# The Schnapsen Log

## A Multitude of Options (solution)

#### Martin Tompa

What is the first thing you should consider when on lead at the last trick before the stock is exhausted? That’s right, you should first consider closing the stock, because having to follow suit cuts down on the number of plays you have to consider. And it freezes your opponent’s score, which is good in today’s deal because Peter is still below 33 trick points. In today’s deal it also means you would play out the remaining tricks with no trump, which simplifies things further.

If you close the stock, you need to collect more than just your two aces, which will probably only bring your trick point total up to 54. There are two interesting suits, spades and clubs, either of which you would prefer to see Peter open up for you. This suggests an elimination play. You will first cash ♥A in order to eliminate Peter’s safe exit card ♥J (if he holds it). Then you throw Peter in with one of the black suits, forcing him to open up the other one. But which black suit?

If a club is in the stock, you can throw Peter in with a club. Leading ♣T to throw him in will put Peter on lead in a position something like this:

Peter:(41 points)

♠ TQ

♥ —

♣ K

♦ —

You:(40 points)

♠ AK

♥ —

♣ Q

♦ —

Peter can cash his last club, but then you will take the remaining tricks and reach 68 trick points.

The problem is that this throw-in will fail if either spade is in the stock. For instance, if ♠Q is in the stock, here is what it will look like when Peter gains the lead:

Peter:(41 points)

♠ T

♥ —

♣ KJ

♦ —

You:(40 points)

♠ AK

♥ —

♣ Q

♦ —

If Peter cashes his clubs, your only remaining trick is ♠A, which will not give you enough trick points.

In fact, if ♣K is in the stock, throwing Peter in by leading ♣T will also fail. This is what it would look like when Peter gains the lead:

Peter:(41 points)

♠ TQ

♥ —

♣ J

♦ —

You:(40 points)

♠ AK

♥ —

♣ Q

♦ —

From this position, Peter can exit safely with ♣J and you will have to open up the spades yourself. If you are going to throw Peter in with a club, you should lead ♣Q rather than ♣T for the throw-in. That way, if ♣K is in the stock, this is what it would look like when Peter gains the lead:

Peter:(34 points)

♠ TQ

♥ —

♣ J

♦ —

You:(40 points)

♠ AK

♥ —

♣ T

♦ —

Now if Peter exits with ♣J, you will have 52 trick points, and cashing ♠A will bring you successfully to 66.

This isn’t too bad: throwing Peter in by leading ♣Q will fail only if either spade happens to be the last card in the stock, which happens with probability 1/3.

Let’s see what happens if you throw Peter in with a spade instead. In this case, the elimination play is to cash ♥A and ♠A and then exit with ♠K. If a club is in the stock, Peter has to follow suit to each of your first three leads, so the position will look something like this when you are about to throw him in:

Peter:(20 points)

♠ T

♥ —

♣ AK

♦ —

You:(54 points)

♠ K

♥ —

♣ TQ

♦ —

When you throw him in with a spade, he can cash ♣A, but then has to let you win ♣T for 68 trick points.

What if something other than a club is in the stock? If ♥J is in the stock, the only safe discard for Peter on your ♥A lead is ♣J, in which case the spade throw-in works exactly as before. If ♠Q is in the stock, Peter’s ♠T will fall on your ♠A, and cashing your established ♠K will give you enough trick points. The only hitch is when ♠T is the last face-down card in the stock. In that case, you will cash your ♥A and ♠AK, but this will bring you only to 60 trick points, and Peter will take the last two club tricks. So throwing Peter in with a spade fails with probability 1/6, the probability that ♠T is in the stock. This makes it a better play than throwing him in with a club, which we saw fails with probability 1/3.

Since Peter is only 1 game point from winning the game, it doesn’t make sense to calculate your expected gain if you close the stock and throw Peter in with a spade. A better summary is that you will lose the whole game with probability 1/6, and gain 2 game points with probability 5/6.

Next let’s investigate what happens if you don’t close the stock.

© 2013 Martin Tompa. All rights reserved.