Losing-Trick Count, Quantified up to 1

飞雪无痕

Author: Lou shi xin (Wuhan, China)
English translation: Ren chengfeng (Shang hai, China)
Proofread: computer AI system

Since 1935, foreign bridge masters have researched bridge’Losing Trick Count (LTC) . After being systematically demonstrated in the book “Modern Losing Trick Count” by R·Klinger, it has become a final verdict with far-reaching implications. The author affirms that “the key to success at the bridge table is accurate bidding, and a good bid depends on estimating the card strength by a sensitive method”. and he is of the opinion that using the results of LTC“There will be more successes and fewer mistakes, thus increasing your achievements and fun in bridge”.
The basic principle of LTC: on average, 3 high card points (HCP) can get a winning-trick (WT) . But, in trumps contracts, the WT produced by card pattern value has not been reflected.Therefore, when trumps fit, a hand’s WT depends on both HCP value and card pattern value. With the method of LTC, the partner's the HCP and card pattern can be clearly known through bidding. Thus, count the losing-trick (LT) . Finally,back-count the WT .
However, when applying the LTC method in practice, 《the natural systems》 the range of HCP due to its natural attributes, are described too wide (opening bid with 12~21HCP ect) , whose the interval rangeΔP=9 (other bids in most cases,ΔP=3~9) , and the number of the card pattern cannot be precisely described, then the number value (R) by LTC is an interval value, whose the range of difference value (ΔR) = 3 (ΔR = 1~3 in most cases) . 《the precision systems》 due to its restrictive bidding characteristics, the range of HCP described is relatively smaller, thus ΔP=3~6, ΔR=1~2. Them are an interval value all. That is to say, R that we need to determine may be 1 or 3, which is an uncertain value.
In addition, for some special bidding, the estimated HCP can’t reflect the actual the R, only provide a list of the empirical value. Which are basically also an interval value, and it needs to be memorized.
Although the theory of LTC is advanced, but because the bidding system can’t quantitatively describe the range of HCP and qualitatively describe the card pattern, the contract at between making or losing, thus preventing LTC from becoming widely popular. The author of “Modern Losing Trick Count”had no choice but to write at the book end: “In some modern bidding systems, it's an ideal way to solve this problem, which is to find out the card pattern of partner by relay bidding”.
Assume there is a bidding system that can reduce the range of estimate HCP and accurately describe card pattern, it makes ΔP≤3, that is, corresponding to the basic principle of LTC, the counted R is no longer an uncertain value but a constant value, and makes ΔR=1! That is, the contract make or go down is determined 1 trick. Bidder so confidence in the rationality of the contract. Then the impact of LTC will be more profound.
This article holds this assumption, expounds theoretically, optimizes the method and quantifies calculation, so as to make a feasibility study.

WT are an intuitive embodiment of high card value, meaning that with HCP, you can get WT. However, if there is a card pattern and the trumps fit, you can still get long suits WT or ruffing WT. Then the high card WT will be distorted, that is, the card pattern value will not be reflected.
LT, are the embodiment of the comprehensive ability (or card strength) of HCP and the card pattern value. The HCP determines the minimum ability to WT, and the card pattern value reflects the extra-WT capability. Then the LT can predict the maximum ability of WT , which is the extra-WT that cannot be reflected by the value of big cards when the trumps fit.

1. Losing Trick Count（LTC）
1) Counting your LT
R1=∑Ra (a=1,2,3,4) 1
R1 — your total count of LT
Ra — your count of LT for each suit.
In counting LTC, A/K/Q is counted as a WT (but excluding singleton K and double-tons Q) , other small cards are counted as a LT (no more than the fourth card in each suit) .
Under the following case, it needs to use rectifying count rules:
① If there is the trumps fit (hand in hand with 9+) , the R minus 1;
② Balanced hands, the R plus 1;
③ When Qxx (+) without T (10) , the R plus 0.5;
④ AKJ, AJ10, AQJ, KJ10, those J is counts as 1 additional value, and 3 additional value, the R minus 1.

