Opening Urrors

The Royal game of Ur looks simple. Seven identical pieces per side on a small board with just a few simple rules. Typically only two or three possible moves in each turn, and the best one is usually obvious. But looks can be deceiving. For example, getting to start increase your winning chances by 1.5%, from 50% to 51.5%. Say you roll a 1 and your opponent rolls 1 too, and now you roll a 2. Your move?

X: ==@@

[6] ...o  .. (0)
    ........
[6] ...x  .. (0)

If you moved from a to c you squandered your good luck and then some. You shed almost 2% and stand at 49.6%. The correct move is to enter a second piece to b with a 51.5%.

X: ==@@

[6] ...o  .. (0)
    ........
[6] ...x  .. (0)

    e 51.49
 ** a 49.59  (-1.90)

In fact, entering two pieces with 1 and 2 (or 2 and 1) at the start is always correct, regardless of Red dice and moves. This is nice because it makes an easy to remember rule. It would have been even nicer if there was an obvious reason and you did not need to remember anything, but I can’t see anything clear cut. Sure, I can claim than now 2,3 and 4 will get me to d on the next turn for an extra move, but I am pretty sure I could invent something as reasonable if it was the other way round. With that in mind, let’s review the opening moves.

Ur Openings

(1,1),(2,2),(3,3) and (4,4) are forced, and we took care of (1,2) and (2,1) above. What about (1,3)? Typically you move your piece to d for the extra turn (case (a)), but not always. If the 4 if already occupied (your so very lucky opponent started with a (4,4)) you enter two pieces (case (b)), unless there is a juicy target to hit (like case (c)). The same rules apply to (3,1).

     (a)                   (b)                  (c)
[6] ..O.  .. (0)     [5] ...O  .. (0)     [5] ....  .. (0)
    ........             ...O....             .O.O....
[6] X...  .. (0)     [5] .X.X  .. (0)     [6] X...  .. (0)

It seems that after your opponent captured the 4 you move to a defensive stance and retaliate by hitting pieces on the 1-3 squares. (1,3) is not as simple as (1,2) but at least there is a relatively simple way to remember what to do.

(2,3) is next. Like (1,2) you almost always enter two pieces, but there are two exceptions. If your opponent has a piece on 1 (after (4,1)) you hit it, and you advance to 1 if he has a piece on 3, like so.

[6] ....  .. (0)
    X.O.....
[6] ....  .. (0)

But doing so when Green has a piece on 2 is quite an error (-1.5%). I could invent a story why this is so, but honestly I find it slightly mystifying.

X: =@@@  b               X: =@@@  b
[6] ....  .. (0)         [6] ....  .. (0)
    ..o.....                 .o......
[6] ..X.  .. (0)         [6] ..X.  .. (0)

 ** b 47.74                  e 48.10
    e 46.96               ** b 46.62  (-1.48)

The same rules apply to (3,2). With a (4,1) you advance to 1. With (1,4) you enter two pieces, unless there is a Red piece to hit at 1. Not hitting cost 2%!.

(4,2)/(2,4) are identical to (4,1)/(1,4). You obviously advance to 2 with a (4,2), and split (2,4) unless there is a hit on 2. Same for (4,3) and (3,4).

This examination of Green first two turns exemplify the deceptiveness of ROGOUR. Almost every case is clear and obvious, but the few exceptions can cost you none trivial amount of equity if you are unaware of them.