rssN 0267 -36 9X "h ,o, 4r9 G A N/[ E S ,o+o Iqque LZ (Sept-Dec) 1989 @ Editor and Publisher t7 G.P.JELLISS, {_} g 99 Bohemia Road F -ZJ 7' L E St Leonards on Sea, TN37 6 RJ" FINAL ISSUE ,T O LI R. N A T- INCLUDING INDEXES
the conclusion that this will have to be the last issue of the Journal. Regretfully I have come to - The number of.subscriptions has unfortunately not compensated for the editorial labour expended. I did consider carrying on into a second volume as a series of special issues, but now that I have a word processor it is possible to expand these into more ambitious works, even full-scale books. Details of these will be announced in my new quarterly Variant Chess, and elsewhere, in due course. The chess problems and chess variant games have already been transfened to Variant Ch9ss, which has been appearing regularly and attracting a good readership. I had hoped to tie up all the loose ends, but it will be found that there are still a few items that are incomplete. It has been necessary to avoid any further delay in publication - my apologies for the six month delay in making this issue. Authorship note: All articles that do not carry a by-tine are written or compiled by the editor.
Looking back through my notes from chessics I found this diagram of the 6-dimensional 'cube' with ."tl e@n to the same length by means of V0S moves, that is (1,8) or (q,7) moves on a 25x39 board. See Chessics I4 1982. page L82 THE GAMES AND PUZZLES JOURNAL
For special cases the formula gives: TYING UP SOME LOOSE ENDS Rook: 2m-1, Bishop: 2m-gcd(m,2) Queen: 4m-gcd(n,2)-1, Nightrider: Panta gounting Verse-Forug. Stefanos zis 4m-2(gcd( m, 3 )+gcd(m,4)+gcd(m, 5 ))+3. writes to correct an error: p.94, issue 5+6: rrwhere P. Cohen claims that the Catalan The p1 42. numbers count rhyme schemes. This is Centres of a Tliangle. false (as explained exactly in the reference f the medial circle F poiilt, given: i.e. M. Gardnerrs Time Travel and and the incircle is at the euerbach other Math. Perplexitie where the;i tquch. The medial circle also ( ne Bell numbersrt touches all three ex-circles. K.lV. Feuerbach, and is significantly larger. For n=4 (quatr- Eigenschaften einiger merkwurdigen Punkte geradlinigen ains) we get 4th Catalan no: L4 and 4th Bell des Dreiecks, Erlangen L822). f grow no: 15 and then the dif erences larger. Nimmily. The 'Black Spott variant can be The Catalans fail to count all rhymes like ffi-singte move, So further conditions ababc or abcab or abbab etc. where the are necessary. Specifying onIV conYex circ- lines joining identical letters have to cross. lets (i.e. no internal angles greatei than 1800) Thus abab is the difference of 1 for the 4th may be sufficient. (See p.67). nos in the sequences." Neither of these two sequences answer the original problem (p.14 A Gnomonic Question (p140) of issue 1) of the number of rhyme forms ?tstrictly T.H.Willcocks notes a correction: interpretedtf (i.e. a rhyme occurs the progression illustrated is at least twice) and t'irreducible" (i.e. the verse cannot be split into two sepaarate un=3un-1-un-2 strict verses - see p.30, issue 2). Solutions for some multiples seem not to fit into any easily recognisable series. The . This has Game lnventi Here are two examples for 6 and 31: PathYr so iS ca{rcelled. I had several ideas of my own for games on the circular board (p.117) but will save them up for publication elsewhere (hopefully in a book on new games). Michael Keller rePorts a game with the black pieces on black and white pieces on white, but on a L 0x8 boardr called rrPonentsrr Copyright Reiss Games I97 4, This is basically a type of Halma.
Riders on a Toru,s p162. E-tmar eartel wrote 26 viii 1989: tfEight years ago I developed a general fortn- ula for (a, b) riders on a (m, m) torus, i.e. a quadratic torus, not the general (m, n) torus. The formula is: z(a, b, m) = 4m z*lgcd(a, b) + 3gcd(2, m, a+b) where A = gpd(n, a2-O2) + gcd(B, 2ab) + gcd(B, a2+b'2) anct B = gcdG, b).m this is valid when gcd(a, b, m) = l and bla. Car Park Patienee. The diagram on the right The last two statements are no constraint to an illegal Position, since the generality of the formula. ... f cannot the thorizontal' card missing from the second give a proof of it, but there is no counter- row could not have got out. This error was example ... the computer didnrt"find any up introduced by the Editor in combining two tom=60. diagrams of the original text. The moves illustrated however are all quite legal. THE GAMES AND PUZZLES JOURNAL page 1 83
Hubert Philtips. Alan Parr has consulted Eddi+gtqnls Cricket Problem. Answers (t g61-1 9?0) which gives his Jenkins fell to speedwen, eodkins to Toss- dates as: 13 xii 1891 I i 1964 and much well, and wilkins was not out. Score at the else! ft...& remarkably talented man (t either g, fall of each wicket: 6, 12, 1 ZJ, 31 , 4L , 44, hadn't realised or had forgotten about his 59' 59, 60. For full solution I wilt have to political career and also his 200 detective refer you to F'red Hoyle The Nature of the stories...). My local library tracked down Universe pages ZZ-24. Journgy to ,Nowhere ltst and onlv completed volume of autobiographyl but...it covered onlv the first 20-25 years of his life and gave Broken Chessboards. Len Gordon sends the no insight into his games interests." His les that have unique detective was Inspector Playfair. solutions, They use the T-piece of Tsquares.
Professor Hoffmann. Len Gordon queries my g House Puz zle" to the ttProfessor't, saying that Slocumrs compendium does not irnply this, and he is probably right. tlHoffmannft was the pen name of a collectort noted in the the Rev. Angelo John Lewis, as Fi.rst, Lqst and Only Numbers. (p138). to a reprint of the Foreword by L. E. Hordern Michael Keller and Gareth Suggett each (puUtished by Puzzles Otd ancl New of 1893 recommend David Wells' Park Road, @ + Kensington Dictionary of Curious London W11 zER). ?n@ ffinly number which is the product of three numbers and also the sum of the
, Pqntangle Books. The puzzle manufacturers, same. AIso the square root of the sum of Pentangle, also have a mail-order book list, their cubes! G.S. mentions that 33 is the which includes Hubert Phillipsr My Best Puzz- largest number that is not the sum of les in Logic and. Regsoning, Prof Floffmann's distinct triangular numbers. The other election of other examples they mention are cases of what I defined as rather than of uding Martin Gardner. Address: rrNumerology'f"Digitology" Pentangle Mail Order, Over Wallop, Hamp- it is easy to extend the shire, SO20 8HT. list vastly if we allow digital properties.
