Problem list（11 problems）
- How many triangles are there in the figure? To count shapes, do not chase them with your eyes — find out what has to be chosen to pin one down
- How many squares are there altogether? Do not count in the order you spot things — count in the order they are determined
- How many diagonals can be drawn in total? Count from one corner, multiply by the number of corners, then halve
- How many lines of symmetry does it have? Many people count only the corner-to-corner lines and stop at half the answer
- Into how many pieces do 4 lines cut the paper? Do not try to count the regions
- How many matches are needed? "The first one is special; after that the growth is steady
- How many squares does the diagonal pass through? Counting crossings is the quick way
- In how many ways can the three chosen rods form a triangle? Whether three lengths make a triangle rests on one test: the longest must be less than the other two added
- How many cubes have 2 painted faces? For N×N×N: 3 faces is always 8, 2 faces is 12×(N−2), 1 face is 6×(N−2)², 0 faces is (N−2)³
- What do the three "?" faces add up to? Hunting for which faces are opposite means fresh work for every net
- Through what angle must it turn to match itself again? The turn that brings a shape back onto itself is "a full turn ÷ the number of corners"
Other ideas
- Shearing — moving a shape without changing its area
- Ratios — answers without fixing a single number
- Add, then subtract — an overlap is a counting problem
- Circles and sectors — multiply by π once, at the very end
- Chasing angles — there are only four tools
- Flatten the solid before you think about it
- Write the same thing two different ways
- Fold it, turn it, copy it across
- Back to why the formula works