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It is vital to play 3M as a splinter, to avoid this sort of guess. With typical 3145 or 3154 shape, 1♦-3♥ suggests 13-15 hcp. Here we stretch slightly; the hand is quite strong at diamonds, of course, and we hope that the sixth diamond will be enough for 3NT if partner chooses that.

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Craig: You called rebidding 1NT a big lie about shape. I responded, not for the first time in reply to you, that rebidding 1NT with a singleton in partner's suit is routine expert practice. Now you gave actual reasons for preferring to promise at least a doubleton in partner's suit for a 1NT rebid. Fine – next time this topic comes up, make those arguments rather than calling a method you disagree with a lie.

I prefer being able to rebid 1NT freely because it avoids playing 5-1, 4-2, or 3-3 fits; because I am not convinced a 5-2 major fit is, on average, substantially better than 1NT; and because it allows one to open freely based on hand evaluation, not rebid considerations. Your mileage may vary.

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Even the strength is not really an issue if you are willing to pass awkward 11 counts as opener: x A9xx Kxx A10xxx looks like a normal opening to me, but if you must make the same rebid on that and x AQ9x KQx A10xxx the range is somewhat wide. Responder must either pass or otherwise go low with an 11 count, potentially missing a 26 hcp 3NT, or push on to 2NT, where a combined 22 count will often fail.

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Once again, Craig, since a majority of experts polled by The Bridge World chose to allow a rebid of 1NT with a singleton in partner's suit, 1NT is no lie at all regarding shape, for any partnership that adopts that style. That leaves the strength, which is not unexpected for a 1NT rebid but not easy to reveal, and the suit orientation, Bidding minors does nothing to clarify the strength, it simply prevents us from playing in notrump.

I think any partnership should agree on one of three approaches for opener's rebid, not try to wing it at the table:

(1) Raise responder on three trumps and a ruffing value. Rebid 1NT possibly with a singleton in responder's suit. Avoid rebidding a five card minor when a 1NT rebid is available.

(2) Require four trumps to raise responder's one of a suit. Require two or three cards in responder's suit to rebid notrump. Open 1♦ when 4-4 in the minors. Rebid a five card minor as needed.

(3) Modify either of the above by tending to open 1♦ with 4-5 in the minors and less than 16 hcp.

Using #3 does not take care of all awkward hands: x Kxxx AQx K10xxx must still rebid either 2♣ or 1NT after partner's expected 1♠ response; opening 1♦ or 1♥ (in a non-Precision, 5 card major context) are absurd distortions.

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Opener's pass suggests balanced 22-24, so we have enough for game, but opener does not have five spades or a penalty double of hearts and did not choose to bid 2NT. That appears to leave game in a minor, so I'll start bidding my minors. First, though, I'd better ask what the double showed – lead directing? Clubs? Majors?

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I played Precision for several years with several partners. At no time would I have given a thought to opening such junk. System need not trump judgement – if you downgrade a hand like this, you upgrade others. If this IS an opening bid in first or second seat, I assume Kxx K10xx xx K9xx would be also.

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Stayman followed by 2♠ over partner's likely 2♥ shows 5-5+, spades and a minor, about 6-8 points. If that hasn't been discussed or you've defined that sequence as something else, I would settle for transferring to diamonds and passing. There are 19 working points out of 35, so partner rates to have about 8, but the variance is very high – anything from grand slam to a part score.

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I don't buy either of your objections, Magnus. I've been doing combinatorics since I was 12. We aren't calculating by some card-by-card method. There are 13C4 possible 4 card suits; there are 10C3 four card suits which include the four and do not include the 3 or 2. So 10C3/13C4 is the probability any four card suit will be a hit. The probabilities each suit will be a hit are independent, so we can multiply those together to get the probability that a specific shape, say 4432, will have four hits. Finally, we know the probability of getting any particular shape so we can multiply by that to get the combined probability of getting four hits on 4432 shape.

As for the Monte Carlo sim, I assume the method was simply to generate random hands and count the hits. There could be problems with generating random deals and counting hits in all four hands, as those are clearly not independent, but it might not affect the results too much.

