Symmetry and Grid Heuristics

A line of symmetry folds the shape onto itself. For a regular polygon there is one through each corner and one through each side's midpoint, which for an even number of sides pair up. A rectangle folds only along the two lines joining midpoints of opposite sides.

1. Regular hexagon: a line through two opposite corners folds it onto itself; there are 3 such lines.
2. A line through the midpoints of two opposite sides does too; there are 3 more.
3. Total for the hexagon: 3 + 3 = 6 lines of symmetry.
4. Rectangle: the line through the midpoints of the two long sides, and the line through the midpoints of the two short sides: 2 lines.
5. A diagonal of a rectangle is not a line of symmetry: folding along it sends a short side onto a long one.

answer6; 2

Turning a shape about its centre, the order is the number of positions in one full turn where it looks exactly as it started, counting the finish. A regular polygon with n sides has order n; a rectangle matches itself twice.

1. A regular pentagon turned by 360^° ÷ 5 = 72^° looks the same; that happens 5 times in a full turn.
2. Order of a regular pentagon: 5.
3. A rectangle turned by 180^° looks the same; turned by 90^° it does not, since its long sides would stand where the short ones were.
4. Order of a rectangle: 2.

answer5; 2

The punch goes through two layers, so it makes two holes, and unfolding reflects one in the fold line. The two holes sit at the same height, one on each side of the fold, each 2 cm from it.

1. Two layers, one punch: 2 holes.
2. Unfolded, the two holes are mirror images in the fold line, each 2 cm from it and 3 cm from the top.
3. Distance between them = 2 + 2 = 4 cm.

answer2 holes; 4 cm apart

Four layers make four holes. Unfolding the second fold mirrors the hole across the horizontal fold line; unfolding the first mirrors both across the vertical one. All four sit 1 cm from the centre lines.

1. Two folds make 4 layers, so one punch makes 4 holes.
2. The two folded edges are the sheet's two centre lines. The hole is 1 cm from each of them.
3. Unfolding gives a hole in each quarter, 1 cm from both centre lines: the four are the corners of a square of side 2 cm about the centre of the sheet.
4. So: 4 holes, at the corners of a 2 cm square.

answer4 holes; a square of side 2 cm

Under the fold, corners B and D coincide, so one cut through the folded sheet removes both. Each corner cut off replaces one corner by one new edge.

1. Folding along AC puts B exactly on D; the cut goes through both layers.
2. Unfolded, the corners B and D are both cut off; A and C are untouched.
3. Each cut replaces a corner with a new short edge: the 4-sided square gains 2 sides.
4. The shape now has 4 + 2 = 6 sides.

answer2 corners; 6 sides

The left end travels from 0 cm to 10 cm along the strip, and the fold is halfway along that journey. A hole 1 cm from the fold is mirrored 1 cm on the other side of it.

1. The left end lands 12 − 2 = 10 cm from the left end's original place.
2. The fold is halfway: 10 ÷ 2 = 5 cm from the left end.
3. One hole is 1 cm from the fold in the double layer: at 5 − 1 = 4 cm and, its mirror image, at 5 + 1 = 6 cm.
4. So the fold is at 5 cm and the holes at 4 cm and 6 cm from the left end.

answer5 cm; 4 cm and 6 cm

Every horizontal step edge can be slid up to the top and every vertical one slid out to the right without changing their total lengths, which turns the outline into the rectangle's. The area is the rectangle less the cut-outs.

1. The horizontal step edges add to 4 + 4 + 4 = 12 cm, the length of the rectangle; the vertical ones add to 3 + 3 + 2 = 8 cm, its height.
2. So the perimeter equals the rectangle's: 2 × (12 + 8) = 40 cm.
3. Area: the rectangle is 96 cm². The cut-out above the first step is 4 cm by 5 cm = 20 cm²; above the second, 4 cm by 2 cm = 8 cm².
4. Area = 96 − 20 − 8 = 68 cm².

answer40 cm; 68 cm²

The shape is a rectangle with a roof on top; the roof splits into two right-angled triangles. For the dots inside, count row by row.

1. The rectangle from (0, 0) to (5, 3): 5 × 3 = 15 cm².
2. The roof: triangle (0, 3), (2, 5), (2, 3) has area 12 × 2 × 2 = 2 cm²; triangle (2, 3), (2, 5), (5, 3) has area 12 × 3 × 2 = 3 cm².
3. Area = 15 + 2 + 3 = 20 cm².
4. Dots strictly inside: rows y = 1 and y = 2 have 4 each (x = 1 to 4); row y = 3 has 4 (x = 1 to 4, the corners at x = 0 and 5 being on the boundary); row y = 4 has 2 (x = 1 and 2; x = 3 is on the roof edge).
5. Total inside = 4 + 4 + 4 + 2 = 14.

answer20 cm²; 14 dots

Extend the two legs of one trapezium and they meet at the centre of the ring. The angle there is what is left of the triangle's 180^° after the two 75^° base angles, and that wedge must go into 360^° a whole number of times.

1. The legs, extended, make a triangle with the longer side: its angle at the centre is 180^° − 75^° − 75^° = 30^°.
2. Round the centre the wedges add to 360^°: 360^° ÷ 30^° = 12 trapeziums.
3. The shorter parallel side is parallel to the longer one, so its angles are co-interior with the 75^° ones: 180^° − 75^° = 105^°.

answer12; 105^°

Covering with no gaps or overlaps means the tile areas add up to the rectangle's. One L-tile has eight unit edges on its outside.

1. Rectangle = 6 × 4 = 24 unit squares; each tile is 3: 24 ÷ 3 = 8 tiles.
2. Two L-tiles fit together to make a 2 × 3 rectangle, and four such rectangles tile the 6 × 4: so 8 tiles really do fit.
3. One L-tile: three squares have 12 edges; two pairs of squares share an edge, hiding 2 × 2 = 4 of them.
4. Perimeter = 12 − 4 = 8 units.

answer8 tiles; 8 units