Checking several successful examples does not prove a rule about every possible case. It shows that the rule worked on the cases checked. A universal claim needs an argument that covers its whole domain, or a genuinely exhaustive check when the domain is finite. One valid counterexample can disprove it. One successful example can establish an existence claim. The task is to recognise which kind of statement you are trying to justify.
This guide repairs the moment a learner writes “therefore it always works” after a short list of calculations. It connects mathematical proof with careful testing, scientific claims and precise writing. The classroom scenes and practice situations are illustrative, not reports of actual students.
Four answers that make a bad rule look promising
Consider the expression n² + n + 11. For n equal to zero, one, two and three, its values are 11, 13, 17 and 23. Each is prime. An imaginary student, Kai Kai, writes, “This expression always gives a prime number.”
The calculations are correct. The conclusion is not. At n = 10, the expression becomes 100 + 10 + 11 = 121. Since 121 = 11 × 11, it is composite.
The four successful cases did not become incorrect when the fifth selected test failed. What failed was the leap from those cases to all non-negative integers. The counterexample belongs to the claimed domain, and its output contradicts the claimed property.
This is an important emotional as well as mathematical distinction. Finding the counterexample is progress. The learner has discovered a limit that the early pattern concealed. A good lesson rewards that discovery instead of treating it as the destruction of an attractive answer.
Write the claim before choosing the evidence
“The expression gives primes” is incomplete. Does it mean for some inputs, for the four inputs checked, for every positive integer, or for every real number? Those are different statements.
A mathematical claim needs its objects, conditions and conclusion. In the example, the objects are non-negative integers n. The proposed conclusion is that n² + n + 11 is prime for every one of them.
Words such as “all”, “some”, “always”, “never” and “there exists” determine what would count as adequate support. They are not decorative language added after the calculation.
MIT’s introduction to mathematical proofs begins with propositions, axioms and logical deduction. The useful school-level lesson is that an argument must establish the proposition actually stated, not a nearby statement that is easier to demonstrate.
Why one failure can outweigh many successes
Suppose a rule says that every member of a set has a property. To make the rule false, it is enough to find one member without that property. That member directly contradicts “every”.
To make the rule true, showing that one member has the property is insufficient while other members remain uncovered. A hundred examples face the same logical problem if the domain contains more than those hundred cases.
This asymmetry is built into the claim. It is not unfair treatment of evidence. “Every locker is empty” is contradicted by one occupied locker. Finding ten empty lockers does not settle the remaining lockers.
Notice the boundary: when the claim is “at least one locker is empty”, one verified empty locker is enough. The same observation can be sufficient for one statement and insufficient for another. Before asking for more examples, ask what the examples are supposed to establish.
A counterexample must satisfy the conditions
Take the claim, “Multiplying a positive number by a number greater than one makes it larger.” Someone proposes 8 × 0.5 = 4 as a counterexample.
The calculation is correct, but it is not a counterexample to that statement. The multiplier 0.5 is not greater than one. It falls outside the stated conditions.
It is, however, a counterexample to the broader claim, “Multiplying a positive number by any positive number makes it larger.” This revised claim permits 0.5, and the result is smaller than eight.
A usable counterexample therefore has two jobs: enter through the conditions and break the conclusion. Checking only the second job produces impressive-looking objections that do not address the actual claim.
The repair is to write both parts explicitly: “This input is allowed because…; the conclusion fails because….” That small habit prevents many confused disagreements in Mathematics and analytical writing.
Search the boundaries, not only the comfortable middle
A learner who tests only small positive whole numbers may miss failures involving zero, negatives or fractions. Those cases should be considered when the stated domain allows them, not added mechanically to every problem.
For example, “If a² = b², then a = b” works for a = b = 3. But a = 3 and b = −3 gives equal squares and unequal numbers. The broader statement fails.
Changing the conditions to non-negative a and b removes that particular counterexample. A proof is still needed for the corrected statement; excluding one failure is not automatically a proof that no other failure remains.
Useful test choices follow the structure. For a denominator, inspect whether it can be zero. For a statement involving size, inspect equality and sign. For a shape classification, inspect a case that satisfies the definition without resembling the usual classroom picture.
This is deliberate testing rather than guessing until something breaks.
Turn a pattern into a reason
Now consider a true claim: the product of two consecutive integers is even. Several examples support it: 3 × 4 = 12, 8 × 9 = 72 and 12 × 13 = 156. But the proof is not hidden inside the length of that list.
Let the integers be n and n + 1. If n is even, the product contains an even factor. If n is odd, n + 1 is even, so the product again contains an even factor. In either case the product is divisible by two.
Those two cases cover every integer n. The argument therefore covers the entire stated domain, including zero and negative integers. It does not depend on which particular integer the reader chooses.
The examples helped us notice the claim. The proof explains why an untested allowed case cannot escape the reason. Ask the learner to point to that coverage step. It is the part that turns a persuasive pattern into a general argument.
Letters are useful because they keep the case arbitrary
Suppose a student notices that the difference between the squares of consecutive integers is odd. Instead of listing more calculations, represent the integers as n and n + 1.
