When a SASMO problem feels like a locked room with no key, proof by contradiction can be the tool that breaks the door down. Instead of trying to prove something is true directly, you assume the opposite and show that it leads to an impossible situation. This technique appears often in the Singapore and Asian Schools Math Olympiad because it forces you to think backwards, which is a skill that separates strong competitors from the rest. Mastering this method will help you solve problems that look impossible at first glance, especially in number theory, geometry, and logic.

What Is Proof by Contradiction and Why Does SASMO Love It?

Proof by contradiction, sometimes called reductio ad absurdum, starts with a simple idea. You want to prove that a statement is true. So you assume it is false. Then you reason logically from that assumption. If you reach a contradiction, something that cannot be true under the given rules, then your original assumption must be false. That means the original statement is true.

SASMO problem setters use this technique because it tests your ability to think logically and to spot hidden constraints. Many olympiad problems are designed so that a direct approach gets messy. But when you flip the problem around, the solution becomes clean and elegant.

Think of it this way. You are a detective. You have a suspect. Instead of proving the suspect is guilty directly, you assume they are innocent. Then you find evidence that makes that assumption impossible. That contradiction proves guilt.

How to Apply Proof by Contradiction in Three Steps

Here is a simple process you can use for any SASMO problem that seems to call for this method.

1. State what you want to prove. Write it down clearly. For example, “The square root of 2 is irrational.”
2. Assume the opposite. In this case, assume that the square root of 2 is rational. That means it can be written as a fraction a/b in lowest terms.
3. Work logically until you hit a contradiction. Square both sides, multiply, and you will find that both a and b must be even, which contradicts the assumption that the fraction was in lowest terms.

That contradiction means your original statement is true.

Let us look at a specific SASMO-style example.

Example Problem: Prove that there is no integer n such that n squared plus 2 is divisible by 4.

Step 1: We want to prove that for all integers n, n^2 + 2 is not divisible by 4.

Step 2: Assume the opposite. Suppose there exists an integer n such that n^2 + 2 is divisible by 4. That means n^2 + 2 = 4k for some integer k.

Step 3: Consider the parity of n. If n is even, then n = 2m, so n^2 = 4m^2. Then n^2 + 2 = 4m^2 + 2. This is not divisible by 4 because 4m^2 is divisible by 4, but adding 2 gives a remainder of 2. If n is odd, then n = 2m + 1, so n^2 = 4m^2 + 4m + 1. Then n^2 + 2 = 4m^2 + 4m + 3. This leaves a remainder of 3 when divided by 4. In both cases, n^2 + 2 is not divisible by 4. That contradicts our assumption.

Therefore, no such integer n exists.

When Should You Reach for This Method?

Proof by contradiction is not the right tool for every problem. But it shines in certain situations.

• When a direct proof is too messy. Sometimes trying to prove something directly involves too many cases. The contradiction approach can simplify things.
• When the problem involves “no,” “not,” or “impossible.” These words often signal that a contradiction proof will work well.
• When you are dealing with prime numbers or divisibility. Many number theory problems yield to contradiction because you can assume a factorization exists and then find a conflict.
• When the problem asks you to prove something is unique. Assume there are two different solutions, then show that leads to a contradiction.

Common Mistakes Students Make

Even strong SASMO competitors slip up when using this method. Here are the most common errors and how to avoid them.

A Classic SASMO Number Theory Example

Let us walk through a problem that often appears in SASMO practice materials.

Problem: Prove that the sum of a rational number and an irrational number is irrational.

Proof by contradiction:

Assume that the sum of a rational number r and an irrational number i is rational. Let us call that sum s. So r + i = s, where r and s are both rational.

Since r is rational, we can write r = a/b. Since s is rational, we can write s = c/d. Then i = s – r = c/d – a/b = (bc – ad) / (bd).

This means i can be written as a fraction of two integers. That would make i rational. But we started with i being irrational. That is a contradiction.

Therefore, our assumption was false. The sum of a rational and an irrational number must be irrational.

This proof is clean and uses only basic algebra. It shows why contradiction is so powerful in number theory.

How to Practice for SASMO Success

You cannot master this technique by reading about it alone. You need to practice with real problems. Here are some suggestions.

• Start with simple statements. Prove that there is no largest prime number. Prove that the square root of 3 is irrational.
• Move to SASMO-style problems. Look for questions that ask you to show something is impossible or that a certain condition cannot happen.
• Time yourself. In the real SASMO exam, you have limited time. Practice writing proofs quickly but clearly.

If you want more structured practice, check out our grade-by-grade SASMO problem sets to find questions at your level. You can also try our weekly SASMO problem challenge for fresh material every week.

A Geometry Example to Solidify Your Understanding

Proof by contradiction is not just for number theory. It works beautifully in geometry too.

Problem: Prove that a triangle cannot have two right angles.

Proof by contradiction:

Assume that a triangle has two right angles. Let the angles be A, B, and C. Suppose A = 90 degrees and B = 90 degrees.

The sum of the angles in any triangle is 180 degrees. So A + B + C = 180. Substituting, we get 90 + 90 + C = 180, which means 180 + C = 180. That forces C = 0.

But a triangle cannot have an angle of 0 degrees. That would mean the triangle is not a triangle at all. This contradicts the definition of a triangle.

Therefore, a triangle cannot have two right angles.

This example is simple, but it shows the logic clearly. For more geometry techniques, read about 7 geometry theorems that appear in nearly every SASMO paper.

Building Your Logical Thinking for Competition Day

Proof by contradiction is one of several key problem-solving techniques you should master. It connects closely to other logical methods. For example, understanding how logical reasoning connects all SASMO math topics together will help you see when to use contradiction versus direct proof.

You should also practice reading problems carefully. Many students miss the contradiction opportunity because they do not notice the words “no,” “never,” or “impossible.” Our guide on reading SASMO questions carefully can help you train your eye.

The best mathematicians are not the ones who never make mistakes. They are the ones who learn to check their assumptions. Proof by contradiction forces you to examine your assumptions directly. That is a skill worth developing.

Putting It All Together for Your SASMO Preparation

Now you understand the core idea behind proof by contradiction. You know the three-step process. You have seen examples from number theory and geometry. You know the common mistakes to avoid.

The next step is to practice consistently. Set aside 15 minutes each day to work on one or two contradiction proofs. Over time, your brain will start to see opportunities for this method automatically.

If you find yourself stuck on a problem, ask this question: “What would happen if the opposite were true?” That question alone can unlock the solution.

For more practice, try our number theory practice problems every SASMO competitor should master and our combinatorics problems when you are completely stuck.

Your Next Move: Apply This Method Today

Proof by contradiction is not a magic trick. It is a logical tool that takes practice to use well. Start with small problems. Prove that zero is not a positive number. Prove that there is no integer that is both even and odd. Then move to harder SASMO problems.

Every time you successfully use this method, you build confidence. And confidence is what you need on competition day.

Remember, the goal is not just to get the right answer. It is to understand why the answer must be right. Proof by contradiction gives you that understanding. So grab a pencil, find a problem, and start assuming the opposite. The contradiction will lead you home.