This article originally appeared in the October 2001 issue of GammOnLine. Thank you to Kit Woolsey for his kind permission to reproduce it here.

## The Frozen Cube

By Kit Woolsey
For the most part, proper cube strategy in matches is simply good money cube stragegy combined with a bit of common sense. The leader in the match tends to be more conservative with his doubles and takes, particularly if we are talking about a cube-level higher than 2. He also tends to avoid gammonish positions. Conversely, the trailer tends to be more aggressive with his doubles and takes, and he goes after gammonish positions.

When the leader has a small number of points to go, this concept can be magnified. The trailer may have an automatic redouble which will put the cube on four, increasing the trailer's equity if he take. For example, consider the following simple position:

 66 ``` ``` White 27 point match Blue 5

A straightforward position. White's winning chances are 21.2%. This can be calculated as follows:

In order for White to have a chance, he will need for Blue to not roll a set of doubles on one of his next two rolls. The chance that Blue will not roll a set on his first roll is 5/6, and the chance that he will not roll a set on his second roll is also 5/6. Therefore, the chance that he will not roll a set on either of his next two rolls is 5/6 X 5/6 = 25/36.

Given that Blue doesn't roll a set, White will still need to roll a set on one of his next two rolls in order to win. By the same calculations, the chance that White fails to roll a set on one of his next two rolls is 25/36. Therefore, the chance that he rolls the needed set is 11/36.

For White to win, he needs the parlay of Blue not rolling a set and White rolling a set. From what we have calculated, the probabality of this is (25/36) X (11/36) = 275/1296 = 21.2%.

It is clear that White has a big money pass. However, it might be a different story at the match score. Keep in mind that White will be redoubling to 4 immediately, so when he wins he will win 4 points. For my calculations, I will be using my match equity table. Click here to see the table.

White passes: Behind 6-2 Crawford (1 away, 5 away), 15% equity.
White takes and wins: Ahead 6-5 Crawford (1 away, 2 away), 70% equity.
White takes and loses: Loses match, 0% equity.

Therefore, White is risking 15% in order to gain 55%. His takepoint is 15/70 = 21.4%. White has a pass, but it is a very slim pass instead of the monster pass it would be for money. If Blue's position were slightly weaker (say 3 checkers on each of the ace and two points so 1-1 isn't effective for Blue), that would be sufficient to give White a take.

The odds can change considerably when the last roll of the game (or the equivilant) is reached. The key is that, while the trailer will make the automatic redouble, the leader is not required to take! The cube is frozen, and will stay there. For example:

 26 ``` ``` White 27 point match Blue 5

A quick count shows that Blue has 23 rolls which get him off and 13 which don't. Obviously White has a huge money take, with about 36.1% winning chances. Since White is behind in the match, it would appear that White would have an even easier take at the match score.

Appearances can be deceiving. The key is that the cube is frozen on 2 for all practical intents and purposes. White will never get to make a meaningful redouble to 4, since Blue has an automatic pass which doesn't cost any equity. If we look at the odds on White's take:

White passes: Behind 6-2 Crawford (1 away, 5 away), 15% equity.
White takes and wins: Behind 5-4, 40% equity.
White takes and loses: Loses match, 0% equity.

White is risking 15% equity in order to gain 25%. He is getting much worse than 2 to 1 odds, compared to the 3 to 1 odds on a money take. In fact, his takepoint under these circumstances is 37.5%. Since he only has 36.1% winning chances, the surprising conclusion is that his correct action is to pass the double.

This is quite a remarkable situation. At identical scores, we saw in the previous position that Blue had a borderline take/pass with only around 21% winning chances, while here he has a proper pass with over 36% winning chances. It all depended upon whether or not the cube was frozen after the double.

Since such unusual skewing is involved, one would think that there might be positions where it is proper to double as a clear underdog, even when ahead in the match. That is correct. Consider the following:

 28 ``` ``` White 27 point match Blue 5

Blue gets off with 14 out of his 36 rolls. This makes him quite an underdog. Can it possibly be correct for him to double? Let's see.

Blue doesn't double and wins: Ahead 6-2 Crawford (1 away, 5 away), 85% equity.
Blue doubles and wins: Wins match, 100% equity.
Blue doesn't double and loses: Ahead 5-3, 68% equity.
Blue doubles and loses: Ahead 5-4, 60% equity.

Blue is risking 8% (68% - 60%) in order to gain 15% (100% - 85%). He is getting nearly 2 to 1 odds on his double. He only needs to win 34.8% of the time to justify doubling. With 14 out of 36 rolls to get off, Blue's actual winning chances are 38.9%. Thus, doubling is very clear.

The concept of the frozen cube exists for middle-game positions also. For example, suppose you are behind 8 away, 3 away. Your opponent will be cautious about doubling, of course. However, he still may have some effective cubes, particularly in straight races. He can use the full 2 points and some of the 4 points (if you should take and recube), so with a decent advantage he may well have a double.

Now, let's suppose the same score, but that you have already doubled him to 2 and he has taken. What will his cube strategy be? If he ever redoubles you only need 6% winning chances to take (since you can redouble to 8 for the match), because that would be your match equity behind 8 away, 1 away. Your opponent will never redouble if there is any contact, and he won't double in a straight race until he is virtually gin. Thus, by your initial double you didn't really give him the cube. In fact, you took it away from him, since there would be positions where he would double from 1 to 2 but not from 2 to 4. At this match score, you don't need much of an excuse to double. Any advantage combined with a couple of market losers should be sufficient. Giving him the cube doesn't cost you anything, because he can't make use of it the way he could if the cube were in the center.

Let's see how this concept might be used in a real live position.