The difference is (n + 1)² − n². Expanding gives n² + 2n + 1 − n² = 2n + 1. Since n is an integer, 2n is even, and 2n + 1 is odd.
This calculation establishes the claim for every integer n. For negative n, the difference may be negative, but it is still an odd integer. If the original question concerns positive consecutive integers, state that narrower domain and keep the argument within it.
The letter does not make the answer rigorous by itself. Its value is that it represents an arbitrary allowed integer throughout a valid chain of operations. A page of symbols with an unjustified division remains defective, however formal it looks.
An example can illustrate a proof without replacing it
Imagine explaining the consecutive-integer product with six and seven. Pairing the six objects in each of seven groups shows why that product is even. To establish the general rule, the explanation must make clear why one of any two consecutive integers is even.
A diagram can carry a general argument when its structure represents every allowed case. It does not become a proof merely because it is visual, nor does it fail merely because it is not written in algebra.
Ask which feature of the picture does the reasoning. If it depends on having exactly six objects, it may establish only that case. If it represents arbitrary pairs and a necessary parity relationship, it may express the general mechanism.
This helps younger learners move towards proof without demanding advanced notation before they understand the idea. The destination is complete reasoning, not symbols for their own sake.
When checking every case really can be enough
The title’s warning concerns selected examples, not a complete finite examination. Suppose the claim is, “Every number on this four-card list is even,” and the list is 2, 6, 10 and 14. Checking all four cards establishes the claim about that list.
It does not establish that every card in an unseen box is even. The proof’s reach is set by the domain that was exhaustively covered.
Similarly, a truth table can establish a proposition about all assignments of a finite collection of Boolean variables when every assignment is included and evaluated correctly. The fact that a method uses checking does not make it automatically inferior to algebra.
The central issue is coverage. Were all allowed cases examined, or was the learner hoping that unexamined cases would resemble the convenient ones? A complete finite check and an incomplete sample are logically different activities.
A finite set of cases can cover an infinite domain
Every integer is either even or odd. That is why the earlier two-case proof covers infinitely many integers. The cases are categories containing all possible inputs, not two individual examples.
Consider the claim that the square of an integer leaves remainder zero or one when divided by three. Every integer can be written as 3k, 3k + 1 or 3k − 1 for some integer k.
Squaring gives 9k², 9k² + 6k + 1, or 9k² − 6k + 1. The first is divisible by three. Each of the other two is a multiple of three plus one. Therefore no integer square leaves remainder two.
Checking only the numbers zero, one and two would not be the same proof. The expressions containing k establish what happens throughout each category. The coverage statement tells us that there is no fourth category waiting outside the argument.
Computer testing needs the same coverage question
A program might check a proposed numerical rule for the first million inputs. That can be valuable: it may expose failures and help investigate a conjecture. It does not, just by reaching a large number, cover all integers.
When the intended domain is finite and every case is checked correctly, exhaustive computation can be part of a proof. Sometimes a separate mathematical argument reduces an infinite problem to a finite set of cases.
There are still implementation questions. Did the program generate the intended inputs? Did it evaluate the intended property? Could numerical approximation alter the check? These questions concern the reliability of the computational argument, not whether computers are inherently disqualified.
The appropriate conclusion should name what was established: “No counterexample appeared among these tested inputs” is a useful report. It is not interchangeable with “No counterexample exists”.
Do not quietly move the goalposts
A counterexample often suggests a better rule. That is legitimate mathematical work. The problem arises when the original statement is silently replaced and the learner claims it was correct all along.
Record the change. “The claim was false for all positive multipliers because multiplying by one-half made the number smaller. The revised claim requires a multiplier greater than one.” Now the reader can see why the condition matters.
Then justify the revision. For x > 0 and k > 1, the difference kx − x is x(k − 1), which is positive. Therefore kx > x. The proof explains the corrected boundary rather than merely protecting the rule from one awkward example.
A useful correction leaves a stronger argument behind. It does not erase the evidence that forced the correction.
The converse needs its own argument
Every square is a rectangle under the usual inclusive definitions: it has four right angles, with the extra condition of four equal sides. It does not follow that every rectangle is a square.
A rectangle with side lengths three and five is enough to disprove the converse. It satisfies the rectangle condition but not the equal-side condition required for a square.
In other problems, a statement and its converse may both be true. The point is that reversing the arrow requires justification. A list of examples where both happen together may hide the difference.
Have the learner rewrite “if A, then B” and “if B, then A” separately. Mark the allowed starting condition in each. This makes a common reasoning error visible without requiring a long lesson in formal logic.
Scientific observations answer a different kind of question
A school experiment may find that a material behaved in a particular way under the tested conditions. Repetition, measurement quality and comparison can strengthen that evidence. The resulting scientific claim is not the same kind of object as a deductive theorem about integers.
Do not use “examples are not proof” to suggest that empirical evidence is worthless. Science investigates the world through evidence and models. Mathematics can establish consequences of definitions and assumptions through deduction. The forms of justification differ because their jobs differ.
Also avoid treating one unexpected experimental reading as an unquestionable refutation. It may challenge the explanation, but the observation, apparatus, conditions and assumptions need checking. A mathematical counterexample must itself be valid; an empirical anomaly must also be investigated rather than merely celebrated.
