■1.8 



KNOWLEDGE. 



[February 1, 1896. 



the fundamental positions a, /3, y, etc. ... a, and w &re 

 in any way connected — i.e., whether one position can be 

 obtained from anotlior by a uniform shifting of the pieces 

 in some deliuitr direction. To this i[uestiou we find the 

 remarkable answer tliiit some of tho positions are in this 

 way intimately connected with one another, while the 

 remainder seem to bo entii-ely independent. The positions 

 which, for want of a better word, we shall call " transfer- 

 able " are those labelled a, /3, y, S, ?, i, and >.. The 

 remainder, s, •>;, 0, k, and k, are not transferable by any 

 process which we can discover. And yet the italicized 

 positions, Nos. !) and 10 in Class (', and still more Nos. 1 

 and 5 in Class 1), show how nearly the non-transferable 6 

 approaches to tho transferable a. In the latter instance 

 the six CJuocns to the left remain the same ; fl being 

 obtained from a simply by shifting the (,)ucens on KKtsq 

 and Kll") to KKtS and Klisq. In no other case is such 

 a trifling alteration possible, so to speak, at the last 

 moment. 



5. The connection between a, S, y, S, 1^, i, and X is as 

 follows : — I'o obtain S from a, move the pieces forward one 

 siiiKor. (A Queen on c8 must, of course, be shifted to cl.) 

 Now move the position 8 forward one square and we obtain 

 I. Again take a, and move the pieces (nu'srjunir to thi- right : 

 this results in the position j. 



A is obtained from a by moving the pieces one square 

 forirard ilini/otmlli/ to the rii/lit. 



The rule for diagonal shifting is as follows : — A piece 

 shifted to an imaginary square in continuation of a rank or 

 file must be placed at the ot/ier end of such rank or file, 

 while a piece shifted beyond a I07111 dini/oniil must be 

 placed at the opposite corner of the board. For instance, 

 a piece moved one square forward diagonally to the right 

 from QB8 must be placed on Qsq ; if from KR8, it must be 

 placed on QRsq. 



To obtain /3 from a, shift the pieces one square hackwards 

 diagonally tuthf /('/'?—/.(■., precisely in the opposite direction 

 to that used for obtaining A. 



^ is obtained most easily from S by moving the pieces 

 one square hacliirards iliai/onalli/ to the ri;/lit. 



[lOj is obtained from tw by a curious method. Move the 

 pieces on files a and b one square fonrards : files g and h 

 one square /'"c/."(/)v/s ,■ the two distant pieces one square 

 to the h'i't : and the two nearer pieces one square to the 

 right.'] 



6. We will now endeavour to ascertain why the 

 positions e, ij, 9, k, and cy are not transferable by any 

 shifting of the pieces in a uniform direction. For this 

 purpose we give a list of these five positions. 



abcdefgh 

 (e) 8, G, 2, 5, 8, 1, 7, 4. 

 (r,) 2, 5, 7, 4, 1, 8, C. 3. 

 (9) 2, 7, 5, 8, 1, 4, 6, 3. 

 (k) 2, 6, 8, 3, 1, 4, 7, 5. 

 (cy) 3, 5, 2, 8, 1, 7, 4, 6. 

 [We have placed the names of the files at the head of 

 the columns in order to have the figures themselves close 

 together for comparison.] 



Now, examining these figures, we notice — 

 (i.) That the square el occurs in every case except (e). 

 (ii.) In every case except that of («) the figure 8 is 

 next to the figure 1 : moreover, this will be foimd to be 

 the case ivhidiever rraij the hoard is turned. In the trans- 

 ferable positions the figures 8 and 1 are never adjacent. 



(iii.) In £,■)], and 6 the number of a is within one unit 

 of the number of h. This is never the case in anij of the 

 transferable po.iitio7is. 



We conclude, therefore, that the conditions which do 

 wot favour the simple transference of one position to 



another are: — (i.) The use of el (King's square) when ali 

 ali, a3, or a4 are occupied, (ii.) The use of opposite ends of 

 (iiljiieent files, (iii.) The use oi a number on one Hook's 

 file within a unit of the number used on the other Hook's 

 file. 



7. I'lnally we arrive at the question — why are there 

 ninety-two methods of solving tho jn'oblem, neither more 

 nor less ? Our mathematical abilities proving insufficient 

 for the deductive solution of this question, we tried the 

 device of using smaller chess-boards of IG, 25, 3G, and 4i) 

 squares respectively, hoping to find some recognizable 

 series of numbers which should determine the law. In 

 this we were disappointed, for whereas the numbers 

 increase generally with the size of the board, there is a 

 remarkable exception in the case of the board of 36 

 squares. Our results are recorded below : — 



Hoard of 16 si|U;uvs Acluiil melliuJ.s -2 



25 ., „ 10 



36 „ „ t 



49 „ ,. 40 



64. „ „ 92 



In the Six Queens problem on a 3G-square board the 

 comparatively small number of solutions seems to be 

 atoned for by their symmetrical beauty. It is remarkable 

 also (1) that the corners are never occupied, and (2) 

 that the colours of the squares are never equally divided 

 among the Queens — there are always four Queens on one 

 colour and two on the other. On the full-sized board, four 

 of the Queens must always be on White squares and four 

 on lilack. 



In conclusion, we must apologize if, in our examination 

 of this (to us) intricate problem, we have anywhere 

 sacrificed intelhgibility to condensation. We sincerely 

 hope, also, that any reader who knows or can discover the 

 law which governs the problem will kindly communicate 

 his information. 



lisseiillal met/tods — 1 

 2 

 1 



1^ 



Contents of No. 123. 



Geography as a Science in Enij- 



land. By Hugh Robert Mill, 



D.Hc 



The Great Red S])ot on Jui>iter. 



By E.Walter Maunder.F.R.A.S. 



Jupiter. By Nath. E. Green 



Periodical Comets due iu l?{t6. 



By W. T. Lynn, B.A., F.R.A.S. 

 Waves.— 1. The Waves of the 



Ocean. By Vaughau Cornish, 



M.Sc. (Illustrated) 



The Banana. By Richard Benyon 

 Onr Fur-Producers, — I. Otters, 



Skunks, Badgers, and Gluttons. 



Two Plates. — 1, Jupiter, from a Coloured Drawing by N. E. Green, F.R.A.S.; 

 3, It:ilian Renaissance Medals. 



PAGE 



By K. Lydekker, B.A.Cantab., 



F.R.S. (Illustraied) 10 



Science Notes l:J 



Letters :— David Flauery ; Alfred 



J. Johnson 14 



Notices of Books. {Illustrated).., 15 

 Italian Medals. By G. F. Hill. 17 

 Parasitic Flowering Plants.— 

 ■ The Mistletoe and Dodder. By 



J. Pentland-Smith, M.A., B.Sc. 



(Illustrated) jfl 



The Face of the Sky for January. 



By Herbeit Sadler, F.R.A.S. ... 22 

 Chess Column. By C. D. Locock, 



B.A.Oxon 23 



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