Haff A Cake. (page 153). An approximate . (p IT 7). ffirre area bisector baratlel to the Gareth suggett notes that Tom Marlowfs base is 2/7 of the height which divides the piece on palindromic cubes suggests gen- triangle into two parts in the ratio 25:24. eralisations. The list of palindromic 4th The exact value is 1 L/,{2, A cake that can powers starts: 1, 140 4L, 104060401, be cut into six equal parts by three area 1 004006004001, 10004000600040001. bisectors is as in diagram E. In other number basesr so long as the binomial coefficients ncr are single digits the nth powers of 11, 101, 1001 ... will-ue palindromic. Reverting to base 10 there are plenty of palindromic squar€sr but few of these have an even number ol digits, the smallest being 6g8gg6 = 9362 oih"rs are : 7 9864 42, 6*0306 482, 831631 1 s 4g82, 636083 zgzbSg 92 (ref. Neitn Devlin's column in Computer Guardian a few years Uact< EniFma 1. Further clues. What is: ZgZ and Gymnastically, an exercise: 982?58 25201b 8TZ (ref: Mike Bennett in the Nautically, a measurement: "Micromatterst? column in IEE News lggg). Physiognomically, a smile. page 184 THE GAMES AND PUZZLES JOURNAL
ANKH, ARCH, BEAU, CIAO, ECHT, ESTH, The Language Simulation Problem ETCH, EUOI, EYAS, EYED, EYES, EYOT, INCH, ITCH, LIEU, MAYA, MOUE, ONST, QUAY, QUEY, SCRY, SHRI, SHWA, SKRY, SOYA, SPRY, THRO, TSHI, UMPH, YAUD, twordr YAUP, YEAD, YEAH, YEAN, YEAR, YEUK, Mathematicians define a to be any YOOP, YOUD, YOUK, YOUR, YOYO, YUAN, sequence of letters from a given alphabet. ZOEA (plus plurals like ABBS, ABCS, ACTS)" Thus if the a).phabet has n letters there" The numbers of pronounceable words of m words letter fiZ of two, and are n oi one , letters however still increase exponentially in general nm of m letters. That is, the with m,- viz: l-Ietter, v words; Z-let-ter, nZ- number of words increases 'exponentiallyl -cz vZ + Zvc; 3-tetter, 3v2s * 3ve2; 4'- with the length of the word. = tetter, 2v3c + 6vZcT + icSr; and so on. In a real language the number of words of length m increases until m is about 10, but Second, we must drastically reduce the then falls away rapidly until m is about 20, possibilities for forming long words. One as shown in the following graph, which is obvious way of doing this, though not ver:/ verisimilitudinous, is prohibit the use of based on a rough count of the pages per to section in chambers' words (which lists any letter twice. Thus the longest words would be of n letters (i.e. using each of the English words by lengTn)Lhis list does not include plurals ending in but letters of the alphabet once). The longest -s, an adjust- pronounceable ment to account for this would not alter words would be of 3v+2, the shape of the curve significantlv, just 3c+2 or v*c letters, whichever expression is the smallest. (tf increase its height. No doubt other adjust- c case, the first expression ments are necessary for accur&cy, but is appties, and with it v=6 just the overall picture we,require here. and c=20 we find the maximum length is 20, as required.) An example long word ?a'oc o in this system might be the improbable: ,'lr STALDERVINGOMPUZWYCH WCRrr:- (e vilage somewhere in northern Europe?)
Two incidental puzzle questions emerge here: (1) What is the longest normal word that has no repeated letters? UNCOPYRIGHTABLE, 15 letters, s€ems plausible but is not explic- itly in Chambers. (2) What is the longest that has every letter occurring twice? The best I How can we account for this radical diff- could think of is UNICONSCIOUS, but again erence between the mathematical idea of this is not in Chambersr so I must have made a 'wordt and the real thing? it up, but it seems perfectly feasible. question must introduce A more structured approach to the First, it is eviclent that we form mono-syllabic root pronounceability into our might be to first some improved prefixes and suffixes (such as ACT' twords'. For this we need to worcls, mathematicat RE-, EN-, EX-, -lNG, -ED, -OR, -lVE, -loN, our alphabet into v vowels and c separate -LY) and then to form longer words by com- + 11. pronounce consonants, v c = Let us a bining these monosvllables (nEACT, ENACT' rrpronounceable" (a) word if it contains at ACTING, ACTED, ACTOR, ACTIVE, (b) EXACT, least one vowel and it does not contain ACTION, REACTION, ACTIVELY, REENACT, more than three successive vowels or three ENACTED, REACTOR, EXACTLY, and so on). successive consonants. This gives us quite a This is more akin to the way longer words are good provided facsimile of normal words we actually formed, but I do not see how to countYasavowel. simulate this in a simple quantifiable manner There are just a few three-vowel and three- at present. The number of longer words gets consonat combinations that are pronounce- less and less presumably because many of able, that eould be allowed as exceptions to the possible combinations have no corresp- the general rule if we wished. The short ondence to reality, e.g:. Bluebells exist and words that fail the above tests are: AEON, Harebells, but not Redbells or Dogbells. THE GAMES AND PUZZLES JOURNAL page 185
Grid Dissections
An item I contributed to chessics 23, (page ?g) Autumn 1985 dealt with @gieJ with the hexomino shape E-..rf-' r It gave a solution for the 19x28 rectangle and showed how that solution could be adapted to rect- angles of the form (Zg+7ilx(28+?b) where either a or b or both must be even. (rnis form- ula was previously given in a clumsier form.) It was also stated that rectangles up to z8x2g had been checked and no other solutions found. It now emerges that this check overlooked the 21x26 solution shown in the diagram. This was found by KarI A. DAHLKE and published in Science News L32 (za) p310, L4 November 198?" V to expanO this result to 26x28 by the method shown in chessics. It thus joins the series of solutions p@ found, but has the additional feature of dividing into two congruent parts. The 19x28, using two fewer hexominoes remains the smallest known solution. (December 198 9).
Calculator Keyboard Curiosit ies By R.J.COOK The single-line palindromic numberso.first paragraph p1?3, can be varied sequentially, thus: 591519, 195915, etc, and the same applies to the square and other four-digit patterns; 352536, and so on. After a little experimentation one soon learns to discover new patterns. A few more, contributed by interested friends, are indicated in the diagrams. There are also'degeneratersequences using just 2 or even only 1 key, e.g. 133311 or 444444.
HNNWM@MMNL73937 L72827 151959 14L858 521589 435675 134976 276834 482628
Remember that the sequences can be reversed or circularly permuted to give other eases. There is scope for number-theoreticians to probe the basis of these patterns' and for group-theoreticians to explore the symmetries, ignoring the key numbers. The 6-digit displays are a subset of the set of integers formed by multiplying the generating numbers 1001, ..., 9009 by 11L, and the symmetries of the key-sequences of the generating numbers transform in interesting ways to the symmetries of the 6-digit key patterns. For example, 1321 generates 146631 and its'image'1231 generates 136641. Empiricists also soon notice that the palindromic displays like 789987 are also multiples of 1.1 and therefore of L22L, that 12332t is generated by 1111, and the relation between displays arising from pairs of generating numbers like 4234 and 4324. There are a great many possibilities to discover, and patterns which cannot be recognised without practice. Ron Cook (27 vii 1989). page 186 THE GAMES AND PUZZLES JOURNAL
Swift Solutions to the on Pl?5. 1. There are six arrangements.
7. stephen Taylorts solution is shown first, XH but J.D.B. and G.J.s. both found the second" The ratio of number of long to number of short lines are: A, 1:5; B, Cr 2z4; D, E, 3:3; F, 422. In E the four points are vertices of a regular pentagon (cf. 3 below).
