Problem 5

The Bank of Bath issues coins with an

H

on one side and a

T

on the other. Harry has

n

of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly

k>0

coins showing

H

, then he turns over the

k

th coin from the left; otherwise all coins show

T

and he stops. For example, if

n=3

, the process starting with

THT

THT\to HHT\to HTT\to TTT

, which stops after three operations. (a) Show that, for each initial configuration, Harry stops after a finite number of operations. (b) For each initial configuration

C

, let

L(C)

be the number of operations before Harry stops. Determine the average value of

L(C)

over all

2^n

possible initial configurations.

Step 6 of 6: Solve the recurrence

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In plain words

Add the successive increments from one coin up to n coins.

E_n=\sum_{j=1}^{n}\frac j2=\frac{n(n+1)}4

Detailed analysis

Iterating the recurrence from $E_0=0$ (or from the base case) gives $E_n=\sum_{j=1}^{n}\frac j2=\frac{n(n+1)}4$ . This is the required average, and the induction simultaneously proves finite termination for every initial configuration.