# The Schnapsen Log

## How Much Help Can You Expect? (solution)

#### Martin Tompa

Because of your trump control and nontrump strength, closing the stock is definitely worth considering. The problem is the whereabouts of that nasty ♣A. If it is still in the stock, then you will collect all five tricks after closing. You have 45 trick points in your hand and 13 in your tricks, for a total of 58. Hans must contribute much more than the 8 points you need in order to reach 66. So this is the easy case.

But it is more likely that ♣A is in Hans’s hand: since there are 8
cards you haven’t seen and Hans is holding 5 of them, the probability
is 5/8 that ♣A is among his 5. In this case, even if you run your
other four tricks first, Hans will probably have no difficulty
deciding to retain ♣A for last, given how many clubs are still
outstanding. This means you will only add 35 trick points from your
hand to the 13 already in your tricks, for a total of 48. How many
trick points can you expect Hans to contribute to your four tricks?
You have seen *T* = 19+13+45+2 = 79 trick points so far in all tricks,
your hand, and ♠J on the table. There are *c* = 8 cards you haven’t
yet seen. This means Hans is expected to hold 5(120−79)/8 ≈ 25 trick
points in hand. Assuming one of Hans’s cards is ♣A, you can expect
to capture 25−11 = 14 trick points from Hans, for a total of 48+14 =
62. Not enough to close.

Actually, the reasoning in the previous paragraph is flawed. The
reason is that we are assuming Hans holds ♣A, yet we ignored that
fact when we computed the value of five average cards in his hand,
averaging over arrangements of the cards including those where ♣A
wasn’t in Hans’s hand. But we can correct the flaw using similar
ideas. Just assume that ♣A is one of the cards you have seen, as
though Hans showed it to you in his hand. In that case, you have seen
*T* = 79+11 = 90 trick points, and there are only *c* = 7 cards you
haven’t yet seen. This means that you expect Hans’s other 4 cards to
total 4(120−90)/7 ≈ 17 trick points, not 14 trick points as we
calculated in the previous paragraph. But this still only brings your
expected total to 48+17 = 65 trick points, still not quite enough to
close with much confidence.

Well then, if you’re not going to close the stock, you will have to choose a card to lead. You would like to keep Hans under 33 trick points. With your powerful trump suit, leading ♠A seems best. You will still retain trump control and the lead, and your intention is to close the stock next trick if you draw something advantageous from the stock. The cards you could draw that make closing the stock an easy decision are ♠K, ♥A, ♣A, ♣K, and ♦K, any of which gives you an extra trick. Let’s examine the position if you draw ♦K, the weakest of these cards. Supposing that Hans discarded ♣J on your ♠A, here is the position:

Unseen cards:

♠ K

♥ AJ

♣ AKQ

♦ —

You:(26 points)

♠ TQ

♥ —

♣ T

♦ AK

If you now close the stock, your own spades and diamonds will add 28 trick points to your 26 already collected, for a total of 54. Hans must contribute at least 13 more points to these four tricks, for a total of 67.

© 2012 Martin Tompa. All rights reserved.