2. Dudeneyts trick is to fold the paper so that two sides of the square coincide. Then 8. The proof qequired is: V(a2+O2) V(c2+a2) to insert the pencil and draw both lines simultaneously then open ppperr - out the = c)Z+(ad)Z+(b e)2+(uo)21 and this keeping the pencil-point in place,, and com- Vt(a = Vt(ac+bd)2+(ao-bc)\, or if ad(bc we can plete the diagram, which is now possible. substitute bc-ad. The set includes all the cardinal numbers since n = V(n2+A2)" 3. Half the star is shad€d, as shown by the following dissection: 9. Four elements bracket in b ways, and five elements bracket in L4 ways. tr Bn is the number of ways of bracketing n etem- ents then Bn = Bt Bn-t + BZBn_Z + BBBn_B *:.:.'BrBn-r+ror{Bn_l 81, with 81=82=1. This is the required recurrence rerbtion. To prove it we note that any bracketing of aloa Zo,..o&11 must be in the form AoAt where A is an expression with r elements and Ar has n-r elements. The number of 4. The fraction is 4115. A simple case of ways of bracketing A is Br and of Ar is Bn_r' simultaneous linear equations: (a+1)/b= L /3 so of AoA', without disturbing the leadinf and a/(b+l)=7/4, i.€. 3a+3=b, 4a=b*1. o is BrBn-r, and we m ust sum these over all the n-l positions of the leading o. 5. Possible anwers might be: An explicit formula is: Bn = Qn-Z)I/nl(n-l)l A 5: only one of the form 5k. but it is easier to calculate by iteration. B 7i only one of form 4n-1 . C l, 3: only one of f orm gx+y2. 10. The very neat proof is: D LTz only one not of the form (xo(aob))oy = a(a+1) ! t. ((xoa)ob)oy [by associativity of a] = E 29: only one with an even digit. (xoa)o(boy) tUy associativity of bl = xo(ao(boy)) lby associativity of a] = 6. Stephen Taylorrs solution is: xo((aob)oy) [by associativity of b ] and equality of the first and last expressions shows that aob is associative. It is interest- ing that a{ five. methods of bracketing the four elements occLy in this proof 11. Snakes & Ladders. The following treat- But other solutions are also possible: me Rev. R.A.Dearman: (From John Beasley and Gareth Suggett). The greatest obstacle would be 6 squares ahead of us. Let pn denote the probability THE GAMES AND PUZZLES JOURNAL page t8T that we will land on the square n ahead of number of hands with a of one suit, b of an- where we are now. Thus Pn=0 for n(0, and oth€rr c of a third and d of the fourth. Then p0=1. If we are to land on a given square We N( 4-4 -3-2)/X ( +-S -3 -3 ) must come to it from one of the six preced- ing squar€sr with the appropriate dice-throw. = 1u?!"'3f)ifre3#'[1'3'ii' So pn=suffir=l...6(Pn-t/6), for all n)0. For 1'3#' n=1...6 this gives a geometric sequence With = 3 (to /4) (J/11) = 4s/zz pl= I /6 and pn=( 7 /6)pn-t (t de Bonots L-Game Two players who have worked out (and remembered) all the possible final winning By Malcolm HORNE positions might find the game rather longl winded This is a two-player game (due as they could possibly avoid these to positions! Edward de Bono) using But itrs quite a nice game if not a 4x4 board. Each played player has an L-shaped piece (a tetromino) at that serious level. and there are two other l-cell pieces (mono- minoes) marked with a black spot. Example game: (postal, 1gg6): Malcolm Horne v. Andy Skinner 1. Ftip L-piece to L-piece to 5 /6 /r0 /t+ and 3 /i /tL /r2 (Spot + 1 to 8). (Spot 8 to Z) O 2. 6/L0/14/15 (rO-+) z IetL2/16 (4-5) 3. 9/L0/LL/Ls (S-O) g (2-1 8 /4/8/L2 6) 4. 7 /9 /LA /n (6-2) 1{ -: Beta has nowhere to move his L-pieee, l2 _i_ r3 o Starting Position. . A go consists in a move of your L_ piece to a new position (these pieces can slide, turn or pick up and flip into any open position other than the current positiloni. Tl,tl may be followed by an optional move of just one of the neutral monominoes to any open square. Final Position. page t 88 THE GAMES AND PUZZLES JOURNAL ZIGZAG SOLITAIRES By Leonard J. GORDON look like zig-zagged. Here is what the classic 3 3 and 37 cell cross-shaped solitaire boards And also the circular board, shown on the front of the double issue 8+9 of G&P Journal. z,IGz.AG CROSS SOLITAIRE For a starter, here are two solutions to the 33 cell board. Be eareful to note that there is no line from 4 to 8: 6-171 8-!o, 1-6-3, 1e-lo-8, 7--J, 34-1o-8, ?t-7-9, ge-B-lA, ?E-?4-lt-?€.1 ?7-?3, l3-i-€.-Jrrr f,3-e5r 31-3O-19, 38-95-11, 3-l-3-5-18-?9-?'e-17. (15 rnc,ves, cerrter-cerrter). ?i--?-u. g4-36, ??-?41 l3-E7-93-33' 1l-e0' 7-??-?4t 3l-?3-?5' 9-11-13-3€,-t4' 1-iO, 3-ll-?3-i3-9-6, ?t-7-9-!-3-10-34' 3O-eB' 33-31-e3-e5. (13 n.:'ves) I wonder how the type of field I have described in these zigzag solitaire articles would function as a base for a war-game. I would be pleased to hear from anyone who tries it. It differs from the "Leapfrog't board pattern in that all intersections (except those on the edges of course) are of equal value. I have also experimented with this field for plane tiling puzzles. The pieces are nice looking'ranimalsr'. Another way to allow jumping pieces to escape from a limited set of cells on an ortho- gonal board is to allow a jump over either one or two (or more) pieces in a row. I employed this rule in a puzzle-game sold as ttleaps-N-Bounds" several years ago. Len Gordon (g iii 1989) ZIGZAG CIRCULAR SOLITAIRE If jumps t-}L,2-2L, !g-37, etc are allowed, but jumps 1-20, 31-33 are not allowed, an example solution is as follows. Irm not sure just how good this is. ?4-37, 3l-34, er-37, 36-33, ?7-37-?3t ll-e7, ?3-37t 3e-35 (B maves se far) 13-11, l5-13, l7-15, l-17, 3-1, 5-3, 7-3, 9-7 (16 msves so far) 1O-le-14-l 6-1S-e-4-5-S-33-5*33-e-31 -17-35-14-e5-37. ( l7 total ) If we allow 31-33 type jumps, I have a 15 move solution. Maybe the solution can be shortened to 15 moves using the above rules. The final sweep is spectacular, but being able to circle the field like that sort of weakens the puzzle; hence I suggest ... breaks in the outer circle. (See above) tnis seems to be a difficult puzzle. Remember, jumps 31-33 etc are not allowed. Len Gordon (zz iii 1999) THE GAMES AND PUZZLES JOURNAL page 189 Quick Questions SOLUTIONS on P.194' 19. Squaring the Rectangle. On page 81 we gave diagrams of a 'rgolden'r rectangle' with 13. Wazir Tours. Construct a Wazir tour of the property that if a square is removed at the 4x4 board, showing the square numberso one end the residue has the same ratio of ttsatintt 1, 41 9, 16, in a (i.e. One in each rank sides as the whole. What rectangle has the and one in each file). Two solutions' (G'P'J') similar property that a square removed from it centrally leaves two rectangles like itself ? Numbering Formulas. If the squares of the L4. (x, y) chessboard are specified by coordinates Manoeuvres. Move as few (1,i) (8,8), find formulas for 20. Matchstick ranging from to matches as little as possible to convert (a) possible numberings each of the eight 'rnaturalt' large and two small squares to two (J.g. at (8,8), 2 at (8'7) ." 