-The second – the Point-Countversion – is based on the well-established evaluation presumption that three points correspond on average to one honour-card trick (A’s, K’s and Q’s). It is just as accurate as the basic Honour-Card version.

-The third – the Point-Count Balance variant –provides precisely the same estimate as the Point-Count version, but – as noted above - it has the important advantage that it also provides the players with an exceptionally simple bidding aid, which enables them to make an immediate and precise estimate of their trick-taking prospects as soon as a prospective trump fit has been established.

For example, on the following deal where E/W have the balance of strength - 23 points - and the likelihood of making game with their nine-card heart suit, we see that their Winning Trick Count of seven honour-card tricks (made up in this case of 2A's; 2K's and 3Q's); plus three trump tricks (a nine-card suit minus six) provides us with the correct estimate of ten tricks.

Given the ease and the speed with which such Honour-Card Winning Trick evaluations can be made for both pairs of hands on any exposed deal, simply by glancing at the cards, it is extremely simple to check their accuracy and to compare them with those of more conventional assessment procedures.

A much broader selection of real hands is provided in ATTACHMENT 2 to illustrate this, and other features, of the Winning Trick Count, but for this introductory commentary I will simply note that I judge that the evaluation procedure described above is so easy to use that players are more likely to be convinced of its merits and accuracy by applying it themselves to a selection of deals of their own choice and then comparing the estimate with the number of tricks actually made.

In which case, I have no doubt that they will quickly see that the Winning Trick Count is most certainly not infallible. Like all bridge-hand evaluation procedures it is a guide and not a guarantee of success. Indeed, as we shall see below it will be incorrect by more than one trick on around one deal in ten – typically the exceptional ones which are marked by a unusual distribution; or those which point to the prospect of a high level contract such as a slam or a minor game.

On the other hand I believe that the exercise will also show that this very simple and uncorrected Winning Trick Count is, in fact, significantly more precise than the traditional evaluation methods for the large majority of more typical hands, such as they are likely to encounter in any bridge event. For example, the detailed consideration of over 500 contracts, which is summarised below, showed that the Winning Trick forecast was within one trick of the right result around 90% of the time, whereas the Losing Trick score - although still impressive - was nearer to 80%.

And, as we shall see, the two alternative Point-Count versions of the Winning Trick Count, which are based on the well-established evaluation presumption that three points correspond on average to one honour-card trick (A’s, K’s and Q’s), are just as accurate.

1. The Winning Trick Count and the evaluation of weak hands  - The Winning Trick Count has the advantage that it will provide an equally accurate estimate of the trick-taking prospects for both pairs of hands in competitive bidding situations.

2. The Winning Trick Count and the quality of the trump fit - Conventional evaluation procedures such as the Losing Trick Count have difficulty in coping with deals where one of the hands has less than three cards in the proposed trump suit, as may happen, for example if partner has made a pre-emptive bid.

3. The Winning Trick Count and the Law of Total Tricks - Jean-René Vernes' Law of Total Tricks states that:

However, while it is increasingly evident that this remarkably simple estimate of the overall trick-taking prospects provides an effective and low-risk bidding strategy for competitive auctions, its weakness is that it tells us nothing at all about how the total will be divided. In addition the proponents of the ‘LAW’ offer no plausible explanation for this seemingly improbable link between the total number of tricks and the total number of trumps.

In contrast, we find that we can obtain an insight into both of these questions by just adding together the Winning Trick estimates for both sides on any deal.

The resulting Winning Trick Total Count (the total number of tricks, which would be available if each side were to play in their best contract) is made up of the sum of the trump holdings minus six for each side - but, this is just the sum of their combined trump holdings minus twelve; plus the sum of their honour-card holdings (the A’s; K’s, and Q’s)- but, this must always be twelve.

And when we add these together we see that the total is simply the number of trumps held by both sides, each in its respective suit.

THE POINT-COUNT VERSION OF THE WINNING TRICK COUNT:

However, the problem, which we face when we are actually aiming to make such an evaluation at the bridge table is that the only cards that we can see are those in our own hand and any further judgements which we then make must be based upon such additional information as may emerge from the bidding.

