Archive-name: puzzles/archive/competition/part3
Last-modified: 17 Aug 1993
Version: 4
See reader questions & answers on this topic! - Help others by sharing your knowledge ==> competition/games/rubiks/rubiks.cube.p <== What is known about bounds on solving Rubik's cube? ==> competition/games/rubiks/rubiks.cube.s <== The "official" world record was set by Minh Thai at the 1982 World Championships in Budapest Hungary, with a time of 22.95 seconds. Keep in mind mathematicians provided standardized dislocation patterns for the cubes to be randomized as much as possible. The fastest cube solvers from 19 different countries had 3 attempts each to solve the cube as quickly as possible. Minh and several others have unofficially solved the cube in times between 16 and 19 seconds. However, Minh averages around 25 to 26 seconds after 10 trials, and my best average of ten trials is about 27 seconds (although it is usually higher). Consider that in the World Championships 19 of the world's fastest cube solvers each solved the cube 3 times and no one solved the cube in less than 20 seconds... God's algorithm is the name given to an as yet (as far as I know) undiscovered method to solve the rubik's cube in the least number of moves; as opposed to using 'canned' moves. The known lower bound is 18 moves. This is established by looking at things backwards: suppose we can solve a position in N moves. Then by running the solution backwards, we can also go from the solved position to the position we started with in N moves. Now we count how many sequences of N moves there are from the starting position, making certain that we don't turn the same face twice in a row: N=0: 1 (empty) sequence; N=1: 18 sequences (6 faces can be turned, each in 3 different ways) N=2: 18*15 sequences (take any sequence of length 1, then turn any of the five faces which is not the last face turned, in any of 3 different ways); N=3: 18*15*15 sequences (take any sequence of length 2, then turn any of the five faces which is not the last face turned, in any of 3 different ways); : : N=i: 18*15^(i-1) sequences. So there are only 1 + 18 + 18*15 + 18*15^2 + ... + 18*15^(n-1) sequences of moves of length n or less. This sequence sums to (18/14)*(15^n - 1) + 1. Trying particular values of n, we find that there are about 8.4 * 10^18 sequences of length 16 or less, and about 1.3 times 10^20 sequences of length 17 or less. Since there are 2^10 * 3^7 * 8! * 12!, or about 4.3 * 10^19, possible positions of the cube, we see that there simply aren't enough sequences of length 16 or less to reach every position from the starting position. So not every position can be solved in 16 or less moves - i.e. some positions require at least 17 moves. This can be improved to 18 moves by being a bit more careful about counting sequences which produce the same position. To do this, note that if you turn one face and then turn the opposite face, you get exactly the same result as if you'd done the two moves in the opposite order. When counting the number of essentially different sequences of N moves, therefore, we can split into two cases: (a) Last two moves were not of opposite faces. All such sequences can be obtained by taking a sequence of length N-1, choosing one of the 4 faces which is neither the face which was last turned nor the face opposite it, and choosing one of 3 possible ways to turn it. (If N=1, so that the sequence of length N-1 is empty and doesn't have a last move, we can choose any of the 6 faces.) (b) Last two moves were of opposite faces. All such sequences can be obtained by taking a sequence of length N-2, choosing one of the 2 opposite face pairs that doesn't include the last face turned, and turning each of the two faces in this pair (3*3 possibilities for how it was turned). (If N=2, so that the sequence of length N-2 is empty and doesn't have a last move, we can choose any of the 3 opposite face pairs.) This gives us a recurrence relation for the number X_N of sequences of length N: N=0: X_0 = 1 (the empty sequence) N=1: X_1 = 18 * X_0 = 18 N=2: X_2 = 12 * X_1 + 27 * X_0 = 243 N=3: X_3 = 12 * X_2 + 18 * X_1 = 3240 : : N=i: X_i = 12 * X_(i-1) + 18 * X_(i-2) If you do the calculations, you find that X_0 + X_1 + X_2 + ... + X_17 is about 2.0 * 10^19. So there are fewer essentially different sequences of moves of length 17 or less than there are positions of the cube, and so some positions require at least 18 moves. The upper bound of 50 moves is I believe due to Morwen Thistlethwaite, who developed a technique to solve the cube in a maximum of 50 moves. It involved a descent through a chain of subgroups of the full cube group, starting with the full cube group and ending with the trivial subgroup (i.e. the one containing the solved position only). Each stage involves a careful examination of the cube, essentially to work out which coset of the target subgroup it is in, followed by a table look-up to find a sequence to put it into that subgroup. Needless to say, it was not a fast technique! But it was fascinating to watch, because for the first three quarters or so of the solution, you couldn't really see anything happening - i.e. the position continued to appear random! If I remember correctly, one of the final subgroups in the chain was the subgroup generated by all the double twists of the faces - so near the end of the solution, you would suddenly notice that each face only had two colours on it. A few moves more and the solution was complete. Completely different from most cube solutions, in which you gradually see order return to chaos: with Morwen's solution, the order only re-appeared in the last 10-15 moves. * Mark's Update/Comments ---------------------------------------------- * Despite some accounts of Thistlethwaite's method, it is possible to * follow the progression of his algorithm. Clearly after Stage 2 is * completed the L and R faces will have L and R colours on them only. * After Stage 3 is complete the characteristics of