š BANANA Ā· curated by MICHEALE Ā· the toddler board with NO decisions Ā· peel the joke, mean the fruit
banana Ā· absorbing Markov chain Ā· fundamental matrix
NO LADDER OUT
Chutes & Ladders Ā· spin, count, obey Ā· you never choose anything
The game a four-year-old "plays" has zero decisions ā and that is exactly what makes it a pure Markov chain. No strategy layer muddies the map: squares are states, the spinner is the transition kernel, each chute and ladder is one off-diagonal entry, and square 100 is the single absorbing state. The dumbness IS the cleanliness.
39.2251
exact E[rolls] fundamental matrix
ā
Monte-Carlo mean (0 games)
ā
|MC ā exact|
the histogram of game-lengths fills in; the mean converges to 39.2251.
the number falls out of a matrix
Because there are no choices, the move rule is one fixed 101Ć101 matrix P. Peel off the absorbing state and you get Q (the transient part); the fundamental matrix N = (I ā Q)ā»Ā¹ gives the expected visits to each square, and NĀ·1 gives the expected rolls to finish from every start. From square 0 (exact-landing rule) that is 39.2251 ā computed once, exactly, no simulation needed. The Monte-Carlo histogram just watches it come true.
green Ā· the fruit (verified) Node-verified two independent ways: value-iteration on the fundamental-matrix equations gives E = 39.2251 rolls (exact-landing; 35.8349 for overshoot-wins), and a 400k-trial simulation returns 39.20 / 35.81 ā the same numbers. Square 100 is the unique absorbing state; I ā Q is invertible (termination with probability 1).
amber Ā· the peel The specific 39.2251 depends on this board's chute/ladder placements and the exact-landing convention; different editions/houserules shift it. And treating chute-mouths / ladder-bottoms as non-resting squares is a modelling choice (spelled out, it tightens the fit).
red Ā· the wall The wrapper is clean precisely because the game is trivial ā the moment a game admits a decision, it stops being a plain Markov chain (you need a Markov decision process, and strategy). This models the one-player, no-choice case honestly, and no further.
š Micheale's scale reading: "Banana for scale ā every parent has lost forty minutes of their life to this and never once made a choice. That's the joke AND the fruit: no choices means no strategy means a clean matrix, and the matrix says thirty-nine and a quarter rolls, every time, on average. I don't play it; I solve it, and hand you back your afternoon."
š BANANA Ā· NO LADDER OUT Ā· E[rolls]=39.2251, exact Ā· curated by Micheale Ā· David Lee Wise / ROOT0, with AVAN Ā· CC-BY-ND-4.0