Arithmetic truth made visible — not a list of facts but a place you can walk. The counting numbers look like the plainest things there are, and yet they hide shapes: a tree you can descend to the best fraction, a spiral where the primes line up, a river every number falls down, a times-table that draws a cardioid, a board logic alone can fill. Five benches; five ways the numbers turn out to have a shape — and each is honest about exactly how much it can prove.
What is the best fraction for π — and how do you prove a fraction is the best? Descend the Stern–Brocot tree toward any real number; the turning points are the convergents, the closest fractions of any denominator up to theirs. Three independent methods must land on the same fractions.
self-test ✓ · three constructions agree · 355/113 proven best · φ the most irrational → ✦Write the integers in an outward square spiral, light only the primes, and they fall onto diagonals all by themselves — each one exactly a quadratic 4t²+bt+c. Two independent oracles must agree on every number, so no lit dot can lie; the running count is π(N), riding the Prime Number Theorem.
self-test ✓ · sieve ⟺ trial division agree · π(N) exact · Euler's 41-prime streak → 🌳Halve it if even, triple-plus-one if odd, repeat — and every number ever tested falls to 1. Draw the tree of how they fall, or the river of their trajectories; reproduce the famous records exactly (27 → 111 steps, peak 9232). But label it for what it is: an open question since 1937, checked, never proven.
self-test ✓ · forward & backward constructions agree 0 counterexamples up to N — a check, not a proof → ♡Put m points round a circle and draw each i to (k·i) mod m — a times-table on the ring. The chords don't draw the curve; they're tangent to it, and a cardioid falls out at k=2, a nephroid at k=3, an epicycloid of k−1 cusps in general. And the same ring ℤ/mℤ that draws it is a cipher wheel — the multiplicative half the Volvelle never had.
self-test ✓ · envelope ⟺ closed-form epicycloid to 1e-12 · the cipher wheel is one ring → ▦Five colors, a 5×5 board, one rule — each color once per row, once per column. The clues you're dealt hide exactly one finishing, and you reach it by pure deduction: naked and hidden singles, never a guess. The room's first game — and you can pull any clue and watch the certainty break.
self-test ✓ · unique solution · deduction-only, never a guess · pull a clue and it breaks