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### RC squared

This deal was played in a Junior practice match for which I was giving live, online commentary. After South’s 1NT, he ended in 3NT. (North investigated a possible eight-card major fit, but found none.) The ♦J was led in this layout.

East encouraged and South took the ♦K. Playing on clubs is conceding the contract (the defense will have too many diamond winners). Declarer needs to take nine tricks without losing the lead. That means each major needs to produce four tricks.

In spades, you can hope for either a 3–3 break, or that the J–10 cooperate. In the heart suit, there is a two-way guess for the queen. If the suit splits 3–3, it is just a pure 50–50 guess. What if they are 4–2? Because it is easier to pick up ♥Q–x–x–x with East (you can cash the king first and lead the 10), let’s say you lead a heart to the king at trick two, all following low.

Do you agree with trying the hearts first? Decide before you read on.

You continue with the ♥10. RHO thinks for a moment and covers with the queen. You play the ace, and left-hand opponent follows with the ♥9. You are left with ♥J 7 opposite ♥2. The opponents have ♥8 x. You can lay down the jack, hoping LHO started with ♥9 8 x, but “Restricted Choice” says that ♥9 x was more likely.

The principle goes like this: With ♥9 8 x, you might have seen the 8 on the second round – West would have had a choice. When he played the 9, assume his choice was restricted; he played it because he had to. Mathematicians would prefer that I explain the theory as just plain odds: that exactly ♥9 8 6 opposite ♥Q 4 3 is only one holding and statistically not as likely. Are you still with me?

Say you want to go back to dummy to finesse against the ♥8, which assumes you carefully watched the spot cards and can calculate the odds. You start with the ♠A K. All follow low, but on the second round, RHO follows with the 10. You play a third round, and when LHO plays low, do you finesse the 9, or play the queen to try to drop the jack? Here we go again. Using all the same Restricted Choice analysis (or probabilities), the odds favor the finesse. Let’s look at the Real Deal:

After the ♦K, declarer can indeed take his eight major-suit tricks if he relies twice on Restricted Choice. A heart to the king, followed by the ♥10 covered (East’s best play here), drops the 9. Now, the ♠A K drop East’s 10. A low spade goes to the 9. After the ♠Q, a low heart is led to finesse the 7. Pretty fancy footwork and use of those spot cards.

Remember the question about spades first or hearts first? If you played spades first and then laid down the ♥K and ♥10 covered, there would be no way back to dummy to finesse to the ♥7. If declarer did play spades first, he could recover by starting hearts with the ♥10 without cashing the king first, possibly losing to a singleton queen with West.

This deal was really for the mathematicians and Restricted Choice fans out there, which is why I named it RC squared