168 



KNOWLEDGE 



[July 1, 1896. 



kept out of each other's range, will possess this property, 

 namely, that every set of .c' squares within it that includes 

 .(• Queens will be a solution of the .r {Queens problem — 

 where .r equals the number of squares in the side of the 

 board — but "out of range" will vary according to the 

 length (.r— 1 ) of the Queen's stride. 



Take solution a, and remove the right-hand " initial 

 position," and notice what sijuares arc available for Queens 

 in substitution for those thus removed. The only other 

 vacancies are at H!i and 15, and two Queens there will give 

 solution fl. 



Similarly, solution ^-the left-hand " initial position " 

 being removed (restoring the missing Queen at F8)— with 

 substitutions at C7 and E4, will give solution i). 



At first sight there would appear to be no connection 

 between the symmetrical solution shown on page 24 and 

 the above. But let one of the " doubled positions" be 

 superimposed on another, and one of them rotated half-way 

 round (through 180 ), using the meeting point of D5 and 

 E4 as centre of rotation. The rule as to omitting all 

 Queens where overlapping now occurs must be observed, 

 and the symmetrical solution w will be at once obtained, 

 being the central part of this position. The reason for the 

 symmetry is thus ascertained. It is a little curious that if 

 a board of twenty-live squares, with its unsymmetrical 

 solution (A3, B5, C2, D4, El), be treated in this way, the 

 turning point being easily seen, the solution w for a board 

 of sixty-four squares will be arrived at. 



Take this position w (page 24, second column) as a fresh 

 starting position, and transfer it two S(|uares to the right. 

 This will throw two Queens off the board, but the one on 

 the fifth rank comes on again at A.5. The other, in being 

 brought on to the third rank at H8, pushes HI to Bl (the 

 only available square), and the solution k is obtained. 



Let this solution k be another starting position, and 

 transfer it three squares upwards and one to the left. 

 This will throw four Queens oft' the board, but the two now 

 on the C and F files produced will at once come on again 

 at CI and F2. The two others, which should come on 

 again at D3 and H8, find both those squares " covered," 

 and have to be satisfied with D8 and H3. This gives the 

 remaining solution a. 



The last five solutions read thus on the ordinary 

 board : — 



Ul 

 K 

 S 



The twelve :\bsolutely distinct positions admit of rotation 

 bringing each of the four sides in turn to the top, and 

 giving forty-eight changes. There are two repetitions 

 arising from the symmetrical position, and the number is 

 reduced to forty-six. All these can be turned over from 

 right to left, making a total of ninety-two. The number of 

 distinct ways of actually arranging the Queens so as to 

 solve the problem oh tlic lioani is twenty-lour, and no one 

 of these positions can be changed to any other without 

 moving the Queens to other squares. But from the 

 twenty-four positions the remainder of the ninety-two can 

 be obtained by simply turning the board and Queens 

 en bloc. 



The application of the above principles to boards of 

 other dimensions might produce some interesting results. 



Mr. Ashdown sends also an ingenious automatic diagram 

 for producing positions on a forty-nine-square board. The 

 height of the diagram is seven squares, but the board is 

 unlunited in its extension towards the right. Placing 

 Queens on A7, B3, and H7, and taking these squares as 



starting points, lines of Queens are constructed sloping 

 (Idiniiriinls tiiiviinix the riii/ii, each Queen being the distance 

 of a Knight's move from its neighbours on the same line. 

 Thus, the first position is A7, B3, C6, 1)2, 1<'.5, Fl, (14. 

 Now omit the A file and substitute the II file, and a fresh 

 position is obtained. This process, which is available 

 whenever the height of the board is ((la -|- 3 ± 2) squares — 

 i.f., for boards of five, seven, eleven, thirteen, etc., squares — 

 may be continued until the positions repeat themselves. 



CHESS INTELLIGENCE. 



Mr. E. 0. Jones has again won the Challenge Cup of 

 the Craigside, Llandudno, meeting. 



The Nuremberg International Tournament is fixed for 

 July 20th. It will conclude on August 0th. The prizes 

 are very valuable, and all the leading players except Dr. 

 Tarrasch are expected to compete. 