2) Calculate partner's LT
A. 9 LTs method
R2a=9 - n 2
R2a — partner's LT
n — bidding level
This method is a simple way, that is to judge the number of LT based on the bidding level of partner’s opening or over-calls. It only considers the HCP value, regardless card pattern value, and is only suitable for balanced hands.
B. formula method
R2b=12- θ 3
R2b — partner's LT
θ — card strength=ω+π
ω — the high card value=HCP/3
HCP—high card point
π — the card pattern value.
for short suit: 2/1/0, π=1/2/3
for long suit: 6/7/8, π=1/2/3
In the formula method, both the high card value and the card pattern value are considered. That is, not only must we estimate the HCP range of partner, but also know his card pattern.
In this article, the assumption that ΔP ≤ 3. Therefore, according to the traditional bidding habit, we divide the HCPs into 10 intervals (the limit value for a hand can range from 0 to 27) , with every 3 HCPs are a group (see the left column of Table 1) . Average the each group of HCP to form a multiple of 3 (see the figures in brackets in the left column of Table 1) , and calculate the corresponding R.
No account of the card pattern value, remove the π in formula 3 and becomes as:
R2b=12- HCP/3 4

The calculation results are shown in the right column R2b of Table 1.

As can be seen, all R2b are a constant value, and each range of difference value ΔR = 1 too. This is what we expect.
For each HCP value (0~27) , with the average value of the corresponding group is used for error test (calculation is omitted) . And compared with Table 1, the error rate δ=±0.68, for HCP≤4 or HCP≥14, and δ=±0.34 for 5＜HCP＜13. In other words, no matter what value HCP takes, the error rate does not more then 1.
If we define the ranges of HCP≥14 as 14-16, 17-19, 20-22, 23-25, 26-28, the error rate is all δ=±0.34, so that the counting is more accurate. Here, we still maintain the division of HCP on the based on traditional bidding habits.
Unbalanced hand, taking account of the card pattern value, for the π value in formula 3, determine whether it is the long suits or short suits, and one of them is selected and counted. The R2b of formula 3 is definitely smaller than that formula 4, which reflects the card pattern value.

C. Empirical value
Some special bids, such as the Major support raise (include responses in opp's over-calls) , pre-emptive bid, bids after takeout doubles, which relate to the number of trumps suits, vulnerability and bidding level factors, there may be some deviations from the actual R when counted R with the formula 3. Usually, some books on LTC, use listed in tables actual R ,which is the empirical value of R.
According to the assumptions in this article and Table 1, we can also summarize the empirical values R into Table 2~Table4.

Compare the R2c of empirical value in Table 2 with the counted the R2b in Table 1. When there are 4 trumps support, all the R minus 1. Actually, it is the ① of rectifying count rules.
Since all the R2c is a constant value, and all the ΔR=1, then the empirical value of R2c can be regressed to formula 3 and quantified calculation.
Considering the rectifying count after the trumps fit, then:
R2c= (12- θ) - m 5
m — trumps 4fit=1, otherwise=0

Compare R2c in Table 3 with R2b in Table 1, for the same HCP.
The range of HCP remains unchanged, there is 2~3 less R2c with the increase of pre-emptive bid level (the cards of long suit increase) . Actually, it is the embodiment of the card pattern value π in formula 3.
At the same time, it can be seen that the 9~10HCP (MAX) in vulnerable (VUL) there is one more R than the 6~10HCP in non-vulnerable (NV) .
Actually, this accords with the principle of down 3 in VUL or down 2 in NV.
Then the R2c in Table 3 can be regressed to formula 2 and quantified calculation too. Considering the vulnerability:
R2c= (9 - n) +v 6
v — VUL (MAX) =0, NV (MIN) =1
n — Pre-emptive level (n≥2)

