What is 1 + 2 + 3 + ... + 20?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Pair the first with the last.
2The strategy
Reorder the sum so the pieces make round numbers: 1 with 20, 2 with 19, and so on. Count how many pairs there are before you multiply.
3The full solution
Each pair adds to 21, and twenty numbers make ten pairs. So the total is 10 times 21 = 210.
Solution
The reliable way
Pair the outside in: 1 + 20 = 21, 2 + 19 = 21, 3 + 18 = 21, and so on. Twenty numbers give ten such pairs, each worth 21, so the total is 10 x 21 = 210. The transferable idea: reordering a long sum into equal pairs turns a page of arithmetic into one multiplication.
The elegant way
The sum of 1 to n is n times n plus one, all divided by two: 20 x 21 / 2 = 210.
Why this is on the test: Pairing to round numbers is the difference between a ten-second question and a two-minute one, and the halving is where it goes wrong.