Did you know that one carefully chosen whole number can overturn a claim that looks correct in a hundred examples? Mathematics is not a contest to collect as many agreeing examples as possible. It is a way to check precisely what a statement promises—and whether that promise can survive a case it did not expect.
The immediate answer: a counterexample is one permitted example for which a general statement is false. To disprove a claim that says every, all, or always, first check that your example meets every stated condition, then calculate or explain exactly how the promised result fails. A single valid counterexample disproves a universal claim; a hundred agreeing examples do not, by themselves, prove it.
- Circle the exact word that makes the claim universal: all, every, always or for any.
- Write the domain: positive whole numbers, integers, fractions, shapes or another defined set.
- Separate the condition from the conclusion, especially in an if–then statement.
- Find one permitted case that satisfies the condition but violates the conclusion.
- Show the arithmetic or diagram so another learner can check the contradiction.
- Keep examples that merely agree with the claim separate from a general proof.
- Use the result to improve the conjecture rather than calling all mathematical patterns unreliable.
Start with the question: what exactly is being claimed?
Suppose a worksheet says, ‘Every multiple of six is also a multiple of twelve.’ That is a universal statement about whole numbers. It does not mean ‘some multiples of six are multiples of twelve’ or ‘the first example I wrote happens to work.’ Its domain is whole numbers that are multiples of six, and its conclusion is that each one must also be divisible by twelve.
A learner might try 12, 24, 36 and 48. All four are multiples of both six and twelve. Those agreeing examples are real observations, but they do not settle an ‘every’ claim. Trying the next smaller multiple of six, 6, is more revealing: 6 = 6 × 1, but 6 cannot be written as 12 multiplied by a whole number. One number is enough.
This ability is useful far beyond number puzzles. Students use it when testing a supposed shortcut, challenging an overly broad statement in a science explanation, and examining a claim such as ‘this method always works.’ The point is not to be contrary. The point is to replace certainty based on a few memorable examples with a check that respects the wording.
Worked example 1: test a claim about divisibility
Illustrative question: ‘For every positive whole number n, if n² is divisible by 4, then n is divisible by 4.’ Is the statement always true? We are not describing real student results. The claim is a fictional exercise designed to reveal a precise logical error.
Step 1: identify the condition and the promised result
The permitted input is a positive whole number. The condition says its square must be divisible by 4. The conclusion says the original number must also be divisible by 4. We therefore need a positive whole number whose square is divisible by 4 but which is not divisible by 4.
Step 2: try a small number with a useful property
Choose n = 2. It is a positive whole number. Its square is n² = 2 × 2 = 4. Since 4 ÷ 4 = 1, the square is divisible by 4. But 2 ÷ 4 is not a whole number, so the original number is not divisible by 4.
Step 3: write a checkable disproof
Answer: ‘The statement is false. Take n = 2. Then n² = 4, which is divisible by 4, but n = 2 is not divisible by 4. This is a valid counterexample.’ Notice that every clause has a job. The example belongs to the stated domain; it satisfies the if-part; it breaks the then-part.
Step 4: repair the claim without overclaiming
A mathematically sound repair for whole numbers is: ‘If n² is divisible by 4, then n is even.’ To explain why, an odd number has the form 2k + 1, and its square is 4k(k + 1) + 1, leaving remainder 1 when divided by 4. Thus a square divisible by 4 cannot come from an odd whole number. This last argument is a proof of a different, corrected claim, not proof of the original one.
Why trial examples are valuable but limited
Trial examples are not worthless. They help us understand the pattern, locate awkward boundaries and notice when a rule may need refinement. The danger comes when a student silently turns ‘it worked for the values I tried’ into ‘it must work for all values.’ A conjecture is a statement proposed for investigation. A proof explains why no permitted counterexample exists. The two are not interchangeable.
For some claims, a number table is an efficient diagnostic. For others, a diagram or algebraic argument works better. The method must suit the domain. If a claim concerns all triangles, a numerical example about rectangles cannot test it. If it concerns strictly positive integers, using zero as the alleged counterexample is not admissible. If a claim includes a condition such as ‘nonzero’, do not choose zero merely because it makes a denominator fail.
Worked example 2: an attractive pattern that breaks
A club member notices that 2 + 4 = 6, 4 + 6 = 10 and 6 + 8 = 14. All three sums are even. They then claim, ‘The sum of any two whole numbers is even.’ This statement is much wider than the tested examples. It covers odd–even pairs as well as even–even pairs.
Try 2 + 3 = 5. Both inputs are whole numbers, yet their sum is odd. This is a counterexample. A corrected and provable statement is: ‘The sum of two even whole numbers is even.’ If the numbers are 2a and 2b, their sum is 2(a + b), which is a multiple of two. Here the general algebraic form explains why the repair works for every permitted pair.
The change from ‘any two whole numbers’ to ‘two even whole numbers’ may look tiny, but it is the entire mathematical lesson: words define the cases you must cover. Precise reading is part of reasoning.
A diagnostic for learners before more practice
- Can you underline the exact condition in the question without changing it?
