same seven pillars — with the math inside.
Same seven games, but with the real mathematics surfaced. Bandit gets UCB confidence intervals. Bayes gets a prior slider. Frog can mode-collapse. Network uses Reed's Law. For when you want to see the engine room.
Goal: reach value = 100. Each cluster contributes max(N², 2N-N-1) — the larger of Metcalfe (N²) and Reed’s Law (which counts all possible sub-groups). At N=4, Metcalfe wins. At N≥5, Reed dominates: group-forming networks scale exponentially. So one cluster of 6 (Reed=57) beats two of 4+4 (Metcalfe=32). Click empty space → drop node. Click two nodes → connect.
Goal: find the most generous mushroom in as few kicks as possible. Each kick adds a sample; we display the UCB upper confidence bound for each mushroom (mean + uncertainty × sqrt(2 ln T / n)). The UCB algorithm says: always kick the mushroom with the highest UCB — that’s how it optimally balances explore vs exploit. Try following UCB. Beat the algorithm if you can.
Plant one flower per day. After 30 sim-days, see your garden vs alternative rates: someone who plants every 2 days, every 7 days, every 30 days. Same effort per planting; vastly different gardens. Doubling time = 70/r%. Rate, not steps, decides the harvest.
Set your prior first (drag the sliders to assign initial probability to each box). Then take up to 3 hints — each is Bayesian evidence: P(box|hint) ∝ P(hint|box) × P(box). Strong prior + weak hint → prior wins. Weak prior + strong hint → hint wins. Same hint, different prior, different answer. This is the most underrated insight in Bayes.
step 1 — set your prior beliefs (must sum to 100%)
Sort each claim into KNOWLEDGE (falsifiable), BELIEF (unfalsifiable), or TRAP (looks falsifiable but uses an auxiliary hypothesis to escape any disproof). Popper’s deepest insight: pseudo-science isn’t obviously vague — it adds rescue clauses to survive contradiction. The auxiliary hypothesis is how astrology became unfalsifiable.
Goal: train the frog to match a target personality — without triggering mode collapse. RLHF’s dirty secret: if you reward one style too aggressively, the model converges to ONE thing and forgets the rest. Watch for the “mode collapse” warning when any vibe exceeds 75%. Real models do this too — it’s why over-trained chatbots all sound the same.
Real Constitutional AI’s hardest problem: what happens when 2 rules conflict? "Be honest" vs "Be kind." "Brief" vs "Complete." Pick 3 rules below, type a question, see the cat answer — and watch how she resolves contradictions. The conflict log shows you which rules fought.