the square's group * is clearly visible on all 6 sides. It is harder to understand what * is accomplished in Stage 1. * * --------------------------------------------------------------------- With God's algorithm, of course, I would expect this effect to be even more pronounced: someone solving the cube with God's algorithm would probably look very much like a film of someone scrambling the cube, run in reverse! Finally, something I'd be curious to know in this context: consider the position in which every cubelet is in the right position, all the corner cubelets are in the correct orientation, and all the edge cubelets are "flipped" (i.e. the only change from the solved position is that every edge is flipped). What is the shortest sequence of moves known to get the cube into this position, or equivalently to solve it from this position? (I know of several sequences of 24 moves that do the trick.) * Mark's Update/Comments ---------------------------------------------- * * This is from my file cmoves.txt which includes the best known * sequences for interesting patterns: * * p3 12 flip R1 L1 D2 B3 L2 F2 R2 U3 D1 R3 D2 F3 B3 D3 F2 D3 (20) * R2 U3 F2 D3 * * --------------------------------------------------------------------- The reason I'm interested in this particular position: it is the unique element of the centre of the cube group. As a consequence, I vaguely suspect (I'd hardly like to call it a conjecture :-) it may lie "opposite" the solved position in the cube graph - i.e. the graph with a vertex for each position of the cube and edges connecting positions that can be transformed into each other with a single move. If this is the case, then it is a good candidate to require the maximum possible number of moves in God's algorithm. -- David Seal dseal@armltd.co.uk To my knowledge, no one has ever demonstrated a specific cube position that takes 15 moves to solve. Furthermore, the lower bound is known to be greater than 15, due to a simple proof. * --------------------------------------------------------------------- * Mark> Definitely sequences exist in the square's group of length 15 * > e.g. Antipode 2 R2 B2 D2 F2 D2 F2 T2 L2 T2 D2 F2 T2 L2 T2 B2 * --------------------------------------------------------------------- The way we know the lower bound is by working backwards counting how many positions we can reach in a small number of moves from the solved position. If this is less than 43,252,003,274,489,856,000 (the total number of positions of Rubik's cube) then you need more than that number of moves to reach the other positions of the cube. Therefore, those positions will require more moves to solve. The answer depends on what we consider a move. There are three common definitions. The most restrictive is the QF metric, in which only a quarter-turn of a face is allowed as a single move. More common is the HF metric, in which a half-turn of a face is also counted as a single move. The most generous is the HS metric, in which a quarter- turn or half-turn of a central slice is also counted as a single move. These metrics are sometimes called the 12-generator, 18-generator, and 27-generator metrics, respectively, for the number of primitive moves. The definition does not affect which positions you can get to, or even how you get there, only how many moves we count for it. The answer is that even in the HS metric, the lower bound is 16, because at most 17,508,850,688,971,332,784 positions can be reached within 15 HS moves. In the HF metric, the lower bound is 18, because at most 19,973,266,111,335,481,264 positions can be reached within 17 HF moves. And in the QT metric, the lower bound is 21, because at most 39,812,499,178,877,773,072 positions can be reached within 20 QT moves. -- jjfink@skcla.monsanto.com writes: Lately in this conference I've noted several messages related to Rubik's Cube and Square 1. I've been an avid cube fanatic since 1981 and I've been gathering cube information since. Around Feb. 1990 I started to produce the Domain of the Cube Newsletter, which focuses on Rubik's Cube and all the cube variants produced to date. I include notes on unproduced prototype cubes which don't even exist, patent information, cube history (and prehistory), computer simulations of puzzles, etc. I'm up to the 4th issue. Anyways, if you're interested in other puzzles of the scramble by rotation type you may be interested in DOTC. It's available free to anyone interested. I am especially interested in contributing articles for the newsletter, e.g. ideas for new variants, God's Algorithm. Anyone ever write a Magic Dodecahedron simulation for a computer? Drop me a SASE (say empire size) if you're interested in DOTC or if you would like to exchange notes on Rubik's Cube, Square 1 etc. I'm also interested in exchanging puzzle simulations, e.g. Rubik's Cube, Twisty Torus, NxNxN Simulations, etc, for Amiga and IBM computers. I've written several Rubik's Cube solving programs, and I'm trying to make the definitive puzzle solving engine. I'm also interested in AI programs for Rubik's Cube and the like. Ideal Toy put out the Rubik's Cube Newsletter, starting with issue #1 on May 1982. There were 4 issues in all. They are: #1, May 1982 #2, Aug 1982 #3, Winter 1983 #4, Aug 1983 There was another sort of magazine, published in several languages called Rubik's Logic and Fantasy in space. I believe there were 8 issues in all. Unfortunately I don't have any of these! I'm willing to buy these off anyone interesting in selling. I would like to get the originals if at all possible... I'm also interested in buying any books on the cube or related puzzles. In particular I am _very_ interested in obtaining the following: Cube Games Don Taylor, Leanne Rylands Official Solution to Alexander's Star Adam Alexander The Amazing Pyraminx Dr. Ronald Turner-Smith The Winning Solution Minh Thai The Winning Solution to Rubik's Revenge Minh Thai Simple Solutions to Cubic Puzzles James G. Nourse I'm also interested in buying