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This is very strange. With three of us computing the same overall result, the 1 in 1777 appears to be the correct theoretical odds. If all bridge players were hunting for hits, we might dismiss Magnus as being on the unlucky side of the bell curve. But as far as we know, he's the only one who has been doing this. So that leaves some flaw in our calculations, some flaw in his ability to recognize cases, some flaw in randomness (I would assume in both machine and hand-dealt hands), or inconceivable bad luck. I predict Magnus will report a 4-hitter within the year, unless someone points out a flaw in our computations.

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In Bridge World Standard, as well as every one of my partnerships, rebidding 1NT is allowed with a singleton in partner's suit. Also, partner is aware that the strength may not be quite in the 12-14 range, but in practice can't afford to assume 15 since you might well have 11. So, for me, 1♣ then 1NT avoids distorting the shape, but the values are still a problem. If it weren't for that 10, 2 9's and good five card suit, I'd take the low road. As is I'll risk the 5-1 fit in order to invite the most likely game. 1♦ followed by 2♣ does nothing to clarify the strength.

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While confirming that partner has diamonds would be useful, I can't see how to then set hearts as trumps and explore for slam. Bidding 3♠ seems like it ought to show slam interest in hearts, as we could bid 2NT and then 3♠ to force in the minor.

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For what it's worth, I came up with the same result for four hits, .0056 or about 1 hand in 1777. I started by computing the odds of a hit for each suit length, then typed out the patterns in four cells (example: 5 4 3 1 ), then looked up the frequency of a hit for each of those lengths, multiplied those four together and multiplied by the frequency of the shape (cribbed from the Bridge Encyclopedia.) Totaling yielded the .0056 overall chance of a hand having hits in all four suits.

So, either Magnus has been extremely unlucky, or has overlooked many 4-hit wonders, or both Wayne and I have both made some major mistake in our approach.

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Wayne, let me see if I understand what your table shows. I see distributions, beginning with 4333. Next comes the frequency of that distribution, .105,,, . What comes next? The probability of 4, 3, 2, 1, 0 hits given that distribution? Then do you total things in any way? Oh, that must be the last line, which has a 1 instead of a shape. So your .00056 is nearly three times the .0002 given in the article?

Paul Hightower

Paul Hightower

I prefer being able to rebid 1NT freely because it avoids playing 5-1, 4-2, or 3-3 fits; because I am not convinced a 5-2 major fit is, on average, substantially better than 1NT; and because it allows one to open freely based on hand evaluation, not rebid considerations. Your mileage may vary.

Paul Hightower

Paul Hightower

I think any partnership should agree on one of three approaches for opener's rebid, not try to wing it at the table:

(1) Raise responder on three trumps and a ruffing value. Rebid 1NT possibly with a singleton in responder's suit. Avoid rebidding a five card minor when a 1NT rebid is available.

(2) Require four trumps to raise responder's one of a suit. Require two or three cards in responder's suit to rebid notrump. Open 1♦ when 4-4 in the minors. Rebid a five card minor as needed.

(3) Modify either of the above by tending to open 1♦ with 4-5 in the minors and less than 16 hcp.

Using #3 does not take care of all awkward hands: x Kxxx AQx K10xxx must still rebid either 2♣ or 1NT after partner's expected 1♠ response; opening 1♦ or 1♥ (in a non-Precision, 5 card major context) are absurd distortions.

Paul Hightower

Paul Hightower

Paul Hightower

Paul Hightower

Paul Hightower

As for the Monte Carlo sim, I assume the method was simply to generate random hands and count the hits. There could be problems with generating random deals and counting hits in all four hands, as those are clearly not independent, but it might not affect the results too much.

Paul Hightower

Paul Hightower

Paul Hightower

Paul Hightower

Paul Hightower

Paul Hightower

So, either Magnus has been extremely unlucky, or has overlooked many 4-hit wonders, or both Wayne and I have both made some major mistake in our approach.

Paul Hightower

Paul Hightower

Paul Hightower

Paul Hightower

Paul Hightower