KA-0150Try small cases first
4 points, difficulty 2 of 3Level 3–4about 90sA diagonal joins two corners of a shape that are not next to each other. How many diagonals does a hexagon have?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Try a square first, then a pentagon.
2The strategy
Count the diagonals of a shape small enough to draw, then the next one up. The pattern in those counts is easier to trust than trying to see all the hexagon's at once.
3The full solution
A square has 2 diagonals, a pentagon 5, a hexagon 9. From each of 6 corners there are 3 diagonals, and each is counted twice: 6 x 3 / 2 = 9.
Solution
The reliable way
Start small. A square has 2 diagonals and a pentagon has 5 - both easy to draw and count. For the hexagon, each of the 6 corners joins to 3 others that are not its neighbours, giving 6 x 3 = 18 ends of diagonals. Every diagonal has two ends, so there are 18 divided by 2 = 9. Check against the small cases: the same reasoning gives 4 x 1 / 2 = 2 for a square and 5 x 2 / 2 = 5 for a pentagon, both correct. The transferable idea: build the rule on cases you can draw, then trust it on the one you cannot.
The elegant way
A shape with n corners has n(n - 3)/2 diagonals, giving 6 x 3 / 2 = 9.
Why this is on the test: Forgetting to halve is the same error as in the handshake count, and small cases are what let a student check the rule instead of hoping.