# silly backgammon puzzle

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### Don Banks

Nov 5, 1996, 3:00:00 AM11/5/96
to

Is it possible that at some point in a game of backgammon, at
the end of a player's move, that the board has returned to its
original starting configuration?

What is the shortest sequence of (legal) moves that could do this?

### Alexander Nitschke

Nov 5, 1996, 3:00:00 AM11/5/96
to

I constructed a game which accomplish your goal in 3 moves:

1) 62: 13/5 64: 13/7 24/20*
2) 43: 25/18* 21: 25/24 20/18
3) 32: 18/13 32: 18/13

I believe it isn't possible in fewer moves, but maybe I just didn't
see the way to build such an opening.

Alexander (acey_deucey on FIBS)

### John S Mamoun

Nov 5, 1996, 3:00:00 AM11/5/96
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Don Banks (dba...@gpu4.srv.ualberta.ca) wrote:
: Is it possible that at some point in a game of backgammon, at
: the end of a player's move, that the board has returned to its
: original starting configuration?

: What is the shortest sequence of (legal) moves that could do this?

Here is what I came up with:

12 11 10 9 8 7 6 5 4 3 2 1
0 X X 0
0 X X 0
0 X X
0 X
0 X

X 0
X 0
X 0 0
X 0 0 X
X 0 0 X
13 14 15 16 17 18 19 20 21 22 23 24

1-3 X slots 5 and splits to 21.
4-1 0 hits blot on 24.
1-6 X enters, hits 0 blot on 24, moves 21-15.
5-5 0 enters, hits on 15 and hits on 24, hitting 3 X blots.
1-1 X enters, hits 0 blot on 24, splits on 23.
5-2 0 enters on 5, runs to 7.
3-2 X runs 23 to 18
5-4 0 runs from 7 to 16.
5-3 X runs to from 18 to 10, without hitting a 0 blot.
3-4 0 moves men back to 19.
3-1 X moves man back to 5.

### Peter Bell

Nov 5, 1996, 3:00:00 AM11/5/96
to

In article <55mvct\$a...@pulp.ucs.ualberta.ca>, dba...@gpu4.srv.ualberta.ca
(Don Banks) wrote:

< Is it possible that at some point in a game of backgammon, at
< the end of a player's move, that the board has returned to its
< original starting configuration?
<
< What is the shortest sequence of (legal) moves that could do this?

Here's a candidate game:

2-1: 6/3 2-1: 24/22* 6/5
6-5: b/20*/14 6-1: b/24 22/16
6-2: 14/6 6-4: 16/6

Thanks,

Peter Bell (USRobots)

### Ole Jensen

Nov 5, 1996, 3:00:00 AM11/5/96
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dba...@gpu4.srv.ualberta.ca (Don Banks) writes:
> Is it possible that at some point in a game of backgammon, at
> the end of a player's move, that the board has returned to its
> original starting configuration?
>
> What is the shortest sequence of (legal) moves that could do this?

Here is a way of doing it in five moves:

1) 41: 13/9 24/23 44: 24/16* 13/5
2) 31: bar/24 23/20* 11: bar/24 16/13
3) 52: 20/13

I conjecture it cannot be done in fewer moves than that.

It is easy to see that it takes at least four moves. Calling the
players X and O, let us assume X has the first move. Both sides, on
their first move, must move forward from the starting position, so
both sides need to get at least one checker hit in order to return to
the starting position. On his first move X has no blots to hit.
Hence, starting from O's first move, at least the following events
have to happen (in some order):

(A) O sends back one of X's checkers.
(B) X reenters.
(C) X sends back one of O's checkers.
(D) O reenters.

Assuming (A) happens first (otherwise we waste too much time), the
only two events that can happen in the same move are (B) and (C).
This means it will take at least three moves for all four events to
happen, bringing the total to four moves. So fewer than four is
definitely impossible.

If someone is going to come up with a four-move solution, they will
have to find a way of doing (B) and (C) in one move that also takes
X's checkers into the starting configuration. I claim this is
impossible, but I have not worked out a proof.

(Incidentally, it _can_ be done if we allow doubles for X's first
move, but that would of course be cheating, since it is the opening
move.)

-- Ole (hoegh on fibs)

### Tom Keith

Nov 5, 1996, 3:00:00 AM11/5/96
to

Don Banks wrote:
>
> Is it possible that at some point in a game of backgammon, at
> the end of a player's move, that the board has returned to its
> original starting configuration?
>
> What is the shortest sequence of (legal) moves that could do this?

The best I could do is five rolls:

--- WHITE --- --- BLACK ---
31: 13/9 44: 24/16*, 13/5
32: bar/20* 31: bar/24, 16/13
43: 20/13

Tom