Boxing and the Horizon Principle

 A hand came up recently which I thought was a good example of when (and how) to use the ideas of boxing and the horizon principle. Let's define them.

Whenever you make a limited bid, you have "boxed" your hand. In other words, you have to a greater or lesser degree reduced the number of possible hands that you can hold. No subsequent bid can get you out of the box. So, if an ace was hiding behind one of the other cards when you made your earlier bid, no amount of persuasion can convince your partner that you have that ace. You will have to make a unilateral bid if you feel that it is necessary. [I covered this topic in one of my first blogs: The No-undo principle]

Similarly, if both partners have boxed their hands, then certain contracts are no longer "on the horizon."

For example, you open 1♠️ and partner responds 2♠️.  If you are playing a strong club system, you have "boxed" your own hand to somewhere between 11 and 16 points. Partner's hand has between 5 and 10 points. The most you can have between you is 26 points, but this would only occur when both partners are balanced. Slam is not on the horizon. Both partners know this, so any bid that you make now is a game try and cannot be a slam try.

If you are playing a standard bidding system where you are limited to, say, 20 points, it's conceivable though unlikely that you still might have slam. So, a bid of a new suit (a game try of some sort) might turn out to be an advance control-showing bid in search of slam if partner accepts what he sees as a game try (following the related principle of "game before slam"). 

When partner is unlimited, certain contracts, such as a small slam, maybe on your horizon but, from partner's point of view--when you have boxed your hand--those same contracts may not be on his horizon.

Enough discussion. Let's look at the hand (matchpoints):

♠️K74 ♥️AK984 ♦️AQ82 ♣️8

A nice hand for sure. You deal and open 1♥️ (nobody is vulnerable). LHO gets in there with 1♠️. Partner, predictably, makes a negative double. RHO passes and it's up to you.

Partner is unlimited (he should have at least 7-8 points) and you have boxed your hand somewhat by your failure to open 2♣️ and the fact that you opened in first seat. So, 11 to 20 points or thereabouts and at least five hearts. You could easily have a slam here, although presumably not in hearts. What about diamonds? 

You are about to re-box your hand. If you bid 2♦️, you will have effectively boxed your hand to something like 11 to 16 points, with at least nine cards in the red suits. You may still have visions of slam, but what about partner? He will need substantially more than a minimum to entertain slam now. From his point of view, 2♦️ will likely take slam off the horizon.

What about 3♦️? You will be refining your box to something like 16 to 20 points with the same red suit cards. If partner has a fit for diamonds (as the double suggests he might) and something like 12 or more points of his own, slam may still be on the horizon for both partners.

You decide to rebid 2♦️ and partner cue-bids 2♠️. Partner's hand now has a new box: at least 11 points and, probably (but not definitely), fewer than three hearts, as he would likely have bid 2♠️ immediately with three hearts and 10-plus points.

It looks like we have a diamond fit (with four or five spades, partner may have opted to trap-pass so partner likely has eight minor suit cards). Possible contracts are 2NT, 3♦️, 3NT, 5♦️, 6♦️, 6NT. Partner's sequence is consistent with all of those contracts. At matchpoints, we would tend to favor 2NT over 3♦️ and 3NT over 5♦️.

Is partner's bid forcing? Obviously. But forcing to what? There are several opinions on this, but let's look at the hand from partner's point of view. With our hand boxed into 11-16 points, partner will need something like 16 points for slam to be on the horizon. What about game vs. part-score? If partner only has 11-12 points, he will want to know if we are at the low end or the high end of our box.

The two bids then that could legitimately be passed by partner are 2NT and 3♦️. We decide to bid 2NT and partner passes. We make twelve tricks in notrump for a somewhat embarrassing +240.

Here is the actual hand:

Sacrificing for Dummies

It's ten years since I last wrote something here on the Law of Total Tricks. My goal this time is to come up with something really simple to remember when considering a save.