The guide to repeating investigations explains what repeated trials can and cannot establish. Keep that scientific job separate from the proof task.
The same mistake appears in written arguments
A writer observes two successful group projects and concludes that group work always produces better learning. The examples may be relevant, but they do not establish the universal statement.
The repair is not necessarily a mathematical proof. It may be a narrower claim, broader evidence, attention to conditions and a fair discussion of cases that differ. An example can illustrate a mechanism without measuring how often that mechanism succeeds.
In a comprehension answer, the requirement may be to support an interpretation of this passage, not prove a rule about every person. Read the task before importing the strictest possible standard from another discipline.
The common habit is to match the evidence to the scope of the sentence. The existing Logic guide provides the wider reasoning context; this article concentrates on the specific gap between selected cases and a general conclusion.
A repair conversation that does not give away the proof
Ask the learner, “What exactly have you shown so far?” A good answer might be, “It works for these four inputs.” Then ask, “Which inputs are allowed but not yet covered?”
If the claim survives sensible testing, ask what the successful cases have in common. Is there an even factor, an invariant difference, a definition or an algebraic identity doing the work?
Offer a representation only when needed. Replacing a particular integer with n may help; so may grouping objects into pairs. Avoid giving a complete argument and then mistaking the student’s agreement for independent reasoning.
Finally, use a fresh claim. The student should decide whether to seek a witness, a counterexample, exhaustive cases or a general proof before beginning. That decision is part of the capability being taught.
A short practice set with reasons
Claim: every odd positive integer is prime. Nine is an allowed input, is odd, and equals three times three. It is a valid counterexample. One is also not prime, but nine makes the composite structure especially visible.
Claim: there is an odd perfect square. Nine is enough to establish this existence claim because it is odd and equals three squared. Notice how the same number performs a different logical job.
Claim: the sum of two even integers is even. Write them as 2a and 2b, where a and b are integers. Their sum is 2(a + b), which is divisible by two. This is a general proof, not two chosen additions.
Claim: every item on a stated three-item list has a property. Checking all three can suffice for that list. It does not automatically justify a statement about items outside it.
Four matching terms can belong to different rules
Suppose a sequence begins 1, 4, 9, 16. Squaring the positive integer position gives these four values, so n² is a natural candidate rule. That observation alone does not logically force every possible continuation to follow n².
Consider another rule: n² + (n − 1)(n − 2)(n − 3)(n − 4). At each of the positions one through four, one factor in the added product is zero. The added term therefore vanishes, and this rule produces exactly the same four starting values.
At position five, the square rule gives twenty-five. The alternative gives 25 + 4 × 3 × 2 × 1 = 49. Both rules fit the displayed beginning; they disagree about the next term.
This does not mean a school question asking for the next square number is defective. The stated pattern, diagram, description or classroom context may specify the intended relationship. Use those constraints. The logical warning applies when a learner treats a short unqualified list as though it uniquely determines every future term.
Ask what additional information makes the proposed rule appropriate. If the task defines the nth term as n², the next value follows from that definition. If it supplies a geometrical construction, inspect why that construction produces a square. If it supplies only four numbers, distinguish a plausible continuation from a uniquely forced conclusion.
The example also explains why testing extra comfortable cases is not the same as finding a general reason. A rule can agree with all the cases examined so far while differing elsewhere. The missing work is coverage, not merely a longer list.
A false proof needs its broken step identified
Selected examples can also make an invalid argument appear harmless. Suppose a purported proof begins with a = b = 1. It writes a² = ab, subtracts b² from both sides and obtains a² − b² = ab − b².
Factorising gives (a − b)(a + b) = b(a − b). Up to this point, the equalities are valid. The next step cancels a − b and claims a + b = b, leading to the impossible conclusion 2 = 1.
The fault is the cancellation. Since a = b, the factor a − b equals zero. Cancelling it would require division by zero. The earlier equality simply states that zero equals zero; it does not imply that the remaining factors are equal.
Trying several other equal values of a and b does not legitimise the cancellation. The same forbidden step remains. Nor is “the answer looks wrong” a sufficient explanation of where the proof failed. Point to the operation and state the condition it violates.
Together, these cases give a useful two-part check. Ask whether the argument covers the allowed inputs, then ask whether every step is valid on those inputs. A chain can fail through incomplete coverage, an invalid operation, or both. More examples repair neither problem unless they lead the learner to the missing reason.
Know when the argument has finished
Once a valid counterexample disproves a universal claim, more counterexamples are not logically necessary, although they may help diagnose the failure. Once a valid general proof covers the domain, endless extra examples are not needed to make it more true.
Conversely, failing to find a counterexample does not establish truth, and failing to find a proof does not establish falsehood. “Not resolved by this work” is a legitimate state.
Use the Mathematics Article Directory for the underlying algebra, number properties and reasoning skills. Return to the precise claim whenever a calculation begins to drift away from what was asked.
Examples are valuable. They reveal patterns, expose failures and make reasons visible. The mistake is not using them. It is asking selected examples to carry the word “every” without an argument that reaches the cases still unseen.