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# Horizontally randomly moving particle

## Question

<------- g -------  f ------- e ------ d ------ c ------ b ------ A ------- B ------- C ------ D ------- E ------ F ------ G ------>

A particle originates at point A and can move left one point or move right point for any move.

It cannot be stationary for any move.

For two moves, the ways can be diagrammed like this:

ABC

ABA

Abc

AbA

Two of those ways begin and end with the particle at the starting point A.

Question:  For six moves, how many of the ways begin and end with the

particle at the starting point A?

## Recommended Posts

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6 choose 3 = 20

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gavinksong,

your number of ways for the six moves versus mine is off by two.  I won't state

whether it's by lower or higher.

I don't see your method as fitting here.  However, maybe it (your use of taking

combinations) can be used as part of a larger different method.

The problem is still open.

Edited by Perhaps check it again
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6 choose 3 = 20

Agree.

Six moves, exactly three of which are to the right.

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The ways that 6 moves that begin and end at point A can be diagrammed:

where the moves happen to be completely to the left of A

AbcdcbA

AbcbcbA

AbAbcbA

AbcbAbA

AbAbAbA

where the moves happen to be completely to the right of A

ABCDCBA

ABCBCBA

ABABCBA

ABCBABA

ABABABA

where the moves start to the left of A but then eventually cross A before ending at A

AbcbABA

AbAbABA

AbABCBA

AbABABA

where the moves start to the right of A but then eventually cross A before ending at A

ABCBAbA

ABABAbA

ABAbcbA

ABAbAbA

Does anyone see more ways (paths) for any of the four categories above?

I am not seeing any more than what I listed in the spoilers.

Edited by Perhaps check it again

• 0

AbABAbA

ABAbABA

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