It is time to finish analyzing completely pure hands – hands where both sides hold all the honors in their two long suits. The length of those two suits, combined, is called the second fit number, or SF, and plays a major role in the total tricks available. As I noted in Part 3 of the Theory of Total Tricks, for completely pure hands,
Here is the key Table 2 again:
Table 2: Expected Defensive Winners for Completely Pure Hands
SF |
Length of Longest (Solid) Suit |
Length of Next Longest (Solid) Suit |
Defensive Winners |
14 |
7 |
7 |
4.36 |
14 |
8 |
6 |
4.18 |
15 |
8 |
7 |
3.82 |
15 |
9 |
6 |
3.85 |
16 |
8 |
8 |
3.28 |
16 |
9 |
7 |
3.49 |
16 |
10 |
6 |
3.32 |
17 |
9 |
8 |
2.95 |
17 |
10 |
7 |
2.96 |
17 |
11 |
6 |
3.06 |
18 |
9 |
9 |
2.62 |
18 |
10 |
8 |
2.62 |
18 |
11 |
7 |
2.70 |
19 |
10 |
9 |
2.09 |
19 |
11 |
8 |
2.16 |
Looking at this, we see that, when SF=17, there will be around 20 total tricks available. For SF=18, the trick total climbs to around 20 ¾ and close to 21 ¾ for SF=19.
The next case, SF=16, is much more complicated. First off, the loser count is not consistent – we see around 3 ¼ expected defensive tricks when our suits are 8-8 or 10-6, but 3 ½ winners to cash for the 9-7 case. Interesting. This is also the first time we’ve run into a potential eight card trump fit. As I pointed out in Part II, seven and eight card trump fits are awkward – maybe there are only three losers, but that doesn’t mean ten winners.
Marginal trump fits carry additional handling charges. These don’t quantify easily, but it appears reasonable to add these handling charge losers:
· For an eight card trump fit – half an extra loser
· For a seven card trump fit – a full extra loser.
With this in mind, I will work out, in complete detail, the 8-8 case. First off, we can cash, on average, 3.28 tricks. Since our SF=16, our opponents' SF is also 16. Thus, they too will have either 8-8, 9-7, or 10-6 fits[1], so they can cash against us, either 3.28, 3.49, or 3.32 winners. This averages to 3.36 winners. As to handling charges, we have an eight card fit, which generates an extra half loser. The opponents seem to have an eight card fit in one of the three cases, so I’ll add in another 1/6 of a handling charge loser for them. Adding these up, we get
3.28 + 3.36 +.5 + .17 = 7.31
Our handling charge estimates are pretty loose, so I will round this off to the nearest quarter of a trick, and conclude that there will be around 18 ¾ total tricks available in the 8-8 case. A similar analysis of the 9-7 and 10-6 cases gives us this:
Table 3: SF = 16, Total Tricks for Completely Pure Hands
Lengths of Longest Suits |
Total Tricks, to the Nearest Quarter Trick |
8-8 |
18 ¾ |
9-7 |
19 |
10-6 |
19 ¼ |
So we get SF+3, or 19, as expected, but adjusted up or down depending on our trump length. Indeed, if we can tell from the auction, or our own short suit holdings, the length of their trump fit, we can adjust even more, with total tricks varying from 18½ to 19½ depending on total trumps.
Larry Cohen will be pleased. Total trumps really matter here, but the actual impact of an extra trump is about a quarter of a trick, not a full trick.
[1] I have intentionally ignored one case, where a side has an 11 card fit, and three 5-card side suits.
This is rare, but not impossible.
Similar calculations finish the analysis, and give us
Table 4: Expected Number of Total Tricks for Completely Pure Hands
SF |
Length of Longest (Solid) Suit |
Length of Next Longest (Solid) Suit |
Approximate Total Tricks, to the Nearest Quarter Trick |
14 |
7 |
7 |
15 ¾ |
14 |
8 |
6 |
16 ¼ |
15 |
8 |
7 |
17½ |
15 |
9 |
6 |
18 |
16 |
8 |
8 |
18 ¾ |
16 |
9 |
7 |
19 |
16 |
10 |
6 |
19 ¼ |
17 |
9 |
8 |
20 |
17 |
10 |
7 |
20 |
17 |
11 |
6 |
20 |
18 |
9 |
9 |
20 ¾ |
18 |
10 |
8 |
20 ¾ |
18 |
11 |
7 |
20 ¾ |
19 |
10 |
9 |
21 ¾ |
19 |
11 |
8 |
21 ¾ |
This concludes the analysis of completely pure hands. It is clear, now, that we should estimate total tricks based on the potential defensive winners in our long suits, and losers in our short suits. Such calculations depend primarily on the SF number, and only secondarily on the length of our longest suit. The Law of Total Tricks has it wrong, but works out well anyway, because SF and Total Trumps are highly correlated.
After a short theoretical break in the next installment (Part V of the Theory of Total Tricks), I will move away from pure hands to more typical hands. See you then.
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