8 at One of the board i Iarge anO one small; (b) A clockwise spiral g 64 at (t't) is a ie,t), ui t?,8)0"1 0 at 0,7)... to an anticlockwise spiral. (G.P.J.) natural numbering). (G' P' J') 15. A CrYPtarithm' Another bY l ll I'comPosed to mark T. H. WIiLCOCKS, I l- (!) -l a young brotherts ?Oth birthd&y" rt I l_l t_ TH I RTY (b) ThIENTY (r) TEN 2L. Super-dominoes. Agange the 24 three- SEV EN colour squares in a 4x6 rectangle, with adJa- THREE cent edges matching, and all the border edges WENTY the same colour. Here is a solutiofir not satis- the border condition, but showing the From a sequence fying L6. +l- Permutatioosi- three colours in the same pattern when the We are allowed to (a, b, c, ...) of numbers board is wrapped round a cylinder. This was by adding one elem- form other sequences published in the IMA Bulletin in 1987. ent to anoth€fr or subtracting it from an- other. By this process we can give the Iteventt Sequence any permutation, but not negat- an "oddtt permutatiolL We can also ive any even number of elements. We can combine an odd permutation with an odd number of negatives however, e.g. (a, b)> (a+b, b)) (a+b, -a)) (b, -a) in three stepso also solved by: (a, o)) (a, b-a)) (b, b-a)) io, -a). solve (a, b, c) to (c, ar b). (G'P'J') Five white and f ive L7 . Pied Pentagram. the Lz placed at the inner and 22. Pentomino Sequence. Arrange black counters are so that each pentagram. How many 5-square pieces in sequence outer points of a from its predecessor by different arrangements are piece is derived geometrically moving one square one step, €ither ortho- [freret How many have two of each co]our (G'P'J) ganally or diagonally (i.e. a King move). in each line? ttre solution is uniquel (G.P.J.) 23. Modern Art. Arrange the 35 6-square pieces to form three squar€sr 11x111 8x8' 5x5 arranged one on top of another in a stepped formation. (This was item 8559 by H. D. Benjamin in.Fai-ry Chess.Revi.ew February 1950. Nc solution was gtven.) 24. Blast Offl Arrange the 56 pieces of all sizes 1 to 6 in a 13x23 rectangle, with the players in a 18. All Out Cricket. If all 11 smaller pieces forming a "stepped pyramid" cricket team bat in an innings and they are shape and the hexominoes the surrounding all out, in how many different sequences space and border. (G.P.J.) can they be dismissed? (G.P.J) page 1.I0 THE GAMES AND PUZZLES JOURNAL ON LIBRARY RESEARCH My article on this subJect in the last issue, copies of which I circulated to all the organisations mentioned there, has elicited quite a lot of response - summarised here. The replies received from the BODLEIAN Librarv, Oxford, the CAMBRIDGE University Library and the BRITISH Library were brief and polite but not particularly helpful. Mathematical Collections The review editors of the Mathematical Gazette are reluctant to ask reviewers to donate the books to the Mathematical Association Library because the reviewing Job is purely voluntary and keeping the book, which the reviewer may have spent several months getting to grips with, seems the only tangible way of saying 'thank your to the reviewers. Access to the MA Collection also allows access to the Leicester University Library' which has an extensive mathematical section. Membership of the London Mathematical Society allows access to the mathematical collection at University College London. STRENS Collection (University of Calgary, Canada): Richard K.Guy writes: tYou and your readers are very welcome to 'user the collection, in the sense that we will try to answer any queries in the area of recreational mathematics, and can photocopy reason- able amounts of material at cost (usually about 10d a sheet). In the other direction' as you kindly imply in the article, we very much welcome gifts and donations to the Collection./ Since it's been here, Martin Gardner has donated a few hundred items, and Wade Philpott's widow.has donated his collection, and there have been numerous smaller gifts.r' Chess & Chess Problem Collections ANDERSON Collection, State Library of Victoria! Ken Fraser, Chess Librarian, writes: 'Your experience at Cambridge with loose covers on a book lifted our spirits here; it has a familiar ring in a librarian's ears./ We were honoured to be ranked no. 2 among chess libraries but the facts are different. On our latest information (Hans Bohmrs book Schaken, 1988) the Van der Linde-NiemeiJer Collection stands at 25000 books with an annuaffike of 500. We do not have any recent figures bn Cleveland but it is at least as large as the collection at the Hague./ Ours is a much smaller (and younger) coltection than eitner of these two. We now number about 101000 volumes with an annual increase of about 300 new titles. Even at that figure we are happy to be listed as the third of the world's public chess libraries. Were Lothar Schmid's collection at Bamberg to be included the rankings would change again. By all reports it is a superb library.' WHITE Collection, Cleveland Public Library, Ohio: an interesting reoly: The covering letter reads: tThe Library has completed a reference search on your question. A maximum of 15 minutes research can be devoted to each request. Included is the information found within the seope of this research policy./ If you require more extensive research, please write or call the Cleveland Research Center, a fee-based service of this Library./ If you should need further information do not hesitate to contact Mrs Alice N. Loranth, Head of Fine Arts and Special Collections.r Also included was a copy of an article about the collection from Michigan Chess August/September 19?5. This notes:rrThe White endowment has been erooeo6ffifffifr-I6-the point that it no longer provides for the essentials of the Department. The income from the White Trust Fund was $10,414 in 19?3 and $13,560 in 1gT4. With the cost of books more than doubling recently and the number of chess publications of all kinds multiplving beyond all expectations, this income has fallen seriously behind the needs.,, VAN DER LINDE - NIEMEIJER Collection, Koninklijke Bibliotheek: From this source I was sent copies of interesting articles from New in cies, No. B 19g5 and No. 4 1gg6 in the form of interviews of Lothar Schmid anO UeiildeiT-Nidi--eijer, two great chess book collectors. I even received a telephone call, and subsequently a copy of the section of the cataloptue of the collection (1955) dealing with Mathematics - which includes Knight,s Tours. I have just (July 1990) completed a first draft (30 A,4 pages closely printed) of a biblio- graphy of Knight's Tours (and other chessic paths). Having a word processor now makes it so much easier to compile such a work. A copy will be sent to anyone who can provide further information on the subject, particularly references to obscure sources in unfamiliar languages. THE GAMES AND PUZZLES JOURNAL page 191 Solutions to Chess Problems in Issue 1L. 133. EBERT. (a) 1Kg2 f3+ 2Kf2 Sg4+ 3Kgl f2+ 4Kh1 f1=Q* (u) tKg2 f3+ 2Kgrf.2+ 3Kh2 f1=S+ 4Kh3 Sf4+. WK circuit in (a). Economical and neat. [A.W.I.] Not a real'reflex'but only arsemi-reflextas therets no possibility for White to checkmate. [E.8.] 