Spatial formulas

Every point moves by the same amount. Size, shape and orientation are unchanged.

These are exact special cases of the rotation rule. For any other centre, apply them to each point's position relative to that centre.

So $270^{\circ }$ clockwise is $90^{\circ }$ anticlockwise. Reduce any angle modulo $360^{\circ }$ first: $450^{\circ }$ clockwise is $90^{\circ }$ clockwise.

A vertical mirror (the y-axis) swaps left and right. A horizontal mirror (the x-axis) swaps top and bottom. Reflecting twice in the same line gives back the original figure.

$k>1$ enlarges and $0<k<1$ shrinks. Angles are unchanged. The same factors apply to similar solids: surface area $\times k^2$, volume $\times k^3$.

The mirror stands beside the clock, so it reflects in the line through 12 and 6. Subtract the actual time from 11:60 (that is, from 12:00): 4:25 gives 7:35, and 9:10 gives 2:50. For a time from 12:00 to 12:59, subtract from 23:60 instead (12:10 gives 11:50). Read a result of 0:mm as 12:mm (11:25 gives 0:35, that is 12:35). The rule is exact for both hands.

N is the order of rotational symmetry. A regular polygon with n sides has N = n and n lines of symmetry.

Valid when every fold doubles the whole packet and the punch does not touch any fold line. A punch that touches a fold line gives fewer holes, because holes merge across the fold.

All the folds are parallel and each one folds the whole strip in half, so the packet is $2^n$ layers thick. One straight cut runs right across the packet without touching a fold. It cuts the strip at $2^n$ places, which leaves $2^n+1$ pieces. For example, 3 folds and one cut give 9 pieces.

Use this only when the question says the die is standard or ordinary. For any other die, work out the opposite faces from the views.

Valid for all six faces painted and $n\ge 2$. The four counts add up to $n^3$. Cubes with at least one painted face: $n^3-(n-2{)}^3=6n^2-12n+8$.

Here p = a − 2, q = b − 2 and r = c − 2. Valid for all six faces painted and a, b, c ≥ 2. If some faces are unpainted, count directly by layer, row and column.

x, y and z are the numbers of cuts in the three directions, with no pieces moved between cuts. For a fixed total number of cuts, the number of pieces is greatest when x, y and z are as nearly equal as possible. To cut a cube into $n^3$ equal cubes without moving pieces between cuts, the minimum number of plane cuts is $3(n-1)$.

All sizes counted. For example, a 3 × 3 grid has 9 + 4 + 1 = 14 squares, and an 8 × 8 chessboard has 204.

A k × k square fits in (m − k + 1)(n − k + 1) positions, for k = 1 to n. For example, a 4 × 3 grid has 12 + 6 + 2 = 20 squares, and the closed form gives $\frac{3\times 4\times 10}{6}=20$.

Choose 2 of the m + 1 horizontal lines and 2 of the n + 1 vertical lines. For example, a 4 × 3 grid has 10 × 6 = 60 rectangles, and an 8 × 8 chessboard has 36 × 36 = 1296.

Cubes of every size are counted. A k × k × k cube fits in $(n-k+1{)}^3$ positions.

The k lines run from the vertex to different interior points of the opposite side, and each parallel line crosses the whole triangle. Every triangle uses two of the k + 2 lines through the vertex and one of the h + 1 lines parallel to the base (the base included).

An equilateral triangle is divided into $n^2$ small equilateral triangles by lines parallel to its sides. The formula counts upright and inverted triangles of all sizes: n = 2 gives 5, n = 3 gives 13 and n = 4 gives 27.

V vertices, E edges, F faces. For a cube, 8 − 12 + 6 = 2. A prism on an n-sided base has V = 2n, E = 3n, F = n + 2. A pyramid on an n-sided base has V = n + 1, E = 2n, F = n + 1.

The element starts at position $p_0$ and moves s positions per step, counted in the stated direction. For an angle, the turn after t steps is $t\theta$ reduced modulo $360^{\circ }$.