A formula works on the first several inputs, so a learner writes “it always works.” Replace that sentence with a quantified claim: for every input in a specified set, the algorithm terminates and returns an object with a specified property. Now the claim can be tested, disproved or proved.
Finite examples are valuable for discovering patterns and errors. Their logical force depends on whether they cover the entire claimed domain or only a sample.
Predict
The polynomial n²+n+41 produces primes for many early nonnegative integers. Predict what happens at n=40 by factoring before calculating a large number. Which universal claim does that one value refute?
Experiment
At n=40, the value is 40²+40+41=1681=41². At n=41, it is 41·43. Both are composite.
Tested input
Value
Logical consequence
0
41
One successful case
1
43
Another successful case
40
41²
Refutes primality for every nonnegative integer
The counterexample does not refute a narrower statement restricted to a range that excludes it. Conversely, changing the range after seeing the failure should be recorded as a revision, not presented as the original conjecture.
Understand
Counterexample design should target assumptions and boundaries. For a division formula, test zero and nearly zero denominators. For a graph theorem, test disconnected graphs, small orders and repeated edges if allowed. For a statistical assertion, test dependence or a boundary parameter. A case outside the stated domain is not a counterexample, though it may expose why the domain matters.
A failed identity can suggest a stronger corrected result. The claim mean(x²)=mean(x)² is false for values 0 and 2: the two sides are 2 and 1. Let μ be the mean of a nonempty finite dataset. Expanding mean((x−μ)²) gives mean(x²)−2μ mean(x)+μ²=mean(x²)−μ². Thus the difference is nonnegative and vanishes exactly when every value equals μ. The correction includes an inequality and an equality condition, both proved from the same calculation.
Numerical counterexamples require care. If a claim concerns exact equality and the discrepancy is comparable to roundoff, the computation may not refute it. Use exact arithmetic, a certified interval or an analytic example. If a claim concerns a tolerance, report the error bound on the test itself.
Preserve conjecture history: original statement, tested domain, evidence, counterexample, repair and remaining open questions. A polished final statement without that record can hide how strongly the investigation was guided by the same examples later used as “validation.”
Apply
State one conjecture with explicit quantifiers. Design three adversarial cases, including a boundary case, and check them exactly where feasible. If the conjecture fails, prove a corrected special case or explain the obstruction. If it survives, write “not refuted by these tests” unless you have a general argument.
Continue
Computation can contribute to a proof when its finite domain, coverage and verifier are justified. The next lesson distinguishes that situation from persuasive but logically incomplete numerical evidence.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For p(n)=n²+n+41, find p(41).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For values −2 and 2, find average(x²) minus square(average(x)).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Rewrite “the method always works” as a precise mathematical claim.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Design adversarial cases for a claim about division-based parameter estimates.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Repair the false claim that mean(x²)=mean(x)² for every finite real dataset.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Record one conjecture, its strongest attempted counterexample and a revision.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
