🍌 BANANA · curated by MICHEALE · a playground rhyme that is secretly a RECURRENCE · peel the joke, mean the fruit
banana · the Josephus problem · survivor = 2L+1
ONE POTATO, TWO POTATO
kids in a ring · count around, every k-th is OUT · who is left?
The counting-out rhyme decides who's "it" — and it's the ancient Josephus problem: eliminate every k-th person around a circle, and where the last one stands obeys a clean recurrence. For counting by twos (k=2), the survivor has a closed form: write n = 2^m + L, and the safe seat is 2L + 1.
players n13
count out every k2
—
survivor (simulated)
—
recurrence J(n,k)
—
closed form 2L+1 (k=2 only)
a ring of players; run the count and watch every k-th fall.
the rhyme is a recurrence
Remove one person and the circle is the same problem, one smaller, with the start shifted by k — that's the recurrence J(n,k) = (J(n−1,k) + k) mod n, J(1)=0. For k=2 it collapses to the closed form: the survivor's seat is 2L+1 where n = 2^m + L (the largest power of two ≤ n, plus the remainder, doubled). So at a power of two, seat 1 always survives; just past it, the safe seat marches up by two.
green · the fruit (verified) Node-verified: physically simulating the ring (delete every k-th) matches the recurrence J(n,k)=(J(n−1,k)+k) mod n for all n, and matches the closed form 2L+1 for k=2 across n=1..40 — all three routes agree. (Samples: n=16→seat 1, n=17→seat 3, n=10→seat 5.)
amber · the peel The real rhyme's rules wobble — where you start pointing, whether "it" is the one out or the one left, "one potato… seven potato MORE" counts to eight. Those are just different (n, k, start) — the mechanism underneath is the same recurrence; the costume varies by playground.
red · the wall The closed form 2L+1 is special to k=2; for general k there's no such tidy formula, only the recurrence (or clever number-base tricks for small k). And the historical Josephus story (choosing a safe seat to survive) is legend, not the mathematics — the fruit is the recurrence, not the tale.
🍌 Micheale's scale reading: "Banana for scale — I have been counted out of many circles, being the obvious one to point at. But the pointing isn't random, friend, it's a recurrence: knock one out and you've got the same ring one smaller, shifted by your count. At a power of two, stand first and you live. I measure the safe seat; whether to take it is between you and the potato."
🍌 BANANA · ONE POTATO, TWO POTATO · Josephus, sim = 2L+1 · curated by Micheale · David Lee Wise / ROOT0, with AVAN · CC-BY-ND-4.0