214 



KNOW^LEDGE <► 



[July 1, 1887. 



already counted those in which both a blank suit and a singleton 

 appear). Thus we obtain the addition sum: — 



One-fourth of the Number of 

 H-inds iu whieh arranj^'ement 

 Arrangement. cun appear. 



6, 4, 3, 2,105,429,010 



6, 4, 4, . 



5, S, 3, . 



6, 5, 2, . 



7, 4, 2, . 



7, 3, 3, . 



8, 3, 2, . 



9, 2, 2, . 



7, 6, 0, . 



8, 5, 0, . 



9, 4, 0, . 



10, 3, 0, . 



11, 2, 0, . 



Total ". 

 Numbers already obtained 



Total . 



1,973,839,725 



1,421,164,002 



1,033,574,256 



574,207,920 



421,085,808 



172,262,370 



13,050,180 



8,833,968 



4,969,107 



1,533,675 



245,388 



18,252 



7,730,214,297 

 48,882,413,450 



50,012,627,747 

 Multiply by 4 



Grand total . . . 220,450,508,988 



That is to say, out of the 635,013,559,600 hands which can possibly 

 be held at whist, no less tlian 226,450,508,988 show either a singleton 

 suit or a blanlc suit. Thus the chance of any hand taken at random 

 in any deal showing either a blank or a singleton suit is represented 

 by the fraction 



226,450,50 8,988 



635,013,559,600' 



which differs very little from (it is really slightly greater than) 



7-20, the approximation mentioned by tlie whist editor of the 



Aust?'alasian. 



In passing, I may remark that I cannot understand a comment made 

 in the Australasian upon my approximate estimate that the odds are 

 about 5 to 2 in favour of the appearance of a singleton in one liand at 

 least out of the four distributed at a single deal. The whist editor states 

 that a friend, from the records of an experience extending over five 

 years, had found that the total number of singleton hands is about 

 equal to the total number of deals. " We believe, therefore," he 

 proceeds, " that the approximate result which Mr. Proctor has pub- 

 lished is under rather than over the mark." My approximate result 

 cannot be tested by counting the number of singleton lianils ; but 

 my exact result relates directly to them. I showed that in 

 635,013,559,000 hands, and therefore for 158,753,389,900 deaU, 

 there will be 195,529,653,800 hands in which there is at least one 

 singleton, or about sixteen such hands for every thirteen deals. 

 This is above, not below, the result observed in five years' play, 

 unless tlie " at least " is to be interpreted rather liberally. But a 

 singleton in one or other hand, when singletons are appearing 

 freely, will often escape uncounted. 



The article by the Amtralasian wliist editor is aimed at the 

 fallacy of supposing that if a player holds Ace and three others 

 (not including the King, or both Queen and Knave) he can safely 

 lead a small one where one trick is necessary to save the game. He 

 wishes me to turn my mathematical deduction to practical account 

 by diverting it from the superstition I attacked to one which 

 affects actual play. For, as he points out, when so long-practised a 

 player as " Pembridge " confidently asserts that the odds are ton to 

 one against " a singleton being on the tapis at all," it is evident 

 that very erroneous ideas are held among whist-players on this 

 point, importantly though it should affect whist play. 



However, it is not by an inquiry into the average number of 

 singleton hands, or hands with a blank suit, per hands dealt, that 

 the propriety of an Ace lead from Ace to four, under the condition 

 mentioned, can be decided. Taking it for the moment as decided 

 that where one trick saves the game, the question of playing Ace 

 from Ace to four must be decided by the cliance of the Ace being 

 ruffed second round, we have iu the case of an original first lead a 

 comparatively simple problem — while if other suits have been 

 already opened our problem becomes so complex (having so many 

 varied forms) that no one would care to deal with it. I take, then, 

 the simple problem : — 



If A, the original leader, holds four cards in a suit, what is the 

 chance that one or other of the opponents, T and Z, mill le able to 

 ruff the suit second round? ajiart from the possihility of either 

 {jetting discards in the suit. 



