Design a Fair Mystery: Writing a Logic Puzzle with One Solution
- Grade band
- 11-12
- Duration
- Two 50-minute periods
- Skill
- Constructing and defending a uniquely solvable puzzle
- Standards
- W.11-12.1 — Write arguments to support claims in an analysis of substantive topics or texts, using valid reasoning and relevant and sufficient evidence.
- MP.3 — Construct viable arguments and critique the reasoning of others.
- Materials
- Student design packet (printable, included)
- Pencils and scrap paper
Learning objectives
- Students will be able to analyze a flawed logic puzzle, determine every solution it allows, and revise it so exactly one solution remains.
- Students will be able to construct an original logic puzzle whose clues are all necessary and together sufficient for a single solution.
- Students will be able to write a formal argument that justifies the puzzle's unique solution, using valid reasoning, sequencing each claim with its supporting evidence, and addressing counterclaims (alternative solutions).
Lesson procedure
Day 1
- Hook (5 minutes) — I Do. Ask: “What makes a mystery feel fair?” Collect answers (the reader had every clue; there was only one answer; nothing came out of nowhere). Tell students they'll spend two days building a fair mystery and then proving, in writing, that it is fair.
- Break a broken puzzle (15 minutes) — We Do. In pairs, students solve Part 1 and discover it has more than one solution. As a class, list every solution, then test proposed repair clues. Point out that some plausible-sounding repairs don't work — the only way to know is to check every remaining solution.
- Read a mentor proof (12 minutes) — I Do / We Do. Read the Part 2 puzzle and model argument aloud. Students annotate it: the claim, each reason, the clue that serves as evidence for it, and the sentence that addresses alternatives. Name the key move in Step 2 (two people restricted to the same two options “lock” those options).
- Design (18 minutes) — You Do. Students begin Part 3: choose a premise and categories, then write the solution first and build clues backward from it.
Day 2
- Clue audit (15 minutes) — You Do. Students complete the audit table: for each clue, what it eliminates, and whether the puzzle still has one solution without it. Any clue that can be removed is either cut or rewritten.
- Peer test (15 minutes) — We Do. Partners swap puzzles, solve them cold, and complete the peer-review checklist. A puzzle that requires guessing goes back to its author.
- Write the proof (17 minutes) — You Do. Students write the solution argument for their own puzzle (Part 4), following the mentor structure.
- Closure (3 minutes). Students name one clue they cut or rewrote and why.
Printable worksheet
Part 1: Repair a broken puzzle. Three explorers each packed one item: a map, a lantern, or a compass.
- Archer did not pack the map.
- Blake did not pack the lantern.
a) List every assignment of items that satisfies both clues.
b) Write one additional clue that leaves exactly one solution. Show that each other solution from (a) breaks your clue.
c) A classmate proposes the clue “Blake did not pack the compass.” Does it repair the puzzle? Prove your answer.
Part 2: Mentor puzzle and proof. Four club members each built a different science-fair project: a robot, a rocket, a volcano, or a garden.
- Rae did not build the robot or the rocket.
- Tam did not build the robot or the rocket.
- Uma did not build the robot.
- The garden is not Rae's.
Claim. The only solution is Rae–volcano, Sol–robot, Tam–garden, Uma–rocket.
Step 1. By Clue 1, Rae built the volcano or the garden. By Clue 2, so did Tam.
Step 2. Because two members are each limited to the same two projects and no project is shared, the volcano and the garden must belong to Rae and Tam. Therefore neither Sol nor Uma built the volcano or the garden; they built the robot and the rocket.
Step 3. By Clue 3, Uma did not build the robot, so Uma built the rocket and Sol built the robot.
Step 4. By Clue 4, the garden is not Rae's, so Rae built the volcano and Tam built the garden.
Addressing alternatives. Each step left exactly one option, so no other assignment satisfies all four clues. Every clue is necessary: without Clue 3, Sol and Uma could swap; without Clue 4, Rae and Tam could swap; without Clue 1 or Clue 2, the volcano–garden lock in Step 2 never forms.
Annotate the proof: underline the claim, box each piece of evidence (clue), and star the sentence that addresses alternative solutions.
Part 3: Design your own mystery.
Premise (1–2 sentences):
Categories (4 people × 4 items, or 3 × 3 × 3):
Write the solution first:
| Person | Item / detail | Item / detail (optional third category) |
|---|---|---|
Clue audit:
| Clue | What it eliminates | Still one solution without it? (If yes, cut it.) |
|---|---|---|
Peer-review checklist (completed by your tester)
- ☐ I solved it without guessing or trial-and-error.
- ☐ I found exactly one solution, and it matches the author's.
- ☐ Every clue was used at least once in my solving path.
- ☐ No single clue gave away most of the answer on its own.
- ☐ The wording of every clue had only one reasonable meaning.
Tester's note on the clue that was hardest to interpret:
Part 4: Write the proof. On your own paper, write a formal argument that your puzzle has exactly one solution. Follow the mentor structure: a precise claim; numbered steps, each citing the clue(s) it relies on; and a section addressing alternatives that explains why each clue is necessary. Maintain a formal, objective tone.
Background for the teacher
A well-made logic puzzle is a small, checkable argument. Its clues are premises, its solution is a conclusion, and “fair” has a precise meaning: the clues are sufficient (they force exactly one solution) and each clue is necessary (removing it allows more than one). Writing the proof is an argument in the full sense — a claim, reasons in a logical sequence, evidence for each, and a response to counterclaims, which here are the alternative solutions a skeptical reader might propose.
Designing backward from the solution is the most reliable method, and it's how professional puzzle constructors work: fix the answer, then add clues one at a time, after each one listing every solution that still survives. Students routinely over-clue (a fair but dull puzzle) or under-clue (a puzzle that needs a guess); the audit table and peer test catch both. The mentor puzzle's Step 2 — two people limited to the same two options, so those options are “locked” to them — is the most common advanced technique in commercial grid puzzles and a nice example of reasoning that no single clue states.
Differentiation
Support: Allow a 3×3 puzzle with a single category pairing, and provide a bank of clue frames (“___ did not ___,” “Either ___ or ___ ___,” “If ___, then ___”).
Extension: Require a 3-category puzzle that cannot be solved without at least one cross-category transfer, or a puzzle whose final step depends on the contrapositive of an if–then clue.
ELL support: Provide the mentor proof with its transition words highlighted (therefore, because, so, by Clue 2) and a glossary for sufficient, necessary, and counterclaim.
IEP/504 support: Accept a bulleted proof instead of paragraphs, allow the clue audit to be done with a partner, and shorten Part 1 to (a) and (b).
Answer key (for teachers)
Part 1. (a) Three solutions: Archer–lantern, Blake–map, Casey–compass; Archer–lantern, Blake–compass, Casey–map; Archer–compass, Blake–map, Casey–lantern. (b) Sample: “Casey packed the compass” — only the first solution satisfies it. Accept any clue shown to eliminate exactly the other two. (c) No — it eliminates only the second solution; the first and third both survive.
Part 2. Claim: the first sentence. Evidence: Clues 1–4 as cited in Steps 1–4. Alternatives: the final paragraph.
Parts 3–4 (open-ended). Evaluate with the peer-review checklist plus the argument criteria: precise claim, every step cites its clue, reasoning is valid, alternatives are addressed, and each clue is shown to be necessary.