Counting Figures & Mirror Images

At a glance 10 rules

1. 1Lines from one vertex$\text{triangles} = \dfrac{n(n+1)}{2}$$n = \text{parts of the base}$
2. 2Count by piecesLabel the corners, then count triangles made of 1 piece, 2 pieces, 3 and so on. A triangle needs three drawn sides and no three corners on one line.
3. 3Triangle cut into n rows$\text{upright} = \dfrac{n(n+1)(n+2)}{6}$$+\ \text{upside-down ones, by size}$
4. 4Rectangles in an m × n grid$\dbinom{m+1}{2} \times \dbinom{n+1}{2}$$\text{squares in } n \times n: 1^2 + 2^2 + \dots + n^2$
5. 5Squares on a grid of dotsEvery upright square of side s has s squares with corners on its sides: itself and s - 1 tilted ones. Count upright squares by size and multiply by s.
6. 6Mirror and water imagesSide mirror: left and right swap; A H I M O T U V W X Y look the same. Water: top and bottom swap, order kept; B C D E H I K O X look the same.
7. 7Half-turnA 180° turn is a mirror image and a water image together: the order reverses and each letter turns upside down. Only H I N O S X Z survive it.
8. 8Clock in a mirror$\text{real} + \text{mirror reading} = 12{:}00$$\text{so real} = 11{:}60 - \text{mirror reading}$
9. 9Lines of symmetrySquare 4; rectangle or rhombus (not a square) 2; isosceles trapezium 1; parallelogram 0; equilateral triangle 3; regular polygon with $ sides $.
10. 10Paper foldingUnfold in reverse order, mirroring every hole across each fold line: f folds give $2^f$ holes. A hole beside a crease stays beside it, never near a corner.