The lesson
How do you break a hard problem into familiar parts?
AoPS puts its own method in one line: it "trains students to approach hard problems by breaking them down into familiar parts." Here is how we teach that idea, with three worked examples written by us.
- Name what is unfamiliar. What makes this problem feel new? Usually it is size (huge numbers), shape (an odd figure) or wording.
- Find the familiar core. Strip it down: a smaller case, a simpler shape, a known formula hiding inside.
- Solve the small version. Get a sure answer for the easy case and look at how you got it.
- Rebuild. Apply the same steps to the real problem.
- Check with a second view. A different method, or a case you can count by hand.
Example 1 (ours): a big sum. What is 1 + 2 + 3 + ... + 50? Unfamiliar: fifty terms. Familiar core: 1 + 2 + 3 + 4, which is 10, and also (1 + 4) + (2 + 3), two pairs of 5. So pair the ends: 1 + 50, 2 + 49, and so on, 25 pairs of 51, total 1,275. Check with a small case: 1 to 4 gives 2 pairs of 5, which is 10. Correct.
Example 2 (ours): an odd shape. How many diagonals does a hexagon have? Unfamiliar: drawing them all gets messy. Familiar core: each corner connects to every other corner except itself and its two neighbors, so each of the 6 corners sends out 3 diagonals. That counts 18 ends, but each diagonal has two ends, so there are 9. Check by drawing a square: each corner sends 1, 4 ends, 2 diagonals. Correct.
Example 3 (ours): wording that hides algebra. The sum of three consecutive whole numbers is 87. What is the largest? Unfamiliar: three unknowns. Familiar core: the middle number is the average, so it is 87 / 3 = 29, and the numbers are 28, 29 and 30. The largest is 30. Check: 28 + 29 + 30 = 87. The familiar part was "average of evenly spaced numbers is the middle one".
Why this transfers
The three examples share no topic: a sum, a shape and a sentence. They share a move, and that move is what lets a student face a problem they have never seen on a contest.
This is why problem-solving programs of every kind, AoPS included, focus on method more than on any list of topics.
How we practice it
Before solving, students must say what the familiar core is. Only then may they calculate.
At the end, they write the core in one sentence. Over a term those sentences become their own toolkit.