134. EBERT. (a) 1Kc8 c7 2Kb7 c8=Q+ 3KaT eb?+ (b) rKcA d7 zKbT d8=e BKa6 eb6* (c) tXcZ e7 2Kb6 gg=e 3Ka5 ebb+. Exact echoes. 135. EBERT. lb? KeT /Kd7 /Kd5 2b8=Q Kd7 /Ge9/Gca 3QdS/Qc?/Qe5*. Why go to such extremes Just to have a line problem? [S.P.] S-fotd Echo. For comparison: Erich Bartel & Hans Gruber, Feenschach xii 1983: WPa?, BKd6, BRs d?,e?. H+2,4 ways. Captureless play. 1Rb?/e5le6ld8 a8=Q 2Rbc?/Rde7 /Ke? /Kd7 Qds/c6lf8le6* 4-fold Echo. tE.B.l 136. WONG. The BK should be at e3 for: Retract 1Ke8xQd8(Kel) and second Black Qd2xQdl forPlay:1Q8d4Qf3+.IntheoriginalsettingwithBKe4tneintetTjffiidE-xBc8(Ke1) Q(or BX3xQdl for 1Bf5 Qd4. This is a Circe Mate but not a Kamikaze-Circe Mate. One point for this solution, or for claim of no solution. 137. HOLLADAY. 1Kc3 Rb5 2Sb4 Ka5 3Kb3= (b) 1Rc3 Rb3 2Sb2+ Ka3 3Kb1=. Exact echo pin ideal stalernates. tE.H.l Please note stipulation shoutd be STALEMATE, else 1...Ras*. 138. STEUDEL. 1al=C KaZ 2b1=C Kxal 3c1=C Kxbl 4d1=C Kxcl ... 8hL=C Kxgl 9Cg4 Cxg4= Very droll! [A.W.I.] Wonderful idea. [E,8,] Nicely forced sequence. tD.N.l 139. BARTEL. 1NPbz b8=NC 2a5 cxbS=NG(NCbl)+ and 1Ka5 b8=NG 2a6 cxbS=NC(NGb1)+ BK in check in diagram. Nice switch. tA.W.I.l 140. BARTEL. 1dl=NC b8=NN 2exd1=NG(NCdS) cxbS=NZ(NNb1)+. Promotion to 4 fairy neutrals. 141. BARTEL. Lb1=NB d8=NB 2cxbl=NN(NBc8) exd8=NN(NBc1)*. Solvers and Scores: A.W.Ingleton & S.Pantazis 16 (maximum), E.Bartel 15, D.Nixon 11. T. R. Dawson Centenary Nightrider Tourney N5. E. HOLLADAY N6. A. MOCHALKIN N?. Z. HERNITZ N8. Z. HERNITZ '%, %, ,*, '% %,% %, % ig,fr%,* %, % ?Ys% %, %%, ,p/,L/&, ,M, % '%, %A'%, % '%, ,g % 7w 7#, '.% %, % %%v%% //M 'm '% %, 'f, % ',ry, ('$t in 2, Mate *Z (version B also) *z (b) Ne5-)e 3 N9. D. MIJLLER v Zvonimir Hernitz sent an alternative version of N?: fl l$ v {rl 2 Move WK-g7, BShT-f?, make Pe6 Black, then move the whole position one file right, for *2, The solution CI 7 is the same - which setting isbest? F Romeo Bedoni A corrects N2: Add wPil5 to stop 18i15+ (t,o)nnr + 2Bxh14+. @ I I & Solutions and comments are invited as usual. These A 0 v will be Dublished in Variant Chess, together with the Award for the Tournev. -I v ns I h2, i9, i10, k3; k8 = (1, 2)R e9; Jg = (2,3)R Mate in 2 Board 1 1x1 0 j3; k7 = (0, 2)R i2; g'9 = (1, 3)R page 1' 92 THE GAMES AND PUZZLES JOURNAL Mathematics in Music - The Reversed Keyboard and a Revised Stave Notation by Graham LIPSCOMB My synthesizer can be programmed to play any note on any key, which led me to setting it up so that all the notes were bactcwards or "mirror-image". Obviously if you look at a keyboard inamimornDrrand"Aflat"areinterchanged,C=E,F=BrG=Aretc.Playingthereversed keyboard while standing behind the synthesizer is not too difficutt since the notes are effect- ivety as normal, but playing from the front leads to some interesting realizations. Now of course b fvfaj6r scale is symmetrical about D (our mirror note) since all the white (F notes in the mirror still equal all the white notes. But what about G major scale sharp instead of F). F sharp is B flat in ine mirror so now we've got F maJor in the mirror as a reflection of G major. so ine key with one sharp becomes the key.with one flat. Similarly with two sharps (B flat), three flats (E flat) and so on. (D), t"hree sharps (n), etc. becoming-chords two flats forming of is quite interesting: All major triads become minor Furthermor",,ih" Major triads (and vice versa). Dominant sevenths become minor sevenths with flattened fifth' sevenths' sevenths become other major sevenths. Minor sevenths also become other minor any- Nintn, become other ninthi. Diminished and Augmented chords are mathematical chords down the key- way as they consist of repeated minor thirds orhajor thirds respectively up and board. This pattern is obviously unchanged in the mirror' In all of these cases the new chorc will be rooted on a different note depending on the major ?th type of chord: c major = A minor, c minor = A malor, C7 = F sharp minor ? flats, c =Fma'or?th,Cminor?th=Fstrarpminor?th,C0=Dg,Cmajorgth=Dminorgth,Cminor gth = D major gth, C diminished = A diminished, C augmented = A flat augmented. All of this is fairly easy to see on the piano, but very hidden on the staves because musical notation on staves onty"attows for seven notes and not all twelve. The reason of course is that a key can only contain s-even notes naturally, the use of accidentals in a piece is effectively a cninge of kly (i.e. a new set of seven notes). Using lines and spaces enables many notes to be crammed into a small sPace. However, it is annoying that the two staves have notes in slightly different positions. Also' having to deal with up to ni@ leger lines is difficult. My thoughts o_n a better system are as (E'-9' D' F) tottor,is: (a) Two fege'r lines betlieen staves instead of one. (bITrebte stave pattern B' (c) lines above is then repeated foi uass stave (the two leger lines are A and c). There are enough the treble stave for an additional stave *it ing top c conveniently sit two leger lines above it. If another stave were added below the bass end and two leger lines below it you would have a very n""t rungu of four staves and five pairs of legers, giving 8* octaves A - C. Unfortunately grand piano range is only ?i octaves A - C! fopC Footnote by G. P. JELLISS: Years ago, when I was an unsuccessful clarinet player, I devised a simplified scheme of notation for chromatic $Z- + note scale) music, doing away with the sharps and flats. Instead of the usual key signatures, a number is placed on the centre stave, 8 indicating a frequ- ency of 2 to the power 8 = 256 cyles Mid c per second, which, if I remember right, is the frequency of middle C (or near enough to it). Thus I signifies 2 to the gth = 5L2, i.e. the octave above. Notes on and between the lines, including one intermediate leger line, give the L2. page 1 93 THE GAMES AND PUZZLES JOURNAL THE ART OF THE STATE By Leonard J. GORDON Elbridgets last name was Geny of course. Elbridge Gerry was governor of Massachusetts in I8L2, when