We already covered Pólya’s four steps for tackling any problem. Now for the juicy part: how do you decide, in seconds, WHICH strategy to try when the problem doesn’t look like anything you’ve seen before?
The Arsenal (and When to Pull Out Each Weapon)
- Look for a simpler case. Does the problem have 20 elements and it’s making your head spin? Try it with 2 or 3 first. The pattern you find there almost always generalizes.
- Work backward. Instead of starting from the given data, start from the result you need and ask: what would have to be true right before getting there?
- Look for the pattern. Try several concrete cases by hand. If you see a regularity, that regularity is your clue.
- Break the problem into pieces. Solve each part separately, then combine them.
- Think of a similar problem you’ve already solved. A problem is rarely 100% new — it almost always shares structure with something you’ve seen before.
- Test the edge cases. What happens at the smallest extreme? At the largest? That’s usually where the constraints the statement didn’t spell out directly are hiding.
How Do You Train This?
Not by randomly trying all six on every problem — that eats up all your exam time. You train it by practicing them outside of competition, without the clock, until recognizing “this calls for working backward” becomes almost automatic.
“No good competitor tries all six heuristics in order. Within seconds, they know which one to try first. That’s not talent, that’s mileage.”
What If None of Them Work?
It happens. And when it does, the real problem is usually something else: you didn’t fully understand the statement. That’s what we cover in how to read a hard problem without getting stuck.