In Table 4, there are 3 cases:
① Normal over-calls, the range of HCP is wide (8~15HCP) , the R2c is interval value. To make R2c a constant value, for one-level over-calls, use asking bid to subdivide the range of HCP as to 8-10, 11-13, 14-15HCP, and for two to four-level over-calls, use invitational bid .
② Pre-emptive over-calls, has the same meaning as the pre-emptive opening bid, R2c is also an interval value. If it’s possible to distinguish between them MAX and MIN, it can also become a constant value.
③ Over-calls after takeout double, where R2c consistent with Table 1. The meaning is:
One level takeout double, 14~15HCP, 7LT.
Takeout double and bid a suit or raise at the second level, 16~18P, 6LT.
Takeout double , cue-bid again ,and bid a suit at the third level, 5LT.
Takeout double to the four-level bid for a solid long suit with more than 7 cards, less than 4LT.
They are all constant value.
Thus, all three situations R2c all is a constant value. Adjust formula (2) , the R2c in Table 4 can be regressed to formula for quantified calculation.
R2c=8- n 7
n — level of takeout double

2. Cover card number (CCN)
For the LTC with long suit=6+ as the trumps, the method of CCN should be used.
1) for PD's long suit, to count your CCN.
A) trumps: A/K/Q=1;
B) side suit: AQ＝1.5, A/KQ＝1, K＝0.5, Q＝0;
C) singleton card=1, void=2.
2) for your own long suit, use the formula to calculate PD's CCN.
t=11- r 8
t=CCN
R=LT
Formula (8) shows that the relationship between the LT and the CCN, it is eleven. Substituting into formula (4) . If balanced hands, remove π, so then:
t=11- (12-HCP/3) =HCP/3- 1 9
Formula (9) shows that the CCN is less than the LT by 1. Actually, it is the ② of rectifying count rules, that is balanced hands, the R minus 1.

3、Back-calculate the WT
1) By calculating the LT of hand in hand.
Ω=24- (R1+R2) 10
Ω —WT
R1 —your LT
R2 —partner's LT (R2a or R2b or R2c) .
That is to say, when 24 subtract the sum of LT, if Ω=13, it well be a grand slam; if Ω=12, it's a small slam; if Ω=10, it can make a game.
2) By count the CCN.
Ω=13- (R- t) 11
Ω — WT
R — LT
t — CCN
This is to say, use CCN fill in LT, if filled full, it's a grand slam, if missing 1, it's a small slam, if missing 3, it can make a game.

4、To determine the level of contract
n=Ω - 6 12
n —level of contract
Ω —WT

2013湖北大企业团体赛冠军

Sometimes, the LTC will are disturbs by the OPP bidding, you can use the LT bidding rule.
1. Single player rules——the opener opening bid, the level of contract must be 9 minus his losing trick.
n=9 - R 13
Formula 13, that is 9 R rules. Is derived from formula 2 . It means that if there are 9 R, no shall be bid, with 8 R, can be bid at one level.
2. Stepping-up rules——As R minus 1, the level of contract can be increased by one, and so on. For example, the partner opening bid (at least 11~13HCP=8R) , if you have 8R, you can bid at the two-level, 7R can bid at three-level to invite a game, 6R can make a game; 5R can try a slam.
3. Invitation rules—— In stepping-up rules, 7R invite a game, if opener is also 7R (14~15HCP) , accept it. Vice versa.
4. 963 rules——
1) Opener: 9R=don't opening bid; 6R=invitation a game; 3R=try slam.
2) Respond-er: 9R=NF; 6R=GF; 3R=forcing to slam;
3) contender: 9R=don't over-calls; 6R=simple over-calls; 3R=GF.

Reach here, some people may ask whether The LTC , ΔR is quantified up to 1, is only an assume, if it holds valid? or what exactly is the bidding system capable of realizing this assumption? The answer is affirmative! That is, the《New Precision System》.

The 《New Precision System》 can basically realize the full-restricted bidding of precision system, which is reflected in the following eight aspects.
① the MAX and MIN for opening bid Major, and the full card pattern.
② the HCP of 1C opening bid is subdivide in to 16-18、19-21、22-24, and the HCP of 1D response is subdivide in to 0-2、3-4、5-7, and the card pattern in 1D.
③ the MAX and MIN for positive response after 1C opening bid, and the full card pattern.
④ the MAX and MIN for 1NT or 1D opening bid.
⑤ the MAX and MIN for 2C opening bid, and the full card pattern.
⑥ the MAX, MED and MIN at 8~15HCP which over-calls 1M or the 3rd player opening bid 1M.
⑦ the MAX and MIN for the two sets cards opening bid or over-calls, and the number of cards suit.
⑧ one-level takeout double, the HCP of the forced bidder is subdivided into 0~4、5~7、8~10、above 11, and the card pattern.

