KA-0121Try small cases first
4 points, difficulty 2 of 3Level 5–6about 90sEveryone at a meeting shakes hands with everyone else exactly once. There are 66 handshakes in total. How many people are there?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Try 3 people, then 4, then 5.
2The strategy
Work out the handshakes for a few small groups and write the results in a list. The pattern that appears lets you climb to 66 without any algebra.
3The full solution
3 people give 3 handshakes, 4 give 6, 5 give 10. The counts are 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 - and 66 is reached at 12 people.
Solution
The reliable way
Try small cases and list the results. With 3 people there are 3 handshakes, with 4 there are 6, with 5 there are 10. Continuing: 15, 21, 28, 36, 45, 55, 66. Counting along, 66 handshakes happens with 12 people. Check: each of 12 shakes 11 hands, giving 132 ends, and each handshake has two ends: 66. The transferable idea: small cases turn a question you cannot start into a list you can read the answer off.
The elegant way
n people give n(n - 1)/2 handshakes, so n(n - 1) = 132 = 12 x 11, giving n = 12.
Why this is on the test: Starting small is the single most reliable way into a question that looks like it needs algebra a student has not met yet.