The progress
you throw away.

The slow attempts are a different crowd.

The walker remembers nothing. It does not learn which way the exit lies. At each tick it moves left or right with equal probability. At the left wall, a leftward coin costs a tick and leaves it in place. Reaching position 20 ends the search.

Most of the corridor lies behind the starting point. Some walks find the nearby exit quickly. Others wander into that long left-hand region. A timer cuts off those unfinished excursions and buys another attempt from the original, favourable position.

It also throws away good positions. A walker one step from the exit gets sent back with everyone else. There is no hidden rule that rescues promising attempts. The policy sees only the clock.

Move the start to 2. Among the timers tested here, starting over no longer helps. Set a timer shorter than the distance to the exit and nobody can ever arrive. More restarts is not a principle.

The useful amount of impatience.

The best tested timer is 12 steps, at 40.52 ticks on average.

The search checks every integer cutoff from 1 through 240, plus the separate option of never restarting. This is a finite comparison, not a global optimum over all timers or all policies. Changing the starting point or reset cost recomputes the entire curve.

The answer is a renewal equation.

Call the completion time of an uninterrupted walk T, the timer τ, and the reset cost c. One attempt consumes min(T, τ) walking ticks. If it fails, pay c and face exactly the original problem again. An arrival on the final step counts as success, before any reset.

M = E[min(T, τ)] + P(T > τ) × (c + M)

M = (E[min(T, τ)] + c × P(T > τ)) / P(T ≤ τ)

The denominator is the chance an attempt succeeds. It cannot be dropped: charging for one truncated attempt while forgetting all the retries would produce an attractive, false answer.

These fractions come from counting coin strings, not from a simulation estimate. Each of the 4,096 twelve-coin strings has the same probability; 1,093 reach the target by the timer. A string that arrives early is charged only for the steps it used.

Without resets, the expected time from position x is (20 − x)(21 + x). Substitution checks the one-step equations at every square, including the wall and the absorbing target. At 16, it gives 4 × 37 = 148.

A reset costs time, not just progress.

For the fixed twelve-step timer, the opening advantage survives only while each reset costs less than 39.12 ticks (exactly 117476/3003). Set the cost to 40 and that policy loses. A different timer may still help; the curve checks that separately.

There is another way to read the threshold. Among walks still unfinished when the timer rings, measure their expected remaining time if allowed to continue. A reset is useful precisely when that remaining time is greater than the reset cost plus the mean time of a fresh, uninterrupted attempt.

At the opening timer, unfinished walks have 187.12 ticks left on average. A fresh attempt needs 148.00. The difference pays for the reset.

The comparison conditions on failure to arrive. It does not claim that elapsed time itself makes a memoryless walker worse, or that every unfinished walker is far away. The timer selects a changed distribution of positions.