the method of creating voting districts so as to be favourable to one political party was used. A political cartoonist of that time noted that the shape of one district looked like a salamander, so coined the name gerrymander. Geny was not responsible for the process since known as gerrymandering. American political cartoonists have never been known for fairness. The solution to the problem can be improved by changing the 2x2 square to an rSr. Changing to a fTrorrl,rshape doesntt help. It is impossible to make a 2-colour map with 12 5-clistrict states as long as the lxb shape is used. Below are two examples of dividing the 8x8 into 13 different parts (iitn sizes 3 to 6). L.J.G. 11 x 1989. Solution to Tangram E, pLZg. wffi Solutions to Word puzzles Edinburgllsms. (p.147) This item was so Cryptic Cros.sword - g named since the question came from R.M.W.M. in Edinburgh. Examples of ,le.I Uj mng sty were offered. Others are: alghan, first, ca.l._mness, cang.y. The lack of a tuv example is surprising. Fourfotd examptes are understudv and gyl!ngpter- ous (Oare-winged, not in ChamOffi Autological ACT CAT Anagrams 1. N II,E LINE DREAD ADDER ERNEST ENTERS BRAISED SEABIRD EDITRESS RESISTED ARGENTINE TANGERINE Enigma 1. BEAM page L94 THE GAMES AND PUZZLES JOURNAL Swift Solutions to the Super'-dominoes- on P1 89" 2!. 13. Wazir Tours. L4. Numbering Formulas. The eight form- ulas required are: 8x+V-8r 8y+x-8r 64+y-Bx, 64+x-8y, 73-8y-x, 73-8x-V, 8x+1-yr 8y+1--x. 22. Pentomino Sequenee. The only order possible is the following 15. A Crlrytarithm. or its reverse: 864583 P V F 827 0B 3 rL C T v s w x 870 17970 86577 HLtrFE=clil&E=qlhh+ L7 97 083 L6. +l- Permutations. The solution takes The possible transformations form the six steps: (a, b, c) ) (a+c, b' c) ) (a+c, a*b*cr network shown below (- Fers move, c) ) (a+c, a*b, c) ) (a+c, a*b, -&) ) (c, a*b, -a) Wazir move). The sequence is a ) (c, a*b, b) ) (c s &t b). Hamiltonian Path (i.e. Tour) of this net. LT. Pied Pentagram. I find gz amangements of which 4 have 2 of each in each line: the one shown below, the one shown in the question, and their complements (formed by changing white to black and btack to white). --€ 23 & 24. Hexominos. Solutions below. 18. A!1,_ Out Cricket. There are IA24 orders, i.e. 2L0 (since actually there are only 10 dis- missals, one always remains ttnot outt'). 19. squaring the Rectangle. A zxl rectangle. 20. Matchstick Manoeuvres. 4 in each. O- - l-f - - - I l_r:| | f---- lr l_ _l I r:l ll PAGINATION There were four unnumbered I I ll l_ pages in issue l-. These are _f shown in the Index as A,B,C,D THE GAMES AND PUZZLES JOURNAL page 1 9b Index of Ideas etc. Chase - 20 Chess & Variants - 6, 22, 38, 54, 76, 104, 107, L26, r27, 133, 136, ABC Sliding Puzzle - 86, 111 159, L7L Abstract Check - 6 Chess Problems 7, 23, 39, 55, 77, Acrostic, Chess - 161 - 105, L32, 156, 17r, 191 Alfil - 107 Alpha Pairs - 59, 92 Chess Libraries - 14i, I90 Alphabetical Cube - 59, 111 Chess Lettering - L28, 178, 180 Alphabetical Topo'l ogy - I44, 162 !!g$_q5 - C, 17, 76, 103 Angle impact 10, 26, 42 - Chessboard Speaks 54, 77 L32 - IZB,180 Antipodean Chess/Men - , Chessics 1, Are you Game? 146 - 9, C, 17, 54, IZ4, 131, 155 - 93, Checkers see Draughts Art, l,iathemati cal - 14, 16, 32, 33, - Cheekerboard Puzzle Compendium IZg 48, 49 , 64, 80, 96, I 16, 148, - 1 50, 178 Chequered Dissections - 9, ZS, !28, Art of the State - 180, 193 152, 153, 'Asterisks - 32,48 I72 Thought, An 14 Astronomi c - Chinese Cross - 767 Atoms - 32, 48 Circe Varieties - 54, 77, 105, 126, 114 Atomic Numbers - 133, 156, 177 Autol og i ca'l Anagrams - L7 9, 193 Circular Board - lI7, 119 Awards for Chess Compositions - 130 Circular Primes - 90, II2 Awkura rd Om'i noe s - L7 2 Circular Saws - 25, 4I Backgammon - 70 Civil isation - I02 Bal anced Magi c Squares - 88 Cloak&Dagger-103 Baroque Games/Men - L2, 169 Clock Words - 93, 146 Battle of Numbers - 66 Close-packed Snakes - 176 Beams - 9. 25 Code Words - 59, 92, 114 Bi-centric Quadrilaterals - 108 Complete Information - 69 Bi furcati on Patterns - 16, 32, 148 Computer Solving - I28 Bj I I i ards, Tri angul ar - 10, 26, 42, 58 Concertina Patience - 52 ' Continued Fractions 81 84, 108 , L42 - 57, Cookjng Game 18 Bi nary Arrays - 25, 44, 62, 64, 82, 96 , -' 160, L72 Cooks/Comments/Corrections - 155 Coral Patterns 49 B'i satirs, 160, L76 - 33, Derangements Bison - 77 Counting - 89 Black Spot NimmitY - 66, LBz Cowboy - 136 Blackmarkit - 35 Creeper - 19 Bodged Chessboards - 9, 25, I72 Cri bbage - 66 Boards - 74,97, LL7, 137,168 Cricket - 150, 166, 183, 189, 194 Cross Game 36 Bri dge 52, 68, I 13 140, 17 5 , IB7 - - , Cross-Point - 29, 44, Bl Broken Chessboards - 57 , 109, 122, L24, Crossing the Parallels - 9, 25 r2B, 152 , 180 , 183 Crossword Cl ub - 31 Buc ket wi th rHol e - 91 , 113 Crossword Puzzle Patterns - 62 Burr Puzzle L67 Cryptarithms - 11, 27, 43, 60, 87, 139 150, 189, 194 Calculator Keyboard - L73' 185 Cryptic Clueing - 46 Calenddr, Lunar - 29, 44 Cryptic Crosswords - 15, 31, 47, 63, 191 Camel/Camelri der - 77, 162, t7I, 95, 115, r47, 163, r7g, 193 Capturel ess P1 ay - L32, 17 L cube(s) - 25, 41, 5g, 111, I77, IgI Car Park Patience - IzL, IBz Custodian Capture - lz Card P1 ay - 5, 28, 35 n 52, 69, 100, LzT, Dabbaba - L07 Carpet Sqr"rares etc 5, 35, 69 Dart Words - 93, 146 Centres of Tri angl es - 85, 109, 14?, Deep Draughts - 53, 71 tltlZ C ha nce 51 , 7 5 Derangements - 59, 89 Change-Chains - 15, 31, 47 Diagrammes - 135 Digitology - 138, I77, 183 page 196 THE GAMES AND PUZZLES JOURNAL Di pl omacy - 2, 3, B Games Theory - 50, 7 4, L37 Di recti onal Draughts - I 18 Ga rde n Game 18 Di rectmates - 39, 77 105, L32, 156 - , Geometri c Jigsaws 80, 116,I22 Di rect Stal emate - 39 - Get My Goat - 40, 86 Di ssecti ons - 9, 25 , 4L, 57 , 65, 80, 81, 109, 116, r22-5, r2B, 149, Gi nger Rummy 52 Gi raffe - 17 I 1 50; L52- 3 L67 17 2, 180 , , Gnomonic 113, 140, IBz 183, 185, 186, 193, L94 Question - 91, Golden Ratio - 81, 91, II2 DNORTY Puzzle L67 - Good & Bad Moves 74 Domi noes - 4, 2L, 45, 60, BB, L23 Grasshoppers 22, 39, 55, 7 6, 105 Double Blank Words L46 ' - L32, 134 17 r Double Draughts - 53 , Greater Lions 161 Doubl e-Move Chess 6, 54 - Grocers, Log'i cal - 13, 29, 44 Doubl e Sati ns - 17 6 Groupoi ds L7 5 Draughts Vari ants - 53, 7 L, 118 Growths & Processes 16, 32, 33, 49, 77 156 Duplex Problems - 23, , 148 Eagl e - 39 Echoes, Exact - 131 Half a Cake - 153, 183 Edge hog - 105 Half-Life - 93 ECinburghisms - L47,, 193 Hal f-Move - 23 Helpcompelmate - 133 Elementary Words - 93, 114, L46 r En Garde - 2 Helpmate 23, 39, 55, 77, 105, 132, Enqlish Quadrilles - 4 156, L7 I Enigmas - 166, 183, 187, 193 Hel pstal emate - 39, 55, 77 , 105, 