How many loops does it take to escape a death loop?

Without memory of past loops, each attempt is independent: the expected number of loops to escape is 1 / pⁿ, where p is survival per decision and n is decisions per loop. With memory, dead branches are pruned and the search collapses to roughly the size of the tree, O(branches) rather than geometric.

Re:Zero · Return by Death

Every death branches a new timeline.

Subaru loops on death. Each loop is a path down a binary decision tree — survive every choice and you escape. Run loops, and watch dead branches get pruned the moment he remembers them.

Loops elapsed

Branches explored

Escape found

Expected loops

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Decisions per loop (n)5

Tree depth. 32 leaf timelines (2ⁿ).

Survival per decision (p)60%

Memory of failures

He remembers each death and avoids known-dead branches — the tree gets pruned.

How it works

Each loop walks a path of n binary decisions. Every decision survives with probability p; one failure ends the loop and that path dies. Reach the bottom alive and you escape.

In Subaru-mode, the search remembers which branches killed him and steers away — backtracking pruning. In Amnesiac mode every loop is an independent random walk from the top.

The math

Without memory, each loop is independent. Success per loop is pⁿ, so escapes follow a geometric distribution: expected loops to escape = 1 / pⁿ. At p=0.6, n=5 that's ~12.86; at n=10, ~165.

With memory, Subaru prunes failed paths. The search becomes O(branches) — bounded by the tree itself, never more than its 2ⁿ leaves — instead of geometric. Far faster. But each death is real to him.

The cost is psychological

The many-worlds interpretation says every quantum branch genuinely exists. Subaru can't tell the other Subarus that their branches were pruned — to them, they simply died.

By the end of Arc 4 he has died hundreds of times, each a complete subjective experience. The branching is computationally cheap. The trauma isn't.

Citations

Russell & Norvig (2020) — AI: A Modern Approach, ch. 3 on tree search & backtracking pruning.
Cormen et al. (2009) — Introduction to Algorithms, 3rd ed., DFS & branch-and-bound.
Everett (1957) — "Relative state formulation of quantum mechanics," Rev. Mod. Phys. 29, 454. Many-worlds.
Feller (1968) — Intro. Probability Theory, geometric distribution: E[loops] = 1/p.
Tappei Nagatsuki (2012–) — Re:Zero kara Hajimeru Isekai Seikatsu, source material for the branching premise.

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