This puzzle is part of the Monthly Topic Challenge #7: Board games.

The famous Backgammon game can be seen as being played on a number axis with one player on the non-negative side (from 0 to 25) and the other on the non-positive side (from -25 to 0). The two sides of the axis interact with each other so that if the difference between two of the numbers (they themselves must not be multiples of 25) is equal to 25, then the two numbers can't hold both players' pieces simultaneously.

Both players move in the direction of the "0" on the axis. The "negative" player starts out with two pieces on -24, five on -13, three on -8 and five on -6. The "positive" player places two of his pieces on 24, five on 13, three on 8 and five on 6.

Hitting will result in a piece being sent to -25 or 25 (the bar). If the player has a piece on the bar, they can't move any piece except the pieces on the bar.

If a player has all its pieces in the range of [-6,6], then he can start to bear off.
And lastly, please be aware that the very first roll in a game must not be a double roll, as implied by the rules. After the first move, players will be allowed to roll double.

                                                 XX                          XX   
     XX                                         XXX                 XXX     XXX   
-25 -24 -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -09 -08 -07 -06 -05 -04 -03 -02 -01 000  
000 001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025  
                        OOO     OOO                 OOO                                         OO
                        OO                          OO  

The "distance from endgame" is obtained by taking the absolute values of the position of each piece and summing up the values. For example, the "distance from endgame" of the starting position is (2x24+5x13+3x8+5x6)x2 = 334.

Now, here are the tasks.

Task 1:

What's the shortest possible game (fewest moves)?

Task 2:

What is the maximum "distance from endgame" that can be reached from the starting position? (hence the title)

Task 3:

Is there a sequence of dice-roll and moves that puts the game into an infinite loop?

Notes on backcammon variants:

Dutch backgammon has all the negative pieces starting out on -25 and all the positive pieces on 25. The bar is also on -25 and 25 but it's different from where the players started. First-turn double rolls are allowed. A player must not "hit" until he has at least one piece in [-6,6].

Longgammon starts with all the negative pieces on -24 and all positive pieces on 24.

Hyper-backgammon starts with only six pieces - three negative on -24, -23 and -22; three positive pieces on 24, 23 and 22.


1 Answer 1


Task 1, 9.5 turns

enter image description here
Some minor shuffling required to handle the bottlenecks at the edges of the far board [7/12/13/18], but otherwise a rather greedy dash for the finish.

Task 2, 720 721 distance

enter image description here
Achieving the initial stage is relatively straightforward with no hitting or special movement required. The work after that divides into three main stages: mopping up the easy targets [Rolls 1-5], colliding the middle groups [Rolls 6-8], and rounding the last pieces up for a couple final captures. [Rolls 9-14]

The above game can be extended to increase the distance to 721, like so:

enter image description here

Task 3

There are several ways to achieve this, even after restricting to forced repetitions.
If Blue leaves the bar with any roll totaling 9, a double-4 from Red will restore the position. A similar position admits repetitions through a 5 from one player and a double-5 from the other.
enter image description here


Instead of struggling with spoiler formatting I have transcribed the games on Pastebin.

  • $\begingroup$ If you are not going to make a "proof game", you might use this backgammon notation instead. By the way, I can't access Imgur images. $\endgroup$ Commented Feb 17, 2023 at 10:47
  • $\begingroup$ I've found a way to connect a variant of the 721 position to the 720 position, so no worries there. Do you want text notations of the games too, to get around your Imgur problems? $\endgroup$ Commented Feb 17, 2023 at 16:37
  • $\begingroup$ Please add the text notation. $\endgroup$ Commented Feb 20, 2023 at 19:30
  • $\begingroup$ I have attached them via Pastebin - hopefully this works for you. $\endgroup$ Commented Feb 22, 2023 at 0:42

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