Thus, it has realized ΔP≤3, R can be quantified calculation, made ΔR=1. Then the assumption of this paper holds, and can made the LTC has been greatly simplified.
Normal bidding, memorize Table 1.
Use formula 3 if considering of the card pattern value, where π selects one of the long suit or short suits.
Abnormal bidding, memorize Table 2~4 or corresponding formula 5~7, and use formula 10 for the WT calculations.
If using CCN, memorize formula 9, and use formula 11 for the WT calculations.

In the introduction, it is mentioned that using relay bidding to find out the accurate card pattern. Here, all the examples, the new precision system uses opening bid 1M and 1NT GF.

Example 1 No better than not knowing

This is a hand we often meet when playing bridge.

North opening bid 1S, maybe three possible card pattern.
(a) the same card pattern as (b) , but with different MAX and MIN;
(b) the same MIN of HCP as (c) , but with a different card pattern。
Let’s discuss (a) first, the bidding process is:

So the LTC is as follows:
South LT: R1=7 (count) .
North LT: R2=12- (15/3+2) =5 (HCP=14~15; Formula 3, singleton π=2) .
Total LT: R=R1+R2=7+5=12
WT: Ω=24- 12=12
This is a small slam. If you exchange North’s long suit, or South's Ace, it would not affect the result.

If North bid 3H (adds one level to answer) , that is 5305 card pattern with H=3, D=single, then:
North LT: R2=12- (15/3+3) =4 (Formula 3, void π=3) ;
Total LT: R=7+4=11;
WT: Ω=24- 11=13.
Wow, this is a grand slam.
Looking at (b) again, South hand card unchanged, North card pattern is still 5521 but MIN, the LTC is as follows.
North LT: R2=12- (12/3+2) =6 (singleton, π=2) ;
Total LT: R=R1+R2=7+6=13;
WT: Ω=24- 13=11, Not enough to try the slam.
Finally, look at (c) , South is the same hand, and North's card pattern is 7321 , but MIN is too, the LTC is as follows.
North LT: R2=12- (12/3+2) =6 (7 cards of long suits, π=2) ;
Total LT: R=R1+ (R2-1) =7+ (6-1) =12 (hand in hand 10 trumps, R2-1) ;
WT: Ω=24- 12=12
There is also a small slam.
LTC can also be completed by using Table 1, which is more convenient and simple. For example (a)
North LT: R2=7-2=5.
When the game ends, we can prove North's LT by counting on 4 dummies.
For (a) , when the cards pattern is 5521, R2=5. If 5305, then R2=4.
In cases (b) and (c) , both R2=6. They are the same as the calculated values.

Example 2 Love to fight to win

An offline match with Guangzhou friends.

Actual combat bidding process and LTC (shown in the figure) .
North's card pattern is 4702 which 9HCP, isn't enough to opening bid. But there are only 5LT, and have the strength to enter the game, thus opening bid 1H.
East bid 2D, South bid 2S, West bid 4D further pre-emptive. Usually, if the opening bid is stronger , the rebid should show weakness, so North 4S is reasonable. However, according to LTC theory, the trumps is very fit, so LT-1. In addition, the suit D of OPP is void, anything is possible, if South doesn't have any wasted high card on D. Let's verify it through LTC.
South LT: R2=7~8 (HCP =11~15 in Table 1) ;
Total LT: R=R1+R2=5+ (7~8) =12~13;
WT: Ω=24- (12~13) =12~11.
In other words, if South is 7LT, there maybe a slam 6S! It should not be a problem for at least 5S, so that bid 5D with safety shown interest in a slam.
Look at South's cards, considering the lacks strength of HCP and partner's opening bid suit H is the void, it’s reasonable to stop at 5S. But if you think from the LTC's angle, that is, partner 5D slam try, should be R1=5LT, you have 7LT, and no waste point in D, it is good for partner's wishes.
Hand in hand is only18HCP, but they use LTC to slam try success.
Love to fight to win！

Example 3 It's all my fault

WeChat official accoun“Bridge ABC”once reprinted a foreign article entitled “Who is at fault for not bidding a grand slam?”