I get the following figures by taking from the table at page 196 

 of my " How to Play Whist " (which, be it noticed, not only shows 

 how the suits may be distributed in a hand, but Low all the cards 



may be distributed in a suit), the various numbers set opposite the 

 different arrangements mentioned : — 



Distribution of cards 



of a given suit among 



the four players, one 



at least holding 4. 



4, 4, 3, 2 . 



5, 4, 3, 1 . 



5, 4, 2, 2 . 



4, 3, 3, 3 . 



6, 4, 2, 1 . 

 i, 4, 4, 1 . 



6, 4, 3, . 



5, 4, 4, . 



7, 4, 1, 1 . 



7, 4, 2, . 



8, 4, 1, . 



9, 4, 0, . 



Total . 



One-fourth the number 



of ways in which 



the suit can be 



so distributed. 



. 34,213,221,900* 



. 20,527,933,140 



. 16,795,581,000* 



. 16,72e,404,010» 



7,464,702,960 

 . 4,751,830,375 

 . 2,105,429,0401 

 l,973,839,725t 

 622,058,580 

 574,207,9201 

 71,775,990t 

 l,533,675t 



I 



. 105,828,585,005 

 Sum of numbers marked *, cases 

 in which no hand is blank or 

 holds but one card in the suit . 67,735,267,000 

 Sum of numbers of cases iu which 

 at least one hand is blank in 

 suit, or holds a singleton . 38,093,317,405 



Sum of numbers marked t, cases 

 in which at least one hand 



is blank 4,720,786,350 



Sum of numbers of cases in which 

 at least one handholds but one 



card 33,438,307,045 



f Sum of numbers of cases in which 

 <. tn-o hands hold either a single- 

 l_ ton or no card in the suit . 695,368,245 



It will be seen that if we neglect as numerically unimportant 

 the cases in which two hands are void, or hold no more than one 

 card in the suit, we may say that in round numbers there are for 

 67,700 cases in which the suit will go round twice, 33,400 cases in 

 which it will go round but once, and 4,700 cases in which it will not 

 go round at all. Of these last-named cases, two-thirdn, or about 

 3,130, in which an enemy holds the hand void in the suit, are of 

 course unfavourable to the Ace lead ; but they do not atloct our 

 judgment, simply because it is certain that the chance of a ruff first 

 round must be risked in any case. (There is, of course, a much 

 greater chance of this happening when Ace is led from Ace to live.) 

 Of the 33,400 cases in which a singleton is held, we may consider 

 at least 400 to be cases where another hand is either void, or holds 

 a singleton, the two hands thus situate being so distributed between 

 T, B, and Z, that A B do not lose through a ruft'. This leaves 

 33,000 cases of a singleton hand in the suit led by A, as against 

 67,700 cases in which the suit will go round twice. Of these 

 33,000 cases, 22,000 will be cases in which either 3' or Z holds the 

 singleton, and therefore the enemy can ruff the second round of 

 the suit. Thus, by leading a small card A will lose the Ace in 

 22,01W cases out of 07,700 + 33,000 or 100,700, or, roughly, about 

 twice in nine trials, or more nearly, five times iu 23. 



How far this 2-9tIi chance of being ruffed second round woidd 

 justify a player in leading Ace first, to save a game (which might 

 be saved otherwise) at the risk of losing command over the suit, 

 must depend on the details of the hand and game. It is to be 

 remembered that even though Z, the fourth player, should be 

 able to ruff, he would not ruff the suit when returned by B in 

 every case, only if weak or very strong in trumps. For as B 

 would return, generally, a small card, to ruff would be rutting a 

 doubtful trick. 



(9ur C&tsisi Column. 



By " Mephisto." 



MATCH BL.iCKBUENE r. ZUKERTORT. 



HE result of this match, 5 to 1 in favour of Black- 

 burne, with 8 draws, has rather taken the Chess 

 world by surprise, and must raise the winner in 

 the estimation of those who apprehended that 

 Blackburne never did hini.*ielf justice when con- 

 testing a match. For nearly forty years, that is, 

 ever since Staunton was in his best form, foreign 

 Chess-players were supreme in this country. By 

 this victory, however, Blackburne has avenged his 

 former defeat by Zukertort in 1881, and thereby an Englishman 

 again occupies foremost rank amongst Chess-players. For the time 