- Can you tell whether a trial number is allowed before doing any calculation?
- Can you produce a counterexample and show the two required checks?
- Can you explain why five agreeing examples do not settle an all-cases claim?
- Can you distinguish disproving a statement from proving a revised statement?
If the learner cannot answer the second question, return to the domain before increasing difficulty. If the learner can find a failure but cannot explain it, practise writing the short ‘belongs / satisfies / fails’ argument. If the learner only lists successful examples, ask them to look deliberately at boundary values, odd and even cases, zero when allowed, negative numbers when allowed, and exceptions suggested by the wording.
A progression from guided to independent thinking
Guided practice: choose between two candidates
Claim: ‘Every whole number divisible by 10 is divisible by 5.’ Compare 20 and 30. Both meet the condition and the conclusion. They do not disprove the statement. Ask the learner what would have to happen for a counterexample to exist: a multiple of 10 that is not a multiple of 5. Since 10k = 5(2k) for every whole number k, such a counterexample does not exist. This claim is actually true.
Supported practice: find an exception
Claim: ‘Every even positive whole number is a multiple of four.’ Write the first five even positive whole numbers: 2, 4, 6, 8, 10. Which value defeats the claim? The value 2 does: it is even but not divisible by four. Ask the learner to write the full explanation in one or two sentences rather than circling an answer.
Independent practice: transfer the skill
Claim A: ‘Every number ending in zero is divisible by five.’ Claim B: ‘Every number that has a factor of three is odd.’ Claim C: ‘Whenever a rectangle’s length increases, its area must increase.’ Ask the learner to classify each claim, prove or disprove it under clearly specified assumptions, and show a counterexample where appropriate.
Answers and important conditions: A is true for whole numbers written in ordinary base-ten notation, because a number ending in zero is divisible by 10 and therefore by 5. B is false: 6 is divisible by 3 and is even. C is not necessarily true if the width may change; a rectangle changing from 4 cm × 3 cm (12 cm²) to 5 cm × 2 cm (10 cm²) has greater length but smaller area. If width is explicitly held fixed and positive, the conclusion changes. That final example shows why checking assumptions matters as much as checking arithmetic.
Common mistakes and how to repair them
- Using an ineligible value: a negative number cannot refute a claim limited to positive whole numbers. Read the domain aloud.
- Testing the wrong half of an if–then: choose a case where the premise is true but the conclusion false.
- Calling an example a proof: replace ‘it worked three times’ with a general argument or a carefully bounded conclusion.
- Assuming a failure means every case fails: a counterexample disproves ‘all’, not ‘some’.
- Ignoring units or shape conditions: if the statement fixes width, angle, or scale, preserve it in the test.
- Silently altering the claim: label your improved statement as a new claim, then check that one independently.
What parents and teachers can do
A parent does not need to announce the correct counterexample. Start by asking, ‘What does the word every make us responsible for checking?’ Then invite the child to find a case that is allowed but awkward. In a short family activity, test an everyday statement such as ‘all square objects are bigger than rectangular objects.’ Agree first on what ‘bigger’ means: area, width, height or perimeter. Ambiguity itself may be the first problem to resolve.
For teachers, use two visible columns: examples that fit and examples that disprove. Only place a candidate in the second column after the student has explained its eligibility and contradiction. Ask different learners to defend opposite classifications. Later transfer the procedure to algebra, geometry and scientific claims while keeping the scope of each subject clear.
Progress is observable when a learner can independently label the domain, pick a useful test case, justify why it is valid, state what fails, and avoid calling a few successes a proof. A score improvement is possible but not guaranteed, and the quality of explanation is more informative than one lucky answer.
Questions students often ask
Does one counterexample prove the opposite claim?
Not generally. One case disproves an all-cases claim, but it does not prove that every case has the opposite property. ‘Not all are even’ does not mean ‘all are odd.’
Can a drawing be a counterexample?
Yes, if it is a permitted object and its features can be verified. A correctly drawn non-square rectangle can refute a claim that all rectangles are squares, provided the definitions and conditions are respected.
What if I cannot find any counterexample?
You have not yet proved the statement. Consider whether a general argument is available, and ask which classes of permitted cases remain untested. When appropriate, seek help with a proof rather than extending a table forever.
How should I write this in an assessment?
Use a compact structure: ‘Take [permitted example]. It satisfies [condition], because [check]. But it does not satisfy [conclusion], because [check]. Therefore the universal statement is false.’ The wording can be shorter, but the logic should remain visible.
Further reading and a useful next step
The University of Waterloo’s treatment of counterexamples explains the formal logic behind this classroom habit. Continue with our guide to testing a number-pattern rule, the Core Aim learning hub and the broader Education Hub. Our Secondary 1 Mathematics learning reference illustrates the same explanation-first and independent-checking approach.
The core aim is not to teach pupils to say ‘wrong’ more quickly. It is to help them ask a better question: Which cases did the statement promise to cover, and what evidence would genuinely decide it? Once a learner can do that, examples become tools for reasoning instead of decorations around a guess.