Logic Simplification Challenge Generator
Each expression here started simple and was made complicated on purpose, by applying one identity backwards — double negation, idempotence, absorption, De Morgan, or the definition of implication. That means the simplification is known rather than searched for, and the page can name the law in play. Equivalence is not argued: with five variables or fewer it is decided by checking all thirty-two assignments.
What this generator does
Takes a small expression, applies one of five identities in reverse to inflate it, and asks you to undo that. The answer, the identity used, and the node count before and after are all shown.
How to use this tool
- Choose how many variables to work with.
- Simplify the expression using the logic laws you know.
- Reveal the answer and the identity that was applied.
- Check your own answer by comparing truth tables if you disagree.
Understanding the controls
- How many variables
- Between 2 and 5. More variables give the identity more room but do not change the method.
- Seed
- Reproduces the same expression and the same identity, so two people can attempt the identical simplification.
- Show the answer
- Reveals the simplified form and names the law that connects the two.
Common use cases
- Boolean algebra practice with a named law behind each question
- Drilling De Morgan's laws on fresh examples
- Showing that equivalence is decidable by table at small sizes
- Homework where the intended route is known
- Setting the same challenge for a group from a seed
How this generator works
The generator works backwards from the answer, which is why the answer is exact. The check then confirms equivalence the only way that is conclusive at this size: evaluating both forms on every assignment and requiring them to agree everywhere, plus confirming the simplification is not larger than what it replaced.
Randomness and fairness
Which identity is used and which expression it is applied to are random; the equivalence is proved. Seeded challenges reproduce exactly and are therefore explicitly not cryptographically secure, and unseeded ones use the browser's cryptographically secure generator.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Exactly one identity is applied per question, so multi-step simplifications do not appear.
- The answer given is the expression the question was built from; another equally simple form may exist.
- Only five identities are used, so the same laws recur.
- There is no step-by-step derivation, only the law's name and the result.
- Challenges are not stored between visits.
Privacy and your data
Expressions are built and checked in your browser. No expression, answer or seed is transmitted or stored.
Related generators
- Truth Table GeneratorThe full truth table for an expression you type — or one drawn for you — with the grouping shown fully bracketed and a tautology or contradiction verdict counted from the rows.
- Karnaugh Map Challenge GeneratorGrey-coded Karnaugh maps filled from a real expression, with the largest valid groups found and each one proved to cover only true cells.
- Logic Gate Circuit GeneratorRandom acyclic gate circuits with their equivalent expression and full input-output table, the two required to agree everywhere.
- Index Laws Question GeneratorSimplification questions on multiplying and dividing powers, in one letter or two, with the numbers and the indices handled separately.