Lawyer Chuck Burger is one of the least-known bridge stars of America. He played for 20 years in most major tournaments with Jimmy Cayne. But here, Burger was playing in the quarterfinals of the Grand National Teams with Allan Falk, who was conveniently placed to record the events at their table.

South reached four spades after showing a weak hand with six spades at his first turn, and Falk led a diamond to the four and Burger’s queen. Things looked bleak for the defense, with trumps behaving and clubs well-placed, but Burger found the unpleasant switch to a low heart at trick two, which went to the 10 and queen. So at least Burger had put himself in a position to get two more heart tricks if he could get his partner on lead. Declarer next played the spade king from dummy, and Burger began his campaign of deception when he unblocked the queen.

Now declarer crossed to hand by playing the diamond ace and ruffing a diamond. When he laid down the spade ace, Burger dropped the jack! Now declarer knew he needed trumps to be 3-3. He played for what he thought was his best chance, that of finding Burger with the spade 10, by playing a third trump. Falk took this with his 10 and played a second heart, for one down.

Of course, declarer could have succeeded in the ending by playing on clubs. But he naturally assumed that this was less likely than Burger’s ingenious defense. He will know better next time.

In today’s deal from the Common Game, I assumed the role of innocent bystander, watching as a defender while declarer missed the best line for his contract.

I sat West, and when my partner opened two hearts in third seat, I heard a double to my left; I raised to three hearts to try to make a nuisance of myself. East doubled, denying spades, and South jumped to four spades to end a competently bid auction.

I led a low heart to the ace, won the next heart with the king and exited with the spade eight. Declarer won in hand, looking unhappy, then crossed to the spade 10, played the club ace and called for the club jack. I ruffed and exited with a spade, and now my side was sure to take the 13th trick, for down one.

Let’s look at declarer’s options here. He could not afford to draw trumps before playing on clubs, since the defenders were threatening to run hearts once his trumps ran out. But if declarer played the club king from hand, followed by another club, I could have ruffed and exited in trump for a safe down one.

The winning line was not too far from what declarer actually did. After winning the spade ace at trick three, he should cross to the spade 10 and run the club jack from dummy. If West wins the queen and gives a ruff-sluff, declarer can trump in dummy and cross to hand in diamonds to draw trumps. But if dummy’s club jack holds, declarer can draw trumps and play for the overtrick.

When the opponents pre-empt, they put you on notice that the other suits may not be breaking.

In today’s auction, our declarer judged wisely to balance with a call of two no-trump, suggesting a fairly balanced hand with significant extra values. Sadly, he bid the hand much better than he played it. After a top diamond lead, it took him no time at all to go down in three no-trump because he had ignored the warning signs from the auction. He captured the opening lead and fired out the club ace, and now when clubs went pear-shaped on him, he could not recover. He did his best to set up clubs, but the defenders held him to two clubs, three hearts and two diamonds.

A far better line would have been to win the diamond lead and follow up with a low club from hand toward the nine. If clubs break 2-2 or 3-1, declarer has nine top tricks now. When they do not, East will win the first club and clear diamonds. Declarer can win, overtake the heart king to take one club finesse, then overtake the heart jack to repeat the process. You finish up with only two heart tricks and two diamond winners, but you also have five clubs, and that is enough for your contract.

You have invested one possible club winner and one possible heart trick, but the dividends are wholly worthwhile — even at matchpoints, you might still follow this line.

Against three spades, West leads the heart queen, and South counts the missing high cards. He realizes that East surely has the heart king and therefore West holds the rest of the deck. The trump king and diamond ace are surely working. Declarer can give up one heart trick and eventually trump his last heart with one of dummy’s small trumps, but he needs to avoid losing two club tricks.

South ducks the first trick, wins the next with the heart ace and immediately leads the diamond jack. The idea is to force out the diamond ace and thus establish dummy’s remaining diamonds as winners. When West takes the diamond ace, East follows with a high spot card, warning South of the possible 5-2 break. West continues with the club king. South ducks, and when East discourages, West reverts to diamonds. Now after putting up the diamond king, declarer turns his attention to trumps. The spade king takes the first trump, and West continues with another high diamond spot.

Playing safe, a necessary precaution today, declarer ducks in dummy and ruffs in hand, then draws trumps, ruffs a heart in dummy and finally pitches his club loser on dummy’s diamond queen.

Can you spot the untapped defensive resource, which is considerably easier to find in the postmortem, but maybe one that we should all add to our repertoires? At trick one, East needed to overtake the heart queen, to make sure he could lead clubs through early on in the deal.

In today’s deal, South’s jump to three spades is not forcing. North might pass with a really bad hand. If he does, South would be better off staying out of game. But as it happens, North has more than enough to bid on. Lacking any spade support, North might try for game at no-trump, in which case South would have no reason to overrule his decision. In this case, though, North is correct to raise spades, since a club lead by East gives the defenders five tricks before declarer obtains the lead.

In four spades, South must win the first diamond in his hand, leaving dummy’s ace as a later entry for a heart trick. He next draws one round of trumps, but must then attack hearts by leading to the 10. West signals count, letting East know he should duck the trick.

