Nov. 21, 1884.] 



♦ KNOWLEDGE ♦ 



433 



(Buv Vm\)iit Column. 



[" Five of Cluhs " has been ill moat of the time since his last 

 papers appeared, with malarial fever, and is but now beginning to 

 be himself again. He trusts that weekly or fortnightly games will 

 now appear regularly. They will chiefly bo fully annotated games 

 from the Westmintfter Papers. The following is a good game. Z 

 was onr esteemed correspondent Mr. F. H. Lewis.] 

 Thb Hands. 

 ,f D. 5, 4, 3. C. 9, 8, 2. 



B 



i S. K, Q, 10, 3, 2. 



H. A, 3. 



SI: 



' 1 H. Q. 0, 8, 7, C. 



None. 

 Kn, 8, C, 



Q, Kn, 4, 3. 



Tr. D. Il-,i 



y z 



r-z, i. 



A Iliads. 



K, Q, Kn, 2. D. ■ 



4. S.I 



10, 5, 4. H. I 



K, 10, 6, 5. C. . 





10, 9, 8, 7, 0. 

 A, 9, 7. 

 B Z 



O O ♦ ♦ O 



♦ ♦ ♦ O 



O O ♦ »! O 



+ ♦ ^^ + 



* + W^ + + ♦ 





9 "? 



l*i 



11 





C. A, 7. ■) 



H. K, Kn, 2. J 



THE PLAT. 



Card underlined wins trick. 



1. The score being at " four all," 

 the trump lead from five was not 

 good. Heart Two would have 

 been the correct card. But, under 

 the circuiti^tanceri, Club Ace, 

 followed by the Seven, would not 

 have been bad. A lead from a 

 short suit at such times, especially 

 when weakness is not thereby dis- 

 closed too quickly, is sound enough. 

 The hand was played before ths 

 days of the penultimate, or 

 Diamond Seven would have been 

 led. Z shows his partner his great 

 strength ; forit is not to be supposed 

 that F having no trnmps, and B 

 being presumably very weak, Z 

 could not have taken the trick 

 with a smaller card than the Ace. 



2. Z plays a surer game than A. 



3. From the fall of B'8 Eight and 

 jl's Seven in trick 2, Z knows T 

 must have the four, and therefore 

 (from his lead) another Club. The 

 play of the Ten is no finesac, as, if 

 J holds the Queen besides the Ace, 

 nothing is lost by letting Queen 

 make. After the Ace has fallen, 

 Z knows Queen to be with Y. 



4. A should have led a Spade, as 

 that is most probably his partner's 

 suit. 



5 and 6. A-B have command of 

 Spades. Z is now numerically 

 weaker in trumps than A, but he 

 is not sure of this till trick 8. A 

 thi'ows Spade Ace, that he may 

 not stop his partner. 



7. Y discards his useless Spade. 

 Had he originally discarded from 

 his longest suite — the general rule 

 now when the adversaries have led 

 trumps — his Spade Knave would 

 have been guarded, and worth 

 keeping. 



8. B has no more trumps ; for 

 had he held any above the Seven, 

 he would have headed A, first 

 round. 



9. If Z goes on with trumps, A 

 will be left with the long trump, 

 will bring in Spades, and A-B will 

 make every other trick. Z there- 

 fore forces ^1. The point is a 

 pretty one, and shows the impor- 

 tance of attention to details. By 

 leading a Heart at trick 2, and 

 afterwards showing that he held 

 Spade Ace, A has left it tolerably 



clear that he holds the King card in Hearts, without which A-B 

 would still have lost, even had Z gone on with trumps. 



10. The trump King can now be safely played, for A has no 

 Spades left. Of course it has to be played anyhow at this stage, as 

 B would have gone on with his winning Spades. 



11, 12, 13. A is forced again ; the Queen of Clabs being already 

 "placed" with X, and A having to lead from his major tenace, i' 

 makes the seventh trick and the game. 



*»• The notes on Whist, by " Five of Clubs," extended and 

 carefully corrected, are now in the press, and will form a small 

 treatise " How to Play Whist." They will be accompanied by forty 

 fully annotated games, from actaal play, illustrating all the 

 principal points of Whist strategy ; and by Whist gossip, and the 

 rules, etiquette, &c., of the game. 



THE VARIETY OF WHIST. 



SiE, — There seems room for another word on the old problem of 

 the hands at Whist, first approached in a satisfactory manner by 

 " Cavendish," in the Field of June 10, 1865. His investigation is 

 undeniable, as far as it goes ; but I think it overlooks one con- 

 sideration of some importance in determining the true number 

 of essentially different hands, i.e., hands so different as to require 

 different treatment. It will be best, however, for me to give my 

 own solution first, and then to compare it with others. 



It is easy to show that the total possible number of different 

 deals, having no regard to dealer or trump-card, is 



|52 



|39 



126 



|52 



|39 |13 |26 |1£ |13 |13^ (\13)* ' 



So far all are agreed. Let us ciU this quantity N, its numerical 

 value being something over fifty-three thousand quadrillions. The 

 question is, what corrections are to be applied to this in order to 

 obtain the true number required ? 



(1.) If we take any one of these N arrangements we can make 

 52 different games by making each card in turn the trump. Hence 

 we must multiply N by 52. 



(2.) The number X includes as different games those cases in 

 which the hands are nnmerically identical, but the suits are inter- 

 changed. But the play is not altered by turning a suit of Clubs 

 into the same suit of Hearts, and the Hearts into Clubs, provided 

 the exchange is similarly performed in each player's hand. We 

 must then divide A' by the number of possible permutations of the 

 four stiits, t.t'. by 4 or 24. 



(3.) Finally, we must allow for the fact that, supposing the 

 hands in any deal to be passed round the table in order, such re- 

 arrangements appear separately in the total A", although the game 

 is not altered thereby, as we have supposed the lead also to move 

 round, to circulate, in fact, thirteen times in each arrangement. 

 Obviously the correction here is made by dividing AT by 4. 



It follows from these considerations that the real number of 

 essentially different games is represented by 



52 



247-4^. 



and this, when worked out, gives the number 



29,057,566,289,639,762,787,920,280,000. 



The result obtained by Cavendish is twenty-four times as great 

 as this, as he omits the second correction given above. It seems, 

 however, quite necessary, the four suits being equivalent, apart 

 from consideration of trumps, which are otherwise allowed for. 

 The agreement otherwise between the two results is satisfactory, 

 as the methods of solution differ in some respect. This is most 

 likely the correction intended by a somewhat obsctu'e passage in 

 the treatise on " Probability " (written, I believe, by the late 

 Sir J. W. Lubbock), in the Library of Useful Knowledge, which is 

 quoted, but apparently misunderstood, by Cavendish. The fijst 

 and third of our corrections do not seem to be made in the essay 

 on '' Probability," which is naturally concerned with the problem 

 rather from the mathematical than from the practical Whist- 

 player's point of view. 



It may assist the mind to grasp the result above stated, if we 

 put it in this form : — Taking the population of the world as sixteen 

 hundred millions, let them all — man, woman, and child — start to 

 play Whist, night and day, at the rate of ten deals per hour ; they 

 will exhaust the variety of Whist in about one thousand billion 

 years. — I am, &c., W. Abnison Slatre. 