Notwithstanding this constraint, it is necessary to make an assessment and the Point-Count version of the Winning Trick Count, which is described below provides a powerful and effective solution for those bridge players who routinely use their High-Card Point Count as an initial measure of their hand strength.

It is based on the well-established evaluation presumption that three points correspond on average to one honour-card trick (A’s, K’s and Q’s) and it is distinguished by the fact that, as soon as a prospective trump fit has been established it provides an immediate and unusually accurate assessment of their prospects (and in many cases those of their opponents) on a wide range of deals:

For example, if we have a partnership count of twenty points (half of the total of forty) this would correspond to a corrected count of eighteen and hence a holding of six point-count tricks. And similarly, of course, these twenty points would correspond, on average, to a holding of six honour cards (half of the total of twelve A’s, K’s and Q’s).

Or, more generally – and as we can see from the table below - if we have a typical combined holding somewhere in the range of 10 - 30 points, the number of point-count tricks that we can expect to make if we play in a suit contract is:

THE POINT-COUNT BALANCE VERSION OF THE WINNING TRICK COUNT:

The somewhat unconventional Point-Count Balance variant of the Winning Trick Count, which is defined below, is intended to meet that need:

The key feature, which distinguishes this novel and unusually simple Point-Count Balance from more traditional bridge-hand evaluation procedures, is that it is based on the relative strength, or weakness, of our partnership point count - or, more precisely, on our combined point holding above, or below, the average of twenty.

For example, if we conclude from the bidding that we have the balance of strength with a joint holding of, say, twenty-six points we just regard this as a positive balance of six points.

Following which, if we continue to use well-established evaluation presumption that three points correspond on average to one honour-card trick, we can judge that this corresponds to a positive balance of two point-count tricks.

Although this notion of the Point-Count Balance may seem somewhat unusual, it is, in fact, a logical and convenient measure of our competitiveness and one, which can be used quickly and effectively at the bridge table in a wide variety of bidding situations.

Thus, as we saw above, with a joint holding of around twenty-six points we simply regard this as a positive balance of twopoint-count tricks.

On the other hand if we have, say, only seventeen points the opposition will be stronger, and we will now have a negative balance of onepoint-count trick.

Or, more generally with a typical combined holding somewhere in the range of 10 – 30 points, our balance of such point count tricks will be:

A singularly important feature of this simple Point-Count Balance evaluation procedure is the strikingly clear measure of the importance of trump strength, which it provides.

For example, with a partnership total of around twenty three points combined with an nine-card fit in one of the major suits, we can judge that that we have a good chance of making game. In this case, nine trump tricks plus our positive balance of one point count trick.

And by the same token, with a ten-card major fit we should consider the prospect of a possible game, even if we only have a total combined count of around twenty points

What is equally important, however, is that this quite uncomplicated method of hand evaluation applies just as effectively in competitive auctions where our opponents clearly have the balance of strength. Thus with only around seventeen points, but again with a nine-card suit, we can judge that we are still likely to make eight tricks if we should compete and win the auction (nine trump tricks minus our negative balance of one point count trick). This really is a forecast, which can be made easily and very quickly during the auction.

This is illustrated in the table below, which shows the application of this Point-Count Balance estimation to a substantial fraction of typical bidding situations (those where we have a total point count somewhere in the range 10 – 30 combined with a trump fit of 8, 9 or 10 cards).

It also shows that the Point-Count Balance shares the broad scope of the other two versions of the Winning Trick Count. Thus:

And by the same token, we can, of course, easily form a view of own balance of point-counttricks in those auctions where the opposition bidding enables us to make an estimate of their point strength.

The main conclusions of the comparison are summarised in the Table below:

Moreover, the overall analysis – which was much more extensive – also showed that the estimates for low-point contracts are quite comparable in accuracy with those for the stronger pairs of hands. A much fuller account of this exercise – and of the related consideration of the prospects of improving the score for that fraction of hands where the evaluation procedure fails to provide an accurate forecast – is provided in ATTACHMENT 2.