Sample · 01-Problem Solving Foundations
Chapter 1 Overview
Book 1 — Olympiad Foundations & Proof · CHAPTER 01: Problem Solving Foundations
"In mathematics the art of asking questions is more valuable than solving problems." — Georg Cantor, doctoral thesis, Berlin, 1867
On \( 23 \) September \( 1999 \), NASA lost the Mars Climate Orbiter. The spacecraft was fine. The rocket was fine. The mathematics was fine. One team had computed thrust in pound-force; the other had read those numbers as newtons. A \( 125 \) million dollar spacecraft burned up in the Martian atmosphere because nobody checked what the numbers meant. Every equation in that mission was solved correctly, and the mission still failed.
That is the entire subject of this chapter. Olympiad problems are not lost because students cannot compute — they are lost because students compute the wrong thing, or start from the wrong end, or drown in a number that was a disguise all along. The nine strategies collected here are what separate the competitor who stares at a problem for forty minutes from the one who, in ninety seconds, says: ah, I know what kind of animal this is.
These strategies are not tricks. They are how mathematics is actually done — by olympiad students, by cryptographers protecting your bank details, by climate scientists who shrink a planet into a toy model, by the engineers who landed a rocket booster on a barge by computing the descent backwards from the landing pad. George Polya, whose \( 1945 \) book How to Solve It became one of the best-selling mathematics books ever written, put the founding idea in a single sentence: "If you cannot solve a problem, then there is an easier problem you can solve: find it." This chapter is nine different ways to find it.
A warning and a promise. The warning: none of these strategies is a formula, and anyone who tells you that olympiad problems have a lookup table is selling something. The promise: they compound. A student who has genuinely absorbed all nine does not become nine times better at problems — they become someone who is never completely stuck, because when one door closes there are eight more in the room.
📋 How this chapter works
Every strategy below is presented in the same rhythm, so you always know where you are. Look for these headings.
| Section | What it gives you |
|---|---|
| 📋 Strategy Corner | the reference box: the move, its variants, the standard consequences |
| Decision flowchart | the procedure as a diagram, with a self-check you must pass |
| 🌍 Math in the Real World | where this exact idea earns its living outside competitions |
| 🔍 Olympiad Detective | a one-line case: spot the structure before you read on |
| ⚠️ Common Trap | the specific way strong students lose this mark |
| Worked Example | a full solution, often with a second and more elegant route |
| 💡 Did You Know? / 🤔 What If? | the surprising fact, and the question that generalises it |
| 🚀 Challenge Yourself | a harder problem with a hint but no solution. That is deliberate. |
Continue reading — 1 chapters, worked examples and practice.
Sign in to buy