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Proof Builder Tap the proof steps in the right order so each one follows from the last. · 16 proofs 
Counterexample Hunt Find the one case that disproves a claim that looks true. · 6 puzzles 
Two-Column Proof Fill in the reason that justifies each statement — the classic school-geometry format. 
Knights & Knaves Knights tell the truth, knaves lie. From what each says, deduce who is which. How are you feeling?
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Practice & play
Practice the concepts 16 kits from logical-thinking basics up to advanced synthesis, with a hint when you get stuck. Proof Duel Head-to-head kit round for 2–4 players — one device or an online room by code. No chat, no accounts. Mixed Practice Shuffles the kits you’ve learned and brings them back when they’re ready to review. Teach the Cast Teach a proof technique to a cast member still learning it — teaching makes it stick.
Concept Kits — 16 kits of proof & reasoning
Kit 1: Logical Thinking Basics Logic & Reasoning · grades 4-5 · 25 questions Kit 2: Patterns and Conjectures Logic & Reasoning · grades 4-5 · 25 questions Kit 3: Deductive Reasoning Direct Proof · grades 5-6 · 25 questions Kit 4: Inductive Reasoning Induction · grades 5-6 · 25 questions Kit 5: Mathematical Arguments Direct Proof · grades 5-6 · 25 questions Kit 6: Visual and Geometric Proofs Logic & Reasoning · grades 6-7 · 25 questions Kit 7: Famous Problems and Puzzles Logic & Reasoning · grades 6-7 · 25 questions Kit 8: Proof Techniques Logic & Reasoning · grades 7-8 · 25 questions Kit 9: Direct Proof Methods Direct Proof · grades 7-8 · 25 questions Kit 10: Proof by Contradiction Proof by Contradiction · grades 7-8 · 25 questions Kit 11: Counterexamples & Disproof Counterexamples · grades 6-8 · 25 questions Kit 12: Mathematical Induction Intro Induction · grades 7-8 · 25 questions Kit 13: Cross-Topic Connections Logic & Reasoning · grades 5-6 · 25 questions Kit 14: Real-World Applications Logic & Reasoning · grades 6-7 · 25 questions Kit 15: Misconceptions & Reasoning Logic & Reasoning · grades 6-7 · 25 questions Kit 16: Advanced Synthesis Logic & Reasoning · grades 7-8 · 25 questions
For grown-ups — progress report →
Cast, stories & more
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😄 Brain break
Meet the cast — the characters who teach this
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Direct-Proof Dora Direct proof: assume premises, derive conclusion by straightforward logical steps -
Induction Ida Weak / standard mathematical induction: base case + inductive step -
Strong-Induction Sten Strong induction: base case + assume all prior cases hold -
Contradiction Cassius Proof by contradiction (reductio ad absurdum): assume the negation, derive a contradiction -
Contrapositive Cara Proof by contrapositive: prove "not Q → not P" to establish "P → Q" -
Construction Cole Proof by construction: prove existence by explicit construction of an example -
Pigeonhole Perch Pigeonhole principle: if n+1 items are placed in n bins, at least one bin contains 2+ items -
Exhaustion Edda Proof by exhaustion / cases: enumerate every case and verify each -
Counterexample Cricket Disproof by counterexample -
Biconditional Bex Biconditional proof -
Uniqueness Una Proof of uniqueness -
QED Closing-mark mentor — the ∎ at the end of every proof; the gentle voice that names completion + invites the next problem
Ensemble stories — the cast together
What next? Pick a door
- More like this ChanceForge What happens after a thousand flips? CardForge How many ways can these cards combine? NumberVerse Where does this number sit on the line?
- A bit harder MasteryForge Ready for the next problem at your edge? ConveyorForge How many machines keep up with the ore? MathLore How did this mathematician figure it out?
- Something different PageForge Chess What is the idea in this position? MapForge Which side of the mountain gets the rain? ReelForge Which shot matches the director’s idea?