Write a simple enumerator that counts every valid object for small inputs, then compare its results with your formula or optimized algorithm on exactly the same cases. This can reveal errors in definitions, edge handling, and implementation—but agreement on a finite test range is not a proof for all input sizes.
What brute-force testing checks
A brute-force test directly generates the candidate objects for a small instance, keeps the ones that satisfy the problem’s rules, and counts them. You compare that count with the result from the solution you want to verify.
The comparison is meaningful only if the enumerator represents the original problem correctly and is sufficiently independent of the solution under test. If both implementations reuse the same mistaken recurrence or interpretation, they can agree and still be wrong.
Settle the counting rules first
Before writing code, specify what counts as one object and when two objects are considered distinct. Make choices explicit rather than letting them emerge accidentally from the implementation.
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- Order: Are arrangements such as (a, b) and (b, a) different?
- Repetition: Can an element appear more than once?
- Labels: Are equal-looking items distinguishable by identity?
- Constraints: What makes a candidate valid?
- Boundaries: What should happen at the minimum size, with no elements, or at other edge values?
These conventions define the count. A program cannot verify an interpretation that has not been stated.
Build an independent reference enumerator
For tiny inputs, prefer the most direct method you can explain: generate all candidates, test each against the stated constraints, and count the survivors. Keep it separate from the optimized method’s key logic. For example, if the proposed solution uses a recurrence, avoid using that same recurrence to compute the reference count.
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The reference implementation should be clear enough that you can inspect its generated objects and see why each is included or rejected. Brute force is useful here because it trades speed for transparency on small instances.
Choose a finite test range
Test a bounded grid of small parameter values that includes the smallest meaningful inputs and important boundary configurations. State the range you actually tested, and identify any cases you skipped. Direct enumeration can grow quickly, so stop where the reference method can still cover each selected instance completely.
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For every test input, run both methods on the same precisely specified values. A simple assertion makes disagreement visible. Python’s official unittest documentation describes test cases and assertions for organizing checks into test suites; a framework is optional, but the comparison should be explicit.
Compare results and investigate failures
- Compute both answers. Run the direct enumerator and the proposed formula or algorithm on identical inputs.
- Assert equality. Make the test fail clearly if the results differ.
- Inspect the smallest failing case. Retain the input and, when practical, the concrete objects generated by the reference enumerator.
- Check the interpretation and boundaries. Look for disagreement about duplicates, ordering, empty cases, or off-by-one limits before changing the formula.
- Reduce and preserve the failure. Find a minimal counterexample and add it as a regression test so the same bug is caught later.
Add hand-checks and generated tests
Tiny examples and structural checks
Include very small cases whose answers you can verify by inspection. If the problem has known structure, you can also check properties such as symmetry or consistency with a recurrence. These checks provide additional ways to catch mistakes, but they do not replace direct enumeration when you need a reference count.
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Property-based testing
Generated tests can complement a deliberately chosen finite grid. The Hypothesis documentation describes strategies for generating inputs and illustrates comparing an optimized implementation with a slower, clearly correct reference. A strategy can exercise cases you did not think to select by hand.
Generated testing is not automatically exhaustive. Hypothesis explains that runs are generally bounded by their settings and behavior; for finite strategies, it may detect that a search space is exhausted and stop early, but it also notes that tracking the search space is imperfect. See How many times will Hypothesis run my test? Treat generated results as test evidence, not as universal verification.
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Know what passing tests establishes
An exhaustive comparison establishes agreement only for the finite inputs you actually covered, and only on the assumption that both programs correctly represent the intended problem. Sampled or generated tests provide evidence on the cases they exercise; they do not establish that every possible input was checked.
Passing tests can make implementation mistakes less likely within the tested range, but no finite collection of examples alone proves a claim about arbitrary input sizes. For that, you need a mathematical proof or an appropriate formal verification argument.
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