156 Enumerations - 11, 27, 30, 32, 43, Hepti amonds - 65, 80, 1 16 48, 57 , 62, 80, B2-3 , 87 , 89, Heterochess 0lympics - 106, 126, 159 90, 94, 116, 137,140, r44 Hexominoes 9, IZS,,l49, IlS, 186, I94 160- 1 , 162, Hole in Bucket - 91, 113 LB?, 194, 186, 1Bg, 194 Hype rc u be - 20 r Equihopper 156 Ide.al Mate Review - 135 Equi-Mover - 23 Imitator - 156 Escalation F 107 Inci rcul abl e Sets - 90 Eure ka - L7 4 I nequaf i ty, Probabl e 13 29 Bl ffiEchoes - 131 - , , I nforma ti on Compl 6g Exchangeabl e Operati ons - 29, 45 , ete - Integral Part Formula - 45, 6l Exchange Pl ay - 133 Interaction F Exponenti ati on - 89, LI?, 139 51, 7S Eye-Spy Enterprises - 103 Jabberwog[y - L4, 31 e Pieces - 133 Fairy Chess Circle - L76 Jigsaw Puzzles a 80, 91, 113, 116, L22 Fanorona - 169 Jujitsu - 35, 69 Fers - 77 , 107 Fibonacci Series - 139 Kami kaze Che s s - 17 I Fi fteen Puzzle - 24, 56, L64 Kentish Draughts - 119 Fi rst, Last & Onl y Numbers - 138 Keyboard Reversal - I92 183 Ki ng- Ci rce - 54 Fluq der Schwalben Der - 158 Ki te s 108 Four (4)-Gon Concl usi- on 85, 108 Kni ght Tours & Paths 1, 1 1, L7 , Four Rs - 45, 61, 87 162, 190 Fractal s - 48, 150 , L7 4 French Quadrilles - 4 Lancers, The - L4 Frogs - 99 Language Simul ati on - lB4 Laws - 50 Game Invent'i on Competition 53, 118 Leapers - 107, 181 LB2 Leapfrog - 97 , 99, 151 Games & Puzzl es Journal - 1, 67 , 98, Letteri ng 178 Letter 71 g-Zags - 86, 111 THE GAMES AND PUZZLES JOURNAL page Lg7 L-Game - L87 Parallels - 9, 25 Library Research - 141. 190 Parallel Time Stream Chess -. 38, lB Lifebelts - 47, 63 Pari ta r 28 Limericks - 14 Patience - 35, 52,69, 100, LZL Lions - I32,136,161 Patrol Chess - I32 Logic - 13, 29, 44 Pentominoes a 57,109, 122, TZ4, 189, 194 Lunar Calendar - 29, 44 Permutable Primes 59, 90, LLZ, 139 Pe rmutabl e Words - 59 , gz Madrasi Chess - L32' 156 Magic Knjght Tours - 1, 11, 17 Pe rmutati ons - 59 , 89 , 189 , L94 Magic Squares - 24, 40, 56, 86, 88, Phenix - 136 96, llz Pi - 30, 94, 139 Malefique Circe - 77, 105, 133, 156 Plakoto - 7L T7T Pl anar Boards' - L37 Mao+Bishop - 156 P1 ayabi I i ty of Vari ants - 159 Marseilles Chess 6, 54 - Postal Soccer - 72 l'latchstick Manoeuvres - 189, 194 Pol ygrams - 16, 32 33, 48' Mathematical Art - 14, L6' 32, Polyhexes - 34, 43, 57 , B0 148 49, 64, 80, 96, 116' Po I yomi noe s - see Pe n t/ Hex/ Hept- l4athematics of Games - 174 Prime Numbers 141, r74 59, 90, LIZ, 138, 139 ffiion- Pri me Words - 59, gz 156 Maximummers - Pri ze Competi ti on r 53, II4,, L46 Mersenne Primes - 138 Probable Inequality - 13, 29, gl 94, 139 Mnemonics - 30, Probabl y Deranged ! 59, 89 79 Moose - 39, Problemist - Cn 134 20 Morri s - ffierver - 134 Moultezim 71 - - 2, 166 Mousse-Mate 133 - Progressive Chess 54 6 - 6, Multi-Check - Progressive Circe Chess .1 126 Multi-Rex - 156 Pronounceabl e Codes - 94, 145 Mul pl i cat'ion Tabl e - 45, 60 87 ti ' Psychol ogi cal Juj i tsu r 35, 69 Music 'L92 Puzzle Box - D, 110 Mutation Chess ' L27 Puzzle Party - 150 Navy Patience - 69 Quadrati c - 30 Neutrals - 54, L32, 156, l7L Quadrilles - 4, ZI, BB Nets - I37 Quick Questions - 175, 189 Nightrider - 55, 105, 134, 156, 165, Quirk - 53 170, i91 r'66 N'im/Nimmity - 66, 118, L82 Rank & File Nimmity Nine-Men's Morris - 20 Rectangles - 57, 81, 109 Number Symbols 145 Recurring Decimals - 139 - Recurrence Numhreri ngs - 64, 189, 194 Relations 82, 89, 160,. Numerical Mnemonics - 30,94,139 L76 Refl ex Mate 39, 105, 135, L7 L Numerology - 138' 183 Retractors - 7,156, L7L 0bituary - 17, 155 Retroanalysis - 77, L32 Octahedron - 96 Retro-0p.posi tion - L57 Odds - 51, 75 156 01ympics, Heterochess - 106 Rhyme Sc heme s 30 , ,94 , LBz 0n. the 'Dole - 100 Riders - 77,107, 162, I7L, 191 One Way Chess - 156 Riddles - 29, 45 Only Numbers 138, 183 Ring Game - 36 Operations - 29, 45 Ri ngi ng the Changes - 36 Origins of Chess - 104 Rook Around the Rocks - 143, 178 Overheard Conversation - 27 Rook Arrangements - 82, 93, L46 Rose 105 Trees 16, 32, 48 - Pair - Rostherne Games C, 103 Palindromic Powers L77, 183 - - 150, 166 Paos - 105, I32 Round Britain Quiz - page 199 THE GAMES AND PUZZLES JOURNAL Tangrams - L23, 150, 193 Royal Carpet Patience - 35, 69 Theorie des Effects 158 Royal Piece r L07 - Iheory of Games - 50, 74, I37 Rul es & Regul ati ons - 50 r Three-U Sopwith - l3 Rummy 5, 52 Torus - 162, LBz Runes r L44, L62 Tours ! 1, 11, 17r 143, 178, lB9, Lg4 Tra peze 1 oyd - 164 Satin Squares - 82, 160, L76 Trapeziums - 108 Saws 13, 4L - 25, 29, Tri angl e Centres - 85, 109, 1 42, L82 Schwalbe - 158 Tri angul ar Bi I I i ards - 10, 26, 42, ffihess - G,54 58, 84, 109, r42 Scrabble Clubs - 31 Triangular Nimmity 1lB Self-Documenting gZ, 66, - 59, 114 Turns - 51, 75 Self-Referential - 139 Twitch - LZI Self-Mate 156 - Two-Colour Checkers - 118 Semi-Progressive Chess - 6 Two-(5)-B-J-A - 100, I2I Sense & Nonsense - 13 Sens i ti ve Ki ngs - 23 Unbalanced Magic - BB, II2 Senti nal Chess - 136 Uncommon English - 63, 95 September - 19 Uni ted - I20 U nsc rabbl eabl e Words 92 I 14 Serieshelpmate - 6, 7, 54, 77, 105 - , Serieshel pstal emate - 7 , 39, 133 Vaos: 105, I3Z Seriesmate - 23, 55, 133 Variant Chess - 6, ZZ, 38, 54, 76, Seriesselfmate - 7, 54 104, 107, 126, 127,133, 136, Shuffle-Link - 115, I47, 163, Llg tl'i Simpl y Strange - 145, 163 ve rr. o:' 10 , e3 Si ngul ar Ri ddl 45 e - 29, Wazir - L07, 143, L7L, I7B Sixteen-15 Puzzle 164 - Westbury - 145 Sliding Block Puzzles Bn 24,40, 56, What's in a Game 50, L37 86, 111, L64 - 74, Wheels within WheeJs 13, Zg Snakes - 176 - ldord- Ladde r Anag rams - 1 15 146 Sna ke s & Ladders - 17 s , Soccer - 72, 1,20 Ze bra - lT Zeroposit'i Solitaire r lln 27,43, 87r 151' 168 on - 77,105, 130 Lig-Zag Letters 111 lBB - 86, 7ig-Tagged Boards - 168, IBB Sol uti ons to Chess Probl ems - 2I, 37 , Zines & Mags - 2, A, B, 73, 98, ll8 78, 79, 106, 129, 154, 170,191 150 Sopwith - 34, 45, 61, 73 I ndex Narnes Sorcerer's Cave - 101 of Space Rummy - 5 Abbott, R. 98 ApSi mon H. 87 Sprookjesschaak - 158 L, , AdaffiS, S.S. IZB Square Pl ay - 20 Adamson Bachet, C. G. 66 Squari ng the Cube 25, 4I , H. A. 13 - 44 Bakcsi, G. 105 Squaring the Rectangle 57, 81, 109 Ainley, S. 4I r29, 1 32, 135 1Bg, Ig4 113, 140 154 Stars & Asterisks - 32, 48 AlbeFt, E. 37 Ball, l^l.Irl.R. 1, L7 Poi Star nts - 26, 58, 85 131, 133, 135 25, 90, 96 Strategy Patience - 100 155 Bansac, P. 158 Strewberries 32, 148 Al fonso a Xth 98 Bartel , EJ. 155.IBZ Sub-Zines 2 A1 may, J . deA. 136 Bartel, Er. 54,55 Superdomi noes - IZ3, 189, 193 , Lg4 77, 79, 104-6 Ananthanarayanan ' Swift Solutions - 186, 194 c. K. 136 r2g- 34 , 1 54- 6 Swordsman r 169 Anderso[, M.V. 141 Beas 1 ey; J. D. 52 Syntheti c Games - 6, ZZ, L3Z 190 68, 76, 113, LzL 186 Andersson , K. A. L. Take - 12 91, 113 Bedon'i , 170, 191 Tangle Tongue - 114, 146 Anstice, J. 89 Beiler, A.H. 138 THE GAMES AND PUZZLES JOURNAL page 199 183 Bell, E; T. LBz Conan Doyl e , A. z Farhi , S. 57 , 109 Hoffmahflo L. 153' r22 75 Bell, R.C. 1, Lz Cook, R. J. 173, 185 Hofstadter, D. Benjamin, H. D. 189 Fe rma t 138 114 CostelJo, M.