When dummy’s 10 holds, South can get back to his hand using dummy’s remaining trump. He then draws East’s last trump and runs a few more for good measure. This cannot hurt him and may embarrass the defenders.

Finally, South leads his second heart to dummy’s jack. This time, East wins and shifts to a club honor, letting the defenders take two club tricks. But declarer now has the rest.

If South had drawn all the trumps before leading hearts, he would have been unable to cross back to hand safely after dummy had won the first heart. He would have had to lead clubs. The defenders would take their two club tricks and knock out the diamond king before South had the chance to lead a second heart.

When this hand arose in the European Championships in the 1950s, the British declarer did well to play five diamonds rather than three no-trump. The defense cashed two rounds of hearts and played a club.

Naturally, declarer took this and led a top trump, on which West played the eight and East the three; then declarer took both top spades. When he led the diamond jack, West followed suit. Now South had to decide whether to play for spades to split and diamonds not to behave (when the right play would be to duck in dummy and draw a third round of trumps), or for the diamonds to be 2-2 and spades to be 4-2 (in which case South should overtake the second trump and would then be able to ruff out the spades).

Spades are slightly favored to break. But how likely is it that diamonds break? Imagine West has the diamond 10-9-8 and East the three, compared to that suit splitting 2-2? In abstract, the individual singleton is less likely than any individual 2-2 split, but in addition there are three doubleton honor-pairs where West must follow with his two cards at his first opportunity, and East similarly has to play his three at his first chance, lest you overtake on the second round with impunity.

So it is clearly right to play to overtake the second diamond, as that suit is much more likely to split than spades. The fact that this was the winning line does not prove anything, but at least virtue was rewarded.

All our deals this week have focused on the principle of Restricted Choice; but this concept cannot be considered purely in abstract, as today’s deal from the later stages of a Vanderbilt Trophy match shows.

Declaring four spades, your chances do not seem so great on a devious low diamond lead, but when the diamond king holds the first trick, things look up. Draw two rounds of trumps, ending in hand, and lead a heart to the jack and king. Now East cashes the diamond queen and returns the heart 10, on which West unblocks the queen.

You ruff the last heart in hand, cross to dummy in trumps, and lead a low club, intending to put in the seven. Naturally, East thwarts you by playing the nine, so you try the queen, losing to the king. When West returns a low club, what should you do?

If you have been following this week’s theme, you may conclude that Restricted Choice suggests playing low. The logic for that would be that East is more likely to have J-8-x or J-9-x than 9-8-x. However, there is a much sounder argument for putting in the six, if you remember the earlier play. East can be assumed to be a true expert player; with three low clubs, as opposed to jack-third, he would not have defended this way when on lead earlier in the hand. He would have shifted to a club rather than returning a heart, to break up the impending endplay.

The Monty Hall problem arose on “Let’s Make a Deal.” The contestant was presented with three doors; behind one was a car, and behind the other two were goats. The contestant selected a door, then the quizmaster opened a different door to reveal a goat. Now should the contestant change his selection to the third door or stick with his original choice?

Interestingly, the contestant doubles his odds by switching his choice. He’d turn his car into a goat only if he had correctly selected the car with his first pick — a 1-in-3 chance. But he would win the car if he had originally picked either of the two goats — a 2-in-3 chance. The reason comes back to Restricted Choice; in each of the latter two cases, the host had only one goat left to reveal. However, if the first pick was the car, the host had a choice of goats to show you.

The biggest caveat for Restricted Choice comes when a player contributes a significant honor or spot-card where that play is not forced. Consider today’s slam, where when declarer cashes the heart king, he sees the 10 fall from East. The singleton 10 might seem more likely than J-10 doubleton, since we must reduce the chance of the latter by half — because East might have followed with the jack from that holding.

That is true, but East wasn’t forced to play the 10 from J-10-7, his actual holding in the diagram. Maybe only an expert would be capable of that false-card, but that is a completely different issue.

This week we are addressing the thorny problem of Restricted Choice in bridge. This dictum says that if (and only if) a player had a choice of equal cards to play, then the probability that he had one of those cards singleton should be compared to half the probability that he had both of those cards — because with the doubleton he might have played the other card. Granted, this does assume that he was equally likely to play the queen or jack from queen-jack doubleton, but unless you know to the contrary, you should indeed assume that.

A deal may make the point more clearly than words. Declaring four hearts, you win the diamond lead for fear of a club shift, after which the defenders might maneuver a ruff. You cash the heart ace, dropping the 10 from East. Should you now lead a heart to the nine or to the queen?

As indicated above, it is correct to play East for the doubleton K-10 rather than for the J-10 doubleton. That is because, with the former holding, he had no choice but to play the 10 at his first turn, whereas with the J-10 doubleton he might have played either of those cards. Thus, one should not compare the initial probabilities of each doubleton holding, which are equally likely, but instead assume that the K-10 is twice as likely as J-10 doubleton.

How does that relate to the Monty Hall problem? We will find out tomorrow.