PuzzleForge ✕
◆ PuzzleForge Learn 🎓 Teach 🔮 Predict 🗺️ Adventure 🤝 Together ⋯ More 🧭 My Journey Teach the Cast Some of PuzzleForge’s cast are still learning. When you teach them, YOU learn it best — catch what they get wrong, then quiz them.
✕ Teach Evan You be the teacher — Evan is still learning. Helping Evan figure it out is the practice. (Evan's progress stays on this device; nothing is sent anywhere.)
🎓 In a minute you'll be the teacher. Get ready to teach Evan: parity / solvability.
Count how many tiles are out of order. That parity tells you whether a sliding-tile scramble can ever be solved. Exactly half of all arrangements can't.
Teach the connections
Show how the puzzle-solving ideas connect You've helped them one at a time. The real understanding is how the ideas CONNECT — teach that next.
Wend (state-space search — map which positions are reachable) × Evan (parity / solvability — count swaps mod 2 to know if it can be solved at all)
Why does Evan run BEFORE Wend on a scrambled sliding puzzle?
Parity (Evan) says whether a solution exists at all; no point in Wend searching a state space with no path to the goal Searching comes first; parity is only checked at the very end Parity and search are unrelated — parity never affects the search
Wend then says: “I'll just start searching moves on any scrambled 15-puzzle — it's always solvable.” — is that right?
No — Evan's parity check first: half of all scrambles are UNSOLVABLE, and Wend would search them forever Right, every scramble is solvable
The link: solvability (Evan) prunes the search (Wend) — don't search what can't be solved.
Check if it CAN be solved, then search how.
Nestor (recursion / decomposition — solve the big tower via the smaller tower) × Minim (optimality — refuse any solution longer than the fewest moves)
How does Nestor's "move the smaller tower first" also satisfy Minim's demand for the fewest moves?
The recursive decomposition (Nestor) yields exactly the minimal 2^n-1 moves Minim requires — structure IS optimality here Recursion always uses MORE moves than necessary, so Minim rejects it Optimality and recursion never coincide
Nestor then says: “Recursion is elegant but wasteful — you can always beat it with clever ad-hoc moves.” — is that right?
No — for Hanoi, Nestor's recursion is provably minimal; ad-hoc moves can't beat 2^n-1 Right, ad-hoc always wins
The link: the recursive decomposition IS the optimal solution — Nestor and Minim agree.
Break it down the right way, and it's already the shortest.
Capstone — plan one puzzle solve together Three moves of a solver's plan. Bring in the cast member for each.
Before touching a scrambled 15-puzzle, decide if it's even solvable. Who?
Who do you bring in?
Evan — the parity / solvability check Nestor — recursion Faro — look-ahead
Solve Tower of Hanoi by first moving the smaller tower. Who?
Who do you bring in?
Nestor — recursion / decomposition Skiff — constraint satisfaction Gill — GCD reasoning
Reject any solution longer than the fewest possible moves. Who?
Who do you bring in?
Minim — optimality Wend — state-space search Gray — binary / Gray-code
The synthesis: check it CAN be solved, decompose it recursively, and hold to the minimum — a solver plans before moving.
Solvable, decomposed, minimal — the solver's plan.
Try it in the puzzle tools →
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Same ideas with a younger band's look: bigger type and buttons, a rounder font, slower motion. For any child who wants it.
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