|,*l, 73 Feuerbach, K. l'.l. IBZ Hol I 39 Bennn -- 94 addy, E. CouFt, N. A. 108 550 106 Bennett n A. L37 Fi bonacc i 16, LI? 77-79, Courtl and-Smi th, 139 131, 133, 156, 191 Bennett, l,T. 183 J. 93 Filipidk, A. 8 Hol t, D. F . 82, 83 Berloquin, 29 P. Coxeter, H. S. M. 10 Fischer, R. 75 Hooper, D.75 Bertinn 158 J. 17, 25, 48, 85 Fixx, J.F. 35 Hopkins, G.M. 30 Besicovitch, A.S" 90 Fleischer, K.J. 73 Horde rn L. E. B 69 , Croes, G.A. 135 ' Fomal haut' 98 24 , 40 , 56 , 150, 183 Beve rl ey, [.l. 1 1 Fortis, A. 6 Horh€ , M. L26, IB7 Bha s ka 87 Crum, G. 1 18 ra Foshee, G, LzB Hoyle, E. 4, 70 Bi d€v, 104 Crumlish, M. 30 P. Foster, 70 Hoyle, F. 150 Blom, I28 -- I. FrankiSS, C.C, 76 Hunte r, J . A. H. 43 Bl onde I D. 54 , Darwin, 70 Fransen, J. LZB B7 136, 157,158 C. 'Dash' 2 FujimuFd, K. 32 Bogdanov, E. 23 Davis, S. 159 Fulpius, J. 135 Ingletort, A.hl. 2L 37 , L32, 154-6 Dawes n J. 87 , LGz 39, 47,79, 104 Bonsdorff, E , 22 Dawson, T.R. 11 Gardner, M. 29r 30 105, 130-2, 155 Booth, C. 72 6, 13, L7 22, 27 94, 118, 139 156, 19.1 BorodatoV, L. 77 , 43, 45, 48, 61 TB2, 106, 132, 154 183 Lr7, 124, 134 Jaenisch, C.F.de 156 Gelpernas, B. 54 1 57, 191 George, A. 55 141 Bosl ey, J. E. L27 Jel TzL Deah€ , E. &D. lZB Gibson, H.P. & Co I'i ss, E.J, Bouton , C. L. 66 Dearman, R.A. 140 3, L02, Jel I i ss , G.P. Bowen, 2 I. 186, 187 Golomb, S. 57, 59 al I i ssues Boyer, J. P. 54 Gordort, Johnson, [^l.hl. 24 de Bono , F. 187 J. 56, 86 136 111, 164 Joustra, E. 73 Del annoy, H. 4, 2L Bra'i n, R. 2I , 47 Gordoh , L. J. 40 Joyce , J. 92 Devlin, K. 183 79, 104, r32 56, 80, 86, Bg Kalotay, A. 131 155 Dickins, A.S.M.D. 96, 98, 111, 114 133 Brokenshi Fo , L. 1, 6, 17, 22r 76 115, 118, L22 Karneugh, -- 44 L28 104, 129, 158 I23, l. 46 , 151- 3 [.arpov, A. 159 Bruc€ , L. 59 92 'Dipole' 87 164,183 o 1Bg, 193 Kasparov, G. 159 93, 115, 145' DittmarIJ, l,l. 158 Gri mstone , C. B9 Keith, M. 139 r47 Dodgson' C.L. 10 IT2 Keller, M, 73r 75 Brugge, J . 17 Donne I I y, T. 10 I Gruber, H. 54, 76 80, 87 , BB, 98 Burns i de , l^l. 82 Donovan, P. L26 130, 131, 155 106, 118, L22 159 156 L26, 127 , 145 Dri ver, J. E. 103 Grubert, H. 23, 37 LBz, 183 7 6, 77 105, 106 Cal dwe I I , C. K. LLz Dudeney, H.E. I , Kemp, C.E.7, 9 'Caliban' 150 13, 24, 40, 86 L29, 131 , 156 13 Cameron, K. 134 L?B, 137 , L52,186 Grunenwald, A. 136 Ki ndred, M. 59, 92 Carmichael , R.D. Dumoht, J.C, 76 Guy, R.K. 141. 190 Kjn9s, P.R.75 89 136 Kirtley, M. 135 10 Dye r, D. 34 'CarrolJ, L.' K'i shon, D. 19 L4, 15, 31 26, Hanazawa, M. 131 Kniest, A.H. 39 Cassano, 159 R. EbeFt, H. 156 133 7B Castelli, A. 6 EckeF, M,|',l. 150 Harri s, J. 86, 150 Kobayashf , S.32 127 Edd i ngton , A. 138 164 \ Koc ke I korn , C. I 58 Catal an , E. C. 94, LBz 150 ) 193 Hawthorne, F,29 Kohtz, J. 158 Charosh, M. 118 Ellinghoven, B. Heaslet, M.A. 138 Kraitchik, M. I Cohefi, P. M. 9 , 25 136 Hernitz, Z. 191 82, 83 54, 75, 76, 79 El y, R. B. 40, 56 Hodg€, L. 31 Kricheli, J. 54 92, 96, 100, 1 14 Euclid 58, 108-9 'Hodos Sphaenodon' Kruszewski, T. Z0 Evarts, 2 LB2 P. 9B Kuratowski, C. I3l page 200 THE GAMES AND PUZZLES JOURNAL Laquiere, -- 4 Murray, H. J. R. I Ri tti rsch , M. 134 Taylor, L.F. 138-9 Lindgren, H. 41 11 Rohrbough, L. 118 Taylor, S.J.G. 35 Ling, J.F. 103 Musson, R.M.l^l. L47 Rosner, E. 135 39, 43 , 55, 69 134 193 Rotenberg, J. 136 76, 78-9, 105 Linss, T. I32, 154 Nash, R. J. 139 Rouse Ball, t,'J.hl. 118-9, L29, 131 Lipscomb, G. 19' 192 Nettheim, N. 7, 2I see Ball, !.l.|^J.R. L32, 1 46 , 150, 186 Locock, C.D.6 47, 55, 79, 131 Roy, S . 94 Ternbldd, H. 37 Long'f and, S. I44 Newing, A. 87 Rubik, E. B Thiess€r'r , H. 20 Loustau, J.M. 136 Niemeijero M. 141 ' Rufus ' 46 Timman, J. 159 Loyd, S. 24, 40 190 Russ, C. 131 Tolkein, J.R.R. 2 56, l2B, 161 Nixon, D. 2L, 30 Tomasevic, M. 54 164 47, 70-1, 79 Sac kson , S. 1 Tresham, F.G. L02 Lucas, E. 4, 145 104, 132, 155 Saul, M. L26 Tulk, P. 2 Luers, H. L28 Noc k, A. 98 Schiffmarr, J.A. Turnbull, R. 134 Lyness, -- 139 Novdk, P. L26 158 TyloF, C.M.B. 38 Lytton, C.C.76 Schmid, L" 190 52, 78, 150, 159 103 0laussofl, M. 156 Schwagereit, J. ;3 r62 TzannesrN.&8. 7 0-1 0rr, A. C. 94 Scovero, G . L27 Mabey, P.H. 122 0sborne, M. I44 Shanahdr, I. 39 I25, I49 0shevn€V, A. 23 48, 55, 76, 78 Uspensky, J.V. 138 Mackie, A.G. 26 37 , L32, 154 79, 130, 131 42, L42 Shankar Ram, N, 37 Valois, P.S. 92 Macleod, N.A.161 Palmer, C. 2L, 44 Shapiro, -- 94 133, 155 van der Li nde L4\ 190 MacMahonn P.A. I22 45, 87 ShoPt, N. I07 Vaughan, C. 39, 7B t23 Pantazis, S. 104 Shuh, F. BB 93 Madachy, J.S. 43 L32, 155, 162,18? Silverman, R. 138 Venn, 44 87 Parr, A, 72, 73 Simonci ri , M. 126 J. 183 Vladjmirov, V. 23 Mandel brot, B. N . r20, 145, 1 50, 159 37 1 54- 6 '48 Pearson , A. C. 45 SimpkirS, P. 2 , L32, Marlow, T.H.1,11 Peery, L. 73 Singmaster, D. B WattS D. 98 17, 40, 59, 61 Pentangle L67, 183 Slocum, J. LZB , 1 53 l'.lells D. 183 69, 86, 92, lLz Petkow, P. A. 37 r52, Smooko 47 Wenzelides, C. 11 114, 139, 143 Phillips, D. 72 R.l^l.ZLr 150, 183, 185 PhillipS, H. 150.i83 79 l,Jhitg, J.G. 141, I90 McDonald, G. 118 Poisson, C. 133 SmuldeFS, K. 158 t^lhitefield, Count McDowell, M. 23 155 Sopwith, T, 34 38, 39, 7B 135 37, 135 Pol I ard, T. G. 2L Sostack, R. [{hitel€y, A. 73 I 14 McGonigal, D. 126 47, 54, 79, 104 SpedF, J. |^l. Co Whyld, K. 75 Speelman, J. 107 McIntyre, S. 101 I32,155 !.lidlert, K. 156 93 159 McWilliam, R.C. Ponsonby, S. 18 |.l'i edenhoff, C. 135 Staffor"d, D. 73 103,146,158 Pri byl i nec , V. 28 !.lil€y, J.I. 40 Stedd, }.l. 9, 30 MeadleV, R. 141 36, 77 , 86, 106 t^lillcocks, T.H. 11 65, 80, 96, 116 Mendel, T. 73 111, 132, 154 17, 2L, 24, 27 l'.J. 98 Merken, J.&P. 73 Pritchard, D, L26 Steinitz, 79 Steudel, Th. 7,2L 43, 47,60, Mersenne 90, Il2 87, 91, 113, 139 38, 54, 55, 105 138 'Querculus' 15, 31 140, 143, LB4 189 r29, 132, 154 M'ihalek, F.M.23 46, 47 , 63, 95 !.li I I mott, A. 7 156 37, 39, 55, 77-9 115, 147, 163 Wong, P. 39, 7B Stohe , J. 126 105_6 , lZ9, L3? L32, I 54- 5 Ramsey, A.72 Storey, l^J .E. 24 154 Woods M. 12 Rehm, 135 141r 190 , Mochal ki n, A. 105 H.P. Strefis, E. Wyatt, E.M, I2B !29, 133, 154-5 Reilly, !.l.H. 37 SturgesS, J. L26 Ribenboim, P. 138 Suggett, G.J, 69 191 Yusupov, A. 159 Rice, J.M. 6, L29 87 90, 92, 94 Monreal, 135-6 , P. Rinderknech, C. rr2, 139, 183, 186 Mul I 126 Tnos ko- Bo ro v s kY age, S. 158 Sukumar, 136 ' K. E. 6 l4ul ler, D. 191 ' Ri ter, C. C. ' 138 Tauber, Th. 105 Zi ma , J. L27 Murby, A. 150 139 r2g, 130, 155