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 Indecisive Walk (Posted on 2023-08-17)
Marilyn vos Savant still has her column in Parade magazine, which introduced everyone to the Monty Hall Problem. Here's her latest (well, a repeat from 1995):

A schoolboy has a test coming up and so wants to study for it, but is conflicted, as there's a movie he wants to see playing at the local theater. He starts out toward the theater, but halfway there he has second thoughts and turns back toward home. But again, halfway home he has third thoughts and turns back again, this time toward the theater, but again, halfway there he turns back toward home, and this indecision continues in the same way with the same switching ad infinitum.

What does this lead to as to his location in the long run?

 See The Solution Submitted by Charlie No Rating

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 Solution | Comment 1 of 2
We can think of this as starting at 0 on the number line and repeatedly halving the distance to 1 and then halving the distance to 0.  (Halving a number in binary is just moving the decimal point.)

In binary this is easy to see:
Start
0
Halfway to 1
0.1
Halfway back
0.01
Halfway to 1
0.101
and so on
0.0101
0.10101
...
0.010101...
0.1010101...

In the long run these are numbers the add to one and one is twice the other.  So these are  the binary equivalents of the common fractions 1/3 and 2/3.

 Posted by Jer on 2023-08-17 08:35:58
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