The article said that a grand slam which completely could be made was not bid, and published the comments by foreign bridge experts on the responsibility between North and South. The process of both hand and bids is shown as follows.
Some said the North is the dominant player. Although know South opening bid 1C of precision, it will stop at 6S for the card strength not enough and the D's control.
Also some said the South is the dominant player. Although I know that the North trump well, I may not be able to play 7S because partner's the MAX and MIN is unknown and the length of my own trump is not enough.

Both sides can't convince the other side, neither of them is wrong, and the final result of the discussion is that North and South played 50 boards each.

If South is the dominant side, ask North's HCP and card pattern by through the relay?

The LTC is:
South LT: R1=6;
North LT: R2=12- (12/3+3) =5 ((HCP=11~13 in Table 1；8 cards of long suit, π=3) ;
Total LT: R=R1+R2=6+5=11;
WT: Ω=24- 11=13.

We can also count it with CCN.
South CCN: t=5 (trump King=1, 2Aces=2, connected Ace and King=2) ;
WT: Ω=13- (r - t) =13- (5- 5) =13.
Both ways come up with 13 WT, so it's safe bid 7 S.
In fact, knowing the MAX and MIN, and card pattern, you can also be proved by WT method. There are 8 cards trumps with Ace and Queen, plus PD’s King, which is 8 WT, and outside 3 Ace and 2 King, which is 5 WT, making a total of 13 WT too.

Example 4 Win or lose is at stake

In 2021, the women’s team of the 44th World Championship, China VS Japan.

East holds 4 Aces and a big card suit in Diamonds. So good a hand, rare to see. Finally bid out 7N. The bidding process is not logical enough, can only say, have confidence, by intuition, and some luck involved also, win or lose is at stake.
For this kind of hands with 5 key-cards, using LTC is the best method.
The bidding process:

The LTC is:
West LT: R1=3;
East LT: R2=12- (12/3) =8(HCP=11~13 in Table 1);
Total LT: R=R1+R2=3+8=11;
Winning-trick: Ω=24- 11=13。
Win or lose not by intuition, and comes from careful LTC, counting 13 WT is the last word.

The establishment of any event stars from the process of assumption to realization. Of course, it requires practice and time for verification. At least it can be confirmed that. This article at least the following can be affirmed:
1. Narrowing down the estimated range of PD's HCP within 3, is in line with the LTC theory that “on average, every 3 HCP get 1 WT”.
2. After quantified calculation reaches a constant value, the range of difference value is 1, which is consistent with the rule of based on 1 as the unit in the LTC theory.
3. The error rate of δ that R value of each HCP's range is not more than 1, which is consistent with the assumed results.

These three clever “coincidences”, or three numbers 1, indicate that it is possible to “losing-trick count, Quantified up to 1” . Then the bidding system that supports this possibility is the 《new precision system》. It can describe the range of HCP quantitatively, made the ΔP≤3, and describe the card pattern qualitatively (including the number of cards suit and the full card pattern) , made the R= a constant value and the ΔR=1.
In addition, the empirical values Rc can be quantified calculation from the list into formulas. After mastering the law, practice makes perfect. Finally, the best contract is counted out. Not only does it enhances the rationality and accuracy of the contract, but it also makes the calculation process more convenient.
Interested bridge players can follow the official account “New Precision Bridge” WeChat to learn and applying. So, you can reach the highest level of bridge bidding and enjoy the fun of bridge.

Ref:
《Bridge Bidding rules》 Zhang Chengwei China.
《Losing-Trick Count method》 R·Klinger Australia.

May 2025

The 44th World Bridge Championship in Wuhan, China in 2019，the author took a photo with former World Bridge Federation President Rona

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