + 12 − 18 = 2.

Curved solids: the lab counts the cylinder as 3 faces (2 flat and 1 curved), 2 edges, 0 vertices; the cone 2, 1, 1; the sphere 1, 0, 0. These give 1, 2 and 1, and the lab points out that Euler's formula applies to polyhedra only, solids with flat faces.

Twelve cards each describe an arrangement of six equal squares joined edge to edge.

Folds into a cube: the cross (row of 4 with squares above and below the 2nd); a row of 4 with one square above the 1st and one below the 4th; the T shape; the 2-2-2 staircase; two rows of 3 shifted so they share one edge; a row of 3 with one square above its left end and a row of 2 below its right end; a row of 4 with squares above and below the 3rd.

Does not fold: a row of 4 with both extras on the same side; six in a row; a 2 by 3 block; a row of 5 plus one; any shape containing a 2 by 2 block.

Useful tests: a row of 5 or more always fails; a 2 by 2 block always fails; in a row of 4, the two extra squares must be on opposite sides of the row.

Seven view descriptions on the left, seven solids on the right. The pairs: top circle and front rectangle is a standing cylinder; top circle with a centre dot and front triangle is a cone on its base; top square and front triangle is a square pyramid; circles from every side is a sphere; top rectangle and front triangle is a triangular prism lying on a rectangular face (like a tent); squares from every side is a cube; top rectangle and front circle is a cylinder lying down with its round end facing you.

Notice that the cylinder appears twice: the same solid gives different views depending on how it is placed.

Fourteen capital letters, three bins, and a 60-second timer.

No lines: F, S, Z, R. Exactly one line: A, M, T (vertical); B, E, C (horizontal). Two lines: H, I, X and O (both vertical and horizontal).

S and Z have no mirror line, but they look the same when turned upside down: that is rotational symmetry, not line symmetry. The answers assume plain block capitals; fancy fonts can change them. An oval O has 2 lines; a perfectly round O, a circle, would have infinitely many.

An 8-round game with eight tile shapes: a triangle (drawn scalene), a square, a regular pentagon, a regular hexagon, a regular octagon, a rhombus, a parallelogram and a trapezium.

Each round shows one shape and asks you to name it or asks a yes-or-no question: Is it a quadrilateral? Does it have a pair of parallel sides? Are all its sides equal? Does it have a right angle?

Useful facts: the square, rhombus, parallelogram and trapezium are quadrilaterals, and all four have parallel sides. All sides are equal in the square, the rhombus and the regular pentagon, hexagon and octagon. Only the square has a right angle.

The link to tiling: every triangle and every quadrilateral tiles a floor (their angles, 180° or 360°, can be fitted round a point), while among regular polygons only the triangle, square and hexagon tile alone. The pentagon (108°) and octagon (135°) leave gaps.