Write a small, direct enumerator for the objects in your problem, then compare its count with your formula or optimized algorithm on a stated set of small inputs. This is a practical way to find errors, not a proof that the solution works for every input: the check is only as reliable as the definition, reference enumerator and cases you actually test.
1. Define exactly what is being counted
Before writing code, specify what counts as one object and when two objects are considered the same. These choices change the answer, so an enumerator cannot check a problem whose conventions remain implicit.
- Ordering: Are sequences with the same elements in a different order distinct?
- Repetition: Can an element appear more than once?
- Labels: Are objects with different labels distinct even if their shapes or values match?
- Constraints: What conditions make a candidate valid?
- Edges: What should happen at the smallest meaningful size, and is an empty object allowed?
Use the same conventions in the proposed solution and the test. If they disagree, matching or mismatching counts may reflect an interpretation difference rather than an implementation error.
2. Build a simple, independent reference enumerator
For tiny inputs, choose the most transparent direct method: generate candidate objects, test each against the definition, and count the valid ones. Prefer clarity over speed. The enumerator’s job is to provide a comprehensible reference result, not to handle large inputs.
#1 Best Overall
Keep it as independent as practical from the solution under test. If both use the same recurrence, algebraic transformation or pruning rule, a shared mistake can make their answers agree. Directly listing candidates and filtering them is often a useful contrast to a compact formula or optimized implementation.
For example, if the problem counts arrangements, generate each candidate arrangement under the stated rules and count the valid arrangements. Make sure the generation step neither omits valid cases nor counts the same object multiple times under the problem’s definition.
3. Choose a finite test grid you can enumerate completely
Test a run of small parameter values that includes the minimum meaningful sizes and boundary configurations. Select a limit the direct enumerator can cover in full; enumeration can grow quickly, and there is no benefit in claiming coverage beyond what you ran.
- Include the smallest allowed inputs and any meaningful empty or singleton cases.
- Cover boundaries where constraints or formulas change behavior.
- Record the exact range and any excluded cases, so the extent of the check is clear.
For each input in the grid, run both methods on precisely the same parameters and conventions. Python’s official unittest documentation describes test cases and assertions for expressing comparisons and organizing them into test suites. A particular framework is optional; the important part is that disagreement produces a visible failure.
PC Slower Than It Used to Be?
A free scan shows the junk files, broken settings and background clutter dragging Windows down - then fixes them in one click.Free scan · Windows 10 & 11Crashes, No Sound, or Screen Glitches?
Random freezes, missing sound and display glitches usually trace back to one bad driver. Find and replace yours safely.Free scan · under a minute4. Compare the results and preserve counterexamples
For every selected input, compute the reference count and the proposed answer, then assert that they are equal. When they differ, keep the smallest failing input and, if feasible, the concrete objects the enumerator generated. Inspect definitions, duplicate handling, order conventions and boundary conditions before changing the formula.
Once you understand a mismatch, reduce it to a minimal example and retain it as a regression test. That way, a later change cannot silently reintroduce the same error.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.5. Add hand checks and generated properties
Hand-checkable cases
Work through a few tiny inputs by hand and compare those counts with both programs. Where the problem has known structural checks—such as a symmetry or a recurrence—check those too. They offer additional ways to catch mistakes, but they do not replace direct enumeration.
Property-based tests
Property-based testing can broaden the set of cases you inspect. The Hypothesis documentation describes strategies for generating inputs and gives comparison with a slower, clearly correct implementation as an example use. You can use a strategy to generate valid small inputs, then check that the optimized solution agrees with the reference enumerator or satisfies a suitable property.
Best Value
Generated tests are not automatically exhaustive. Hypothesis explains that runs are generally bounded by settings and behavior; for finite strategies, it may detect search-space exhaustion and stop, but its tracking is imperfect. Treat generated cases as a complement to a deliberately chosen finite grid, not as universal coverage. See How many times will Hypothesis run my test? for the framework’s explanation.
What a passing brute-force check does—and does not—show
If the enumerator and solution correctly represent the problem, exhaustive agreement over a finite grid shows that they agree on the inputs in that grid. It is evidence against mistakes in those tested cases. It does not establish agreement for untested sizes or prove a statement quantified over all input sizes. That stronger conclusion requires a mathematical proof or a formal verification argument.
When reporting a check, state the tested domain, whether every case in it was enumerated, and any cases omitted. Keep sampled or generated testing distinct from exhaustive testing. In all cases, the conclusion depends on the reference enumerator correctly implementing the intended definition.
Quick Recap
Product prices and availability are accurate as of the date/time indicated and are subject to change. Any price and availability information displayed on Amazon at the time of purchase will apply.
Do these 3 things before closing this tab:
1Clear out junk files and repair common Windows errors2Fix the driver behind crashes, sound loss and screen glitches3Repair Windows errors before they cause bigger problems
