542 



• KNOWLEDGE • 



[Apkii. 21, 1882. 



ANSWKIl TO MATHEMATICAL QUERY. 



[8I3 — Tho^iiimlicr of wnya in wliicli five (firln can bo choipn out 



of oIoTcn >• . i - T- , —402. E<tc)i day a arrongoinont uiios 



icven of theao wnyH, nnmoly, ono in n Bopftmto gronp nf five iinil 

 ■ix in tliu Ki'ooii <>' ■■*■ 'J'hcroforo all tlio wnyn will he used in 



-— — GO, days, irhicli la t]io answer to thoquory, proridod tlie ifirla 



ran bo so arningod ns to make nso of all combinations. I do nut 

 Bco liow tliia can bo determined except by triiil, but by this nicana 

 I fijul tUnt it enn bo done ns follows. Let the groups of Gre for 

 six days b» na follows (it will bo unnoccssnry to (;irc the groups of 

 ■ix which will consi.st of tlio rcmnining girls) — 



1.2.3.4.0-1.2.3.7.10-1.2.3.8 9-1.2.1.5.8-1.2.4.7.9-1.2.0.8.10 

 From coch of these groups mako groups for ten more doys by 

 ■nccossiro additions of 1 to ooch number except 11, which you 

 must reduci> to 1, instead of increasing it to 12. 



1 add the method of conducting the ex])erinient : — 

 If the numbers 1 to 11 are supposed to be arranged in order in a 

 circle BO that tho distance from 11 to 1 is the same as that from 

 any number to tho next, tho Bum of all tho distances between tho 

 conBCCutivc numbers of any five Boloctod will be eleven, e.g., if the 

 five numbers bo 1.3.4.7.11, the differences will bo 2.1.3.1.1, and the 

 same differences in tho same order may be used for eleven different 

 groups of five according to the number cho.sen to begin with. The 

 following list is easily made of all the different arrangements of 

 differences : — 



13112 

 13214 

 13223 

 13232 

 13322 

 14222 

 22223 



Silecting a group of five having one of these sets of diffiTciiccs wo 

 see what sets of differences be'ong to the combinations of five which 

 can be made out of the group of six which was left when the first 

 group of five was made. Seven sets of differences are thus dis- 

 posed of, and it will be found easy to divide the whole forty-two 

 into six such sevens. I believe, however, that there arc only two 

 ways in which this can be done. Of course after this is done each 

 Boven sets of differences can be nscd eleven times, thns solving the 

 problem. 



I hod some donbt whether the true construction of the problem 

 was that the five who did not present themselves to the giver of 

 the bouquets constituted a leparato group. If they did not it 

 would seem as if only six groups of five were used in a day, and 

 that tho answer might be 77 dayr, bnt I do not know how groups 

 of six can be selected out of eleven for 77 days without having the 

 same five in a group twice. It cannot be done by the method I 

 have naed. Algernon Brat. 



#iir Cftcss Column. 



END-GAMES. 



IN one important respect, at least, end-games are of more con- 

 soe|uence than the openings. Any weak move made in the 

 beginning of a game does not necessarily entail its loss, as in the 

 middle-game a player has many chances to re-establish the balance 

 of position, or even to obtain a superiority, notwithstanding his 

 unfavourable commencement ; but the end-play directly influences 

 the result — there is no appeal. A single weak move to compromise 

 a position will have tho loss of tho game as its consequence. This 

 axiom has a twofold application in actnal play — it holds good both 

 " for winning a game " and " defending a game." 



Defending a game naturally includes playing to obtain a draw ; 

 while winning a game, also means playing to prevent a draw. The 

 grcotest iKissible amount of precision is required in either of the 

 above cases, which fact renders play in an ending far more difficult 

 than in the middle or in the opening. Every position has its limited 

 number of probable moves, and if through receiving odils or by 

 any other means a player has a better position than his opponent, 

 ho n-ill not have much difficulty in recognising and following 

 up the natural advantages of his position, as, to a certain degree, 

 the advantage manifests or developes itself. E<|nally it may be 

 said that the player having an inferior game will have gVcat 

 difllrulty in avoiding the natural outcome of bis position. We have 

 played many a game where we plainly saw our defeat impending in 

 ten or twelve moves. Our opponent did not aee it ; nay, perchance 



ho might have oven thought his own game loBt ; but Iho ponitioi 

 played itself. More oftcr move headojitcd tho most promi..:r .; liti' 

 of ploy, till suddenly, to his surprise, he found himself the K.i,i:<..r 

 In tho end the positions are generally less suggestive, and, ilierv 

 fore, a player is thrown more upon his own resources. 



There are two kinds of endings : first being that termination o 

 the gome brought about by a brilliant sacrifice, or a series of more 

 of great power and deep and fine play. Thia ending ia tho mos 

 beautiful and ingenious. From it tho art of problem-making lu 

 sprung — problems ore merely correct endings having a mate in ; 

 certain number of moves. As a fine example of this class, wo giv* 

 tho following end-game, which occurred a few days ago at th' 

 Birmingham Chess Club. Mr. W. Cook gave his opponent ,' 

 Knight, and after eight moves only he arrived at a position whici 

 enabled him to win the game in a brilliant manner. 



Position after the eighth move. 

 Mr. Wilso.n. 



BLACK. 





WHITB. 



Mr. W. Cook. 

 White here announced Mate in four moves, and proceeded as 

 follows : — 

 Q takes Kt B takes Kt (last) 



(Black cannot take the Queen, on account of B takes EP mate.) 

 Q takes KP (ch) K takes Q 



KP takes B (ch) K to Ktsq 



R to 118 (mate) 

 Play of this kind hasverj- truly been called the '' poetry of the 

 game ; " but of far more importance to the learner are those 

 examples where the game is won by correct and strong play only. 

 The most interesting endings are those where the Knight plays a 

 leading part. We give as an illustration an end-game which oc- 

 curred in the match between Messrs. Blackbume and Gunsberp. 

 showing how, with an equal position, tho Knight with correct play 

 did win against a Bishop. 



Position after Black's 52nd move. 

 Mr. Blackbcene. 





WHITB. 

 Gu.VSBERG. 



In this position, which (as can be seen from the number of mOTM 

 matle) was arrived at after prolonged manoeuTring with tli» 



