KA-0177Overcount, then correct
5 points, difficulty 3 of 3Level 11–12about 150sHow many diagonals does a convex polygon with 12 sides have? A diagonal joins two vertices that are not already joined by a side.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count from each vertex, then think about how many times each diagonal got counted.
2The strategy
From any one of the 12 vertices you can draw a diagonal to 9 others: not to itself, and not to its two neighbours.
3The full solution
That gives 12 x 9 = 108 endpoint-counts, but each diagonal has two endpoints, so divide by 2 to get 54.
Solution
The reliable way
From a single vertex, you can reach 11 other vertices, but two of them are its neighbours and those lines are sides, not diagonals. So each vertex is the endpoint of 12 - 3 = 9 diagonals. Multiplying gives 12 x 9 = 108, but this counts every diagonal twice, once from each end, so the true number is 108 / 2 = 54. Equivalently: all C(12,2) = 66 vertex pairs, minus the 12 that are sides, gives 66 - 12 = 54. The transferable idea: whenever you count from each object, ask how many times the thing you want has been counted.
The elegant way
All 66 vertex pairs minus the 12 that are sides leaves 54 diagonals.
Why this is on the test: Forgetting to halve is the standard error, and having a second method that agrees is the fastest way to catch it.