13S 



• KNOWLEDGE 



[Jdly 21, 1882. 



in which the c&rds arc iliiitribntcd among different suits in any i c 

 Aible way : — 



1 cspdf or 12 cards ujay be taken from any suit in 13 ways. 



2 or 11 cards may bo taken from any suit in ^ = 78 ways. 



3 or 10 cards nmv h- taki 

 ways. 



4 or 9 cards ninv lie take 

 "15 ways. 



5 or 8 cards may be mkc 

 -12S7ways. 



6 or 7 card? mav be taken 



■ • 2x3x4x5x6 



The number of ways in wliich a complete hand of thirteen cards 

 can be made, having; the cards distributed amonK different suits in 

 a certain way, is obtained by multiplying together the numbers of 

 the ways in which the numbers of cards in the several suits m:iy be 

 taken, and multiplying that jjrodact by the nnml)er of way's in 

 which the snita can be chosen. For instance,* if the cards were 

 divided among the snits thus— 1, 3, 4, 5^-the nuAibcr of wavs would 

 be 13 X 286 x 715 x 1287 x 24 ; 24 being 4"x 3x2, the number of ways 

 of choosing the suits. A hand dindecl'*llius— 272, 3, G— might be 

 made in 7S x 78 x 286 x 171C x 12 ways, J^2 or 4 x 3 being the number 

 cf ways in which the suits of 3 and can be chosen, the otircr two 

 snits having two cards each. A hand composed of three cards of 

 each of three suits and four of the fourth; may'bo made in 2SG' x 

 715 X 4 ways, there being oiily four ways in whicli tho four suit can 

 be chosen. > . 



The following table, made in tho manner above indicated, will 

 show the chances of holding any sort of hand, tho number of 

 chances in each ease being one-fourth of the number of the ways in 

 which the hand can'be rtiade. The hands are arranged in t^e order 

 of their respective frequencies : — . i , ' ■ 



The total number of chances is 158, 753, 389, 900, which is ono- 

 fonrth of the whole number of ways in which a hand can be made. 

 The chances cannot bo exactly expressed in much simpler terms 

 than these — tho only factors which will diWdo tho whole number of 

 chances and the chances of any particular hand being 100, and tho 

 factors of 100. There are, however, only three hands of which tho 

 chances cannot bo expressed more simply by dividinir bv 2. 4. 10 

 20. 25, or 100. 



If it is desired to know tho chance of having any sort of hand 

 with the condition that it shall contain, or not contain, a trump, 

 the mle for determining it is as follows :— If tho hand comprises 

 only two suits, one-third of tho whole number of chances of having 

 such a hand is the numljcr of chances of having such a hand, ono 

 of those Kuits being trumps ; and two-thirds of tho whole number 

 of chances is tho number of chances of having snch a hand without 

 a trump. In a three-suit hand tho prjportions nro rovei-scd, two- 

 thirds of the possible hands containing a trnmp and one-third not 

 containing one. For, in tho case of a two-suit hand, 26 of the .39 

 cards not in the hand, any one of which may bo the trump card, 

 bojong to the two other suits, and in tho case of a throe-suit liand 

 13" ef the 39 cartls belong to tlio other siiit. It is eqnally 

 easy to determine the chance of having a certain hand of 

 which a certain suit shall be tmmps, for instance, in a 

 10,"?, 1,0 hand, of the whole number of chancci of having snch 

 shand, ,', or -f^ i% tho numtwr of elianres of having such a hand 

 containing ten tmmp», 4; J is tho number of chances of tho hand con- 

 taining two tromfjs, and H or ,*, is tho number of chances of the 

 hand containing only ono trump. <)( course, this applies only to a 

 non-dealer's hand. The dealer Ixjing sure to have a trump, the 

 chances of his having such a hand with 10 or 2 tnimps, or only 

 1 trump, are rcsperfirelr '1. -f,, and ,'j of tho whole number. 



Ai,(;kkni>x Ukay. 



(Pur €l)tS5 Column. 



By Mkpiiisto. 



GAMES BY CORRESPONDENCE. 

 (Confinttrd/iom p. 122) 

 I'osition after Black 10th move, 1' takes I 



There can be no doubt about it that Black is now subjected to a 

 strong attack, and we think that tho movo recommended in our 

 note (c) to a game, p. 597 (Vol.' i.)— viz.. B to B4 — is much stronger, 

 and ought to yield Black tho advantage. In order to test this, a 

 subsidiary game was played'between'the Editors with tho following 

 continuation. Instead of 10. 1' takes B, 10. B to B4. 11. B to Kt3. 



Instead of this move, our correspondent, Leonard P. Roes, thinks 

 that in reply to B to B4, 11. Kt takes BP might bo playml. This, 

 however, would result in the loss of a piece, i.e. : — 



11. Kt takes BP 11. K takes Kt 



12. Q to KB3 (best) 

 to KK5(ch), then 12. P to KKt3, and White has three 



pieces attacked.) 



— 12. Kt takes QP 



13. B takes P(oli) . 13. K to Ksii 

 and White mnst lose a piece. 



luc the subsidiary game, which continued — 

 11. B takes Kt 



12. Kt takes B 12. P takes Kt 



13. B to R4 13. q to (^4 

 ove ; it prevents White from taking tho Knight and 



doubling tho Pawns, it also prevents P to Q5, and, finally, it may 

 also servo for tho purpose of attacking Wliite's Queen's Pawn by 

 B to Qsq. 



' ■ - - 14. Castles KR 



15. Q takes B 



16. Q to QKt3 

 note : " Tho rest of tho game 



requires no comment. Only careful play is needed to ensure 

 Black's winning." The game was continued up to tho 32nd more, 

 when White resigned, thereby clearly proving that 10. B to B4 is 

 tho proper reply to White's 10th move, KKt to Kt5 (for from the 

 above position, Black (Chief Editor) played, as will bo aeon, a 

 losing game). 



(To be continued.) 



(If 



A good 



14. Castles 



15. B takes Kt 



16. R to Bsq 

 Our Chief Editor says 



SOLUTION OF PItODLE.M Ni>. 45, 

 Bv B. Ci. Laws, p. 63. 



1. B takes P ' 1. K toQ5 



2. K to t^2(ch) - 2. K moves 



3. B takes P. mate 



• ' («)• 



1 . K to Q6 

 2. B take's P 2. K to QS 



3."irtoT}2, mate 



(h). 



K toB6 



2. B takes P(ch) 2. K takes P 



3. Kt to K3, mate 