 6984 ``` ``` White 811 point match Blue 3

Blue has 11 aces which virtually claim the game, but if he misses he will be well behind in the race. Assuming Blue rolls about 9 pips with his miss, that will put him six pips behind. The crossover situation make Blue's race even worse. It looks likely that if Blue doesn't hit White will have a quite efficient cube (even at the match score), and Blue probably will have a take. Let's see about what Blue's takepoint is:

Assuming Blue does take, he will be very quick on the trigger with the recube. Let's suppose that Blue makes sure he never loses his market (which is close to his proper strategy anyway), so whenever Blue wins the game he will win 4 points. Of course this aggressive strategy will cost Blue some 4-point losses also. Let's estimate that 1/3 of Blue's losses will be with the cube on 4 due to this agressive redoubling.

Blue passes: Behind 9-3 (8 away, 2 away), 12% equity
Blue takes and wins: Behind 8-7 (4 away, 3 away), 41% equity (remember, Blue always wins 4 points when he wins the game).
Blue takes and loses: By our assumptions he will lose 2 points and be behind 10-3 (8 away, 1 away) for 6% equity 2/3 of the time, and he will lose 4 points and lose the match for 0% equity 1/3 of the time. This comes out to a weighted average equity of 4%.

Therefore, Blue is risking 8% (12% - 4%) in order to gain 29% (41% - 29%). This comes to 21.6% winning chances Blue needs. I would estimate that Blue will be slightly better than this on most of his missing rolls, so the likely cube action if Blue doesn't double and misses the shot is double/take.

Suppose Blue turns the cube! If he hits, obviously he is a happy camper. What if he misses? As we have seen, White won't be close to a recube, since Blue's takepoint is around 6%. White will virtually never redouble.

Thus, for all practical intents and purposes, if Blue misses the shot the cube will wind up on 2 and the position played out whether or not Blue doubles. Given that, Blue might as well double. The double will gain tremendously if he hits the shot, while it costs very little if he misses. By turning the cube Blue freezes it, but if he doesn't turn it White has full access.

How far can this concept be carried. Quite far if the circumstances are right. Jake Jacobs told me about the following situation. First, a few preliminary cube decisions:

 26 ``` ``` White 37 point match Blue 4

Offhand, double and take appear to be the proper cube actions. Blue has 23 rolls which get off. Let's check it out.

From White's point of view if Blue doubles:

White passes: Behind 5-3 (4 away, 2 away), 32% equity.
White takes and wins: Ahead 5-4 (3 away, 2 away), 60% equity.
White takes and loses: Behind 6-3 Crawford (4 away, 1 away), 17% equity.

White is risking 15% in order to gain 28%, which comes to a drop point of 34.8%. He wins 13/36 of the time, which is 36.4%. Therefore White has a take, although it is surprisingly close.

From Blue's point of view (although we can be pretty sure of the answer from the closeness of the take).

Blue doesn't double and wins: Ahead 5-3 (4 away, 2 away), 68% equity.
Blue doubles and wins: Ahead 6-3 Crawford (4 away, 1 away), 83% equity.
Blue doesn't double and loses: Tied at 4-4, 50% equity.
Blue doubles and loses: Behind 5-4 (3 away, 2 away), 40% equity.

Blue is risking 10% to gain 15%. Since he is a favorite in the position and there is no recube involved, his double is very clear.

Now, let's keep the same score and position, but change the cube situation:

 26 ``` ``` White 37 point match Blue 4

We can see at a glance that if Blue doubles, White has a trivial take. If White passes he is behind 6-3 (1 away, 4 away) with 17% equity, while if he takes it is for the match. White's winning chances are 36.4%, so the pass/take decision isn't close. But what about the double?

Blue doesn't double and wins: Ahead 6-3 Crawford (1 away, 4 away), 83% equity.
Blue doubles and wins: Wins match, 100% equity.
Blue doesn't double and loses: Behind 5-4 (2 away, 3 away), 40% equity.
Blue doubles and loses: Loses match, 0% equity.

By doubling, Blue is risking 40% (40 - 0) in order to gain 17% (100 - 83), so he is getting worse than 2 to 1 odds on the double. Since Blue fails to get off more than 1/3 of the time, it is clear that Blue should not double.

With these two results in mind, we are now prepared to examine Jake's position.

 65 ``` ``` White 47 point match Blue 3

Can it possibly be right for Blue to double? He needs 2-2 or better to get off this roll, and if he doesn't roll them he is a sizable underdog. Blue is only behind one point in the match, so the match score doesn't seem too relevant. If Blue were to double for money, this would be a monster beaver.

Let's look at the problem a little more objectively. Obviously if Blue does roll 2-2 or better he will be happy he doubled. Suppose Blue doesn't roll 2-2 or better. Then:

1) If Blue didn't double, White would then double and Blue would take. White would be rolling for a 5-1 with the cube on 2.
2) If Blue did double, White would then not have a redouble. Again, White would be rolling for a 5-1 with the cube on 2.

Therefore, the value of the cube going into White's roll (assuming Blue doesn't roll 2-2 or better) will be the same regardless of whether or not Blue doubles now. The only difference will be who owns the 2-cube, but since White's next roll will be the last roll of the game cube ownership doesn't make any difference. The startling conclusion is that it is clear for Blue to turn the cube. He has everything to gain and nothing to lose. Doubling is a heads Blue wins and tails Blue breaks even proposition. By doubling Blue froze the cube for the following roll, which is all that mattered.

The concept of the frozen cube comes up more often than one might imagine in match play. It is important to look ahead and see what the future ramifications of doubling or not doubling are. Sometimes the conclusions can be very unintuitive.