May 12, 1882.] 



KNOWLEDGE 



597 



MATHEMATICAL PROBLEM. 



41] — Sliow that a proper fraction, which in its lowest terms 



kes the form - (a and b being prime to each other and 



J, . 2"> • 5" 



o 10), when converted into a decimal, recurs from, but not before, 

 be (m + l)"" or (n + 1)"' digit after the decimal point, according as 



> or < >i. — Gradatim. 



["Gradatim" supplies a solution of the above problem, turning 

 in the lemma that if a is prime to b, the fraction — = a decimal re- 







oning from first digit after point. I would submit that writing 



he fraction in the form - . - — or - according as m > or 



b 10'» b lO" 



< fl, the above result follows at once from this lemma. — Ed.] 



[42] — Given the length of three lines, a, b, and c, draivn from any 

 [Kunt within a regular polyg<.'n of n sides, to any three of its con- 



intivc comers. A, B, andC. Required a geometrical determination 

 the polygon, granting that the polygon of »i sides can be con- 

 itmcted when one of its sides is given. — Y. 



0\iv €l)t^5 Column. 



Endings from actual games contested by Leonard P. Rees. 



HIGHGATE CHESS CLUB. 

 President — Professor Tomlinson. 

 CONSVLTATIOX GaME, PLATED JIarhi 6, 1882. 

 Kings' Gambit. 

 Wliite. B)«k. Wbitf. Blwk. 



P to Ki P to Kt I 11. P to K6 P takes P (e) 



PtoKB4 P takes P I 12. Stakes P(ch) B takes B 



Kt to KB3 P to KKt4 ' 13. Q takes B(ch) K to Rsq 



B to B4 B to Kt2 [ 14. Kt to QR3 Q to Q4 



Castles P to Q3 I 15. Q to KKt4 QKt to K4 



P to QBS (a) P to KR3 I IG. Kt takes Kt y takes Kt 

 P to t,»4 Kt to K2 (b) 17. B to Q2 QR to Qsq 



Q to QKtS (<■) Castles 18. QR to Ksq (/) g to QB4(ch) 



P to K5 (<?) P takes P 19. Resigns. 



P takes P QKt to B3 | 



NOTES. 



(a) P to Q4 deserves a slight preference ; it enables 'White to 

 choose different lines of play, in addition to his having the option 

 of arriving at the po.silion in the text by following ivith P to B3. 

 The attack obtained in this variation of the King's Gambit is very 

 indifferent. 



(6) His best. 



(f) Sounder than the usual move of P to Kt3, as Black, by 

 replying to the latter move witli P to Kt5, will obtain a good game. 



(d) Hero we think P to Kt3 might have been played with more 

 safety than before ; there is nothing to bo gained by P to K5, which 

 weakens White's centre. 



(e) This is good enough, but Black might also have played 

 P to B4 in order to be able to win ; the P on K6 play would, how- 

 ever, have become more dithcult. 



(/) This was an ovcrsignt i he now loses the Bishop. Ho might 

 have played B to Ksq with a view of playing B to B2 and getting 

 his Rooks into play, and also bringing his Queen's Knight into 

 active operations. Black had an extra Pawn, but his King's side 

 was exposed, and Wliitt- still had some chance of retrieving his 

 fortunes. 



TIio following bright game illustrates the attack obtained by 

 10. KKt to Kt5 in the Giuoco Piano : — 



White. Dlack. 

 Lioiinrd P. Bees. 



1. P to K4 P to K4 



2. Kt to KB3 Kt to QB3 



3. B to B4 B to B4 



4. P to B.J Kt to B3 



5. P to Q4 P lakes P 



6. P takes P B to Kt5(ch) 



7. B to Q2 B takes B 



8. QKt takes B Kt takes KP 



9. Kt takes Kt («)P to Q4 



10. KKt to Kt5 (!-)P takes B (o) 



11. Q toR5 PtoKKt3(d) 



White. 

 L P. R. 



12. Q to RO 



13. R to Qsq 



14. P to B4 (e) 



15. Q to Kt7 

 10. K to Bsq 

 17. Kt to B3 



Q takes P 

 Q to K4 

 Q takes BP(/) 

 Q to K6(ch) 

 Qto B5(ch) 

 Q to B4 (.j) 



18. Q lakes R(ch) K to K2 



19. K to B2 P to KR4 (/;) 



20. Kt to BO (i) Q takes Kt 



21. KRtoKsq(ch) B to K3 



22. R to Q7(ch) liesigns 



NOTES. 



(a) As we have shown in our analysis of the Giuoco piano, 

 p. 442, White can continue ivith 9. P to Q3, obtaining thereby a 

 very fair game. 



(b) In the same analysis we characterised the move as an attacking 

 style ; it may become very dangerous, but, with correct defence, it 

 will prove less effective. 



(c) This is exactly what nine players out of ten will do, but we 

 demur to this move. White will obtain a strong attack by Q to R5. 

 Black can prevent this by just delaying the capture of the piece 

 for ono move, and playing instead 10. B to B4. It is obvious 

 that the key move of White's attack, Q to R5 would be bad now. 

 as Black would reply with B to Kt3. Q to K2 would be met by 

 Black's castling. 10. B to B4 destroys White's attack entirely, 

 and, in our opinion, even gives Black a superiority ; for supposing 

 now B takes P, Q takes B, White's Queen's Pawn must eventually 

 fall. 



(d) Q to K2 would be met by White Castling, with a good 

 game. 



(e) An attacking move; 14. Castles would also have been good 

 play. 



(/) Black's position is very precarious; 14. Q takes KtP would 

 have prevented for a time the entry of the White Queen on Kt7. 

 14. Q to R5(ch) would have been met by 15. Kt to QB3, and, if 

 Black then proceeded with 15. B to Q2, with the idea of Castling, 

 White wonld play 16. Kt takes BP. 



(3) White has played very well. Black is now compelled to give 

 up his Rook, as otherwi.se he would lose his Queen, i.e., 17. R to Bsq, 

 18. Kt to BG(ch), 18. K to K2, 19. Ktto Q5(ch). 



(h) R to Ktsq was better, as he could then play B to K3. 



(i) This wins the Queen ; KR to Ksq, however, looked stronger. 



ANSWERS TO CORRESPONDENTS. 

 •«* Please address Chess-Editor. 



Leonard P. Rees. — Best thanks for findings, which were 

 ingeniously played. Solutions of No.s. 36, 37, and 38 correct. 



G. Licence, — Solutions correct. Problems received will be 

 examined. Thanks for good wishes. 



H. A. N. — Problem received with thanks. 



Brenton. — Solution of 35, 36, and 38 correct. 



A. McD. — Solution of 38 incorrect. 



J. B. B.— Solutions of Nos. 35, 36, and 37 correct. 



Correct Solution of No. 38 received from G. W. Edward Wilson, 

 Fusee, Moleque. 