My thoughts on this were prompted by a recent hand:

My overcall of 1♦️opposite a passed hand was not a thing of beauty, I'll admit. But, I'm loath to make a sub-standard takeout double when our side is probably out-gunned. South's 3♣️ was described as "weak." What should West do here? I think a responsive double might work out best. If partner has four spades, we'll find it. If not, we'll likely be playing 3♦️ which can't be all bad. At the table(s), many pairs played 3♠️ either by East or West which mostly made given that N/S didn't find the double-dummy lead of ♦️K or ♦️T.

Over partner's 3♦️, North made a crazy leap to 5♣️. I could have been the hero by doubling (+300) but "knowing" that partner cannot bid higher (see Passed Hands may make only one Free Bid), I thought I'd allow him to pass or double, as appropriate. 5♦️ was completely unexpected and, as I'll show below, very unlikely to be the winning action. It's almost never right to take the last guess! And, it's OK to save with the ace of the enemy suit because it's likely to be of value at defense and offense. But kings, queens and jacks in their suit should be a red flag as they may be useful only on defense.

In fact, along double-dummy lines, N/S can make 3♣️, 2♥️, or 2NT. E/W can make 2♠️ or 3♦️.  21 total trumps. 18 total tricks. I would suggest that the shortfall in total tricks is due to the lack of useful shortness: each side has the (short) top honors in the other side's trump suit.

For the remainder of this article, we will consider entirely hypothetical situations. The following table shows the number of total tricks to make a sacrifice profitable at matchpoints, according to the levels of bidding involved:

Levels	Favorable	Equal Red	Equal White	Unfavorable
4/4	17	18	18	19
5/4	18	19	19	20
5/5	19	20	20	21
6/5	20	21	21	22
6♠️/6♥️	18	19	20	21
6/6m	19	20	20	21
7/6M	19	20	21	22
7♣️/6♦️	20	21	21	22
7/7	18	19	20	21

Note that it is assumed in all cases that the opposing contract is actually making. The requisite number of total tricks may be available but if they are distributed too evenly, the save will be a phantom.

Let's remind ourselves that the most common number of total tricks is 17. If the opponents bid 4♥️, and we have a good spade fit and are at favorable vulnerability, we can consider saving in 4♠️. How do we know if there will be 17 (or more) total tricks? The bidding will give clues as to the fits around the table. But, the simple number of tricks in each direction isn't really sufficient information (see "I Fought the Law"). A trick total of 17 will likely involve some shortness (singleton) somewhere at least. Do you have it? Did partner show shortness? Did one of the opponents? If so, you may try it. Otherwise, you might want to hold back until you think there are 18 total trumps.

There are several likely outcomes in 4♠️. Any time 4♥️ was not making, we will get a poor score, unless 4♠️ makes. Even if they didn't double, -100 instead of +100 (or 200 if we had doubled) will not usually score well.

But let's assume that 4♥️ was indeed making. If they didn't double 4♠️, we are guaranteed a good result. If they do double, as long as our estimate of 17 total tricks was accurate, we should be fine. Except when they could have made 650 and we are down four for -800. That's an all-too-common disaster. That's why, even in this situation, you really would like to have 18 total tricks.

And this is, according to the chart above, the most advantageous situation for taking a sacrifice (shown in green in the table).

There are three other situations where we might seriously consider a sacrifice (yellow rows in the table):

• at the 5-level over their 4-level game;
• 6♠️ over 6♥️;
• 7 over 7.

In each of these cases, we require 18 total tricks (not an uncommon situation) and of course favorable vulnerability. Each worsening of the vulnerability situation (see table) requires one additional total trick. Except in goulash-type hands, deals with 20 or more total tricks are rare. Also note that in the second and third of these situations, the all-white and unfavorable situations are particularly dangerous because they require 21 and 22 total tricks respectively (in each case, one more than the 5/4 sacrifices).

From the red rows in the table, we can also see that we should never (well, hardly ever) even contemplate a sacrifice at the 6-level over a game contract, or 7♣️ over 6♦️, as these require at least 20 total tricks. Don't even think about these when not at favorable.

The other situations (amber in the table -- 5 over 5, 6 over 6 minor, 7 over 6 major) should generally be avoided too. To consider any of these at equal vulnerability--especially the last one when all white--is, well, just madness.