A match of one hundred players a side, representing the 

 north and south sides of the Thames, took place on 

 May 9th, at the Cannon Street Hotel. The northern 

 representatives were victorious by r)7l to 42.',. 



On May 15th a team of eight players of the City of 

 London Club, by no means their best team, succeeded in 

 drawing a match against the newly-established Divan 

 Association. The merit of the performance will be apparent 

 from the fact that the latter team consisted of E. Lasker, 

 T. Gunsberg, E. Teichmann, -T. Mason, L. Van Vliet, 

 S. Tinsley, A. (luest, and E. F. Fenton. 



Entries for the Brujhton Societij Two-move Self-mate 

 Tourney should reach the Chess Editor, 101, Queen's Road, 

 Dalston, N.E., before November 1st. Three prizes are 

 offered, and the adjudication will be by experts. 



Contents of No. 128. 



The Nature of tlie X Rtys of 

 Riiutwu. Bv .T. J. Stewurt, 

 B.A.Cilntlxli., B.Sc Lonrl 121 



Brief Description of the Orcliiri 

 Photographs. By H. A. Bur- 

 berry ,'F.R.H.S. '{Illmlraei)... Hi 



A Geographicul Description of 

 the British Island?. By Hu^h 

 Kobert Mill, D.Sn., F.RS.E. .. IJ:'. 



Protective Resemhlauee in tlie 

 Nests and Ei^gs of Birds. Bv 

 Harry F.Witherhy. (Jllus(ratri) 125 



8nn-Syuiltols in Ancient Ei^ypt. 

 ByF. W. Kead. (liiiKtr.ited).,. 127 



The Royal Society of Painters in 

 Water Colours 128 



Science Notes 129 



Letters :— David Flanery ; David 

 E. Hadden; Helios; F. H. 

 Worsley-Beuison .' 130 



PAGE 



Tlie Approaehintr Total Eclipse 

 of the Sun. By A. Fowler, 

 F.R.A.S. (Illustrated) 131 



Photograph of the Cluster Messier 

 24 Clvpei. By Ifaac Iloherts, 

 D.Sc.', F.R.S 13t 



Notices of Books 135 



Waves. — VI. St luding Waves in 

 Flowing Water. By Vaughan 

 Cornish, M.So. (IHustrnleil) .. 13G 



A Rare Metal. By T, L. Phipson, 

 Ph.D 140 



Sertiilarian Polyi«doius ; or 

 " Horny Corallines." By P. 

 L. Addison, F.G.S., Assoc.M. 

 lust.C.E. {lUn^trattA) 141 



The Face of the Sky for June, 

 By Herbert Sadler, F.R.A.S. ... 143 



Chess Column. By C. D. Locock, 

 B.A.Oxon 143 



Two Plates. — 1. Photographs of Orchids. 2. 

 Messier 24 Clypei. 



Photograph of the Cluster 



NOTICES. 



The numbers of Knowledge for January and February of 1S94 can now be 

 had, price One Shilling each. 



Bound volumes of Knowledge, New Series, can be supplied as follows:— 

 Vols. I., II., III., and VIII., 10s. 6d. each ; Vols. VI., VII.. IX., and X. (1895), 

 8s. 6d. each. 



Binding- Cases, Is. 6d. each ; post free, Is. 9d. 



Subscribers' numbers bound (including case and Index), 2s. 6d. each Toltune. 



Index of Articles and lUuBtrations for 1891, 1892, 1894, and 1895 can be 

 supplied for 3d. each. 



TERMS OF SUBSCRIPTION, 



Annual Subscription, 8g., Post Feee. 



" Knowledge *' as a Monthly Magazine cannot be registered as a Newspaper 

 for transmission abroad. The terms of Subscription per annum are therefore 

 as follows : — To any address in the United Kingdom, the Continent, Canada, 

 United States, Egypt, India, and other places in the Postal Union, the 

 Subscription is 8 shillings, including postage ; or 2 dollars ; or 8 marks ; 

 or 10 francs. 



For all places outside the Postal Union, 6 shillings in addition to the postage. 



Commimications for the Editors and Books for Beview should be addressed 

 Editors, " Knowledge" Office, 326, High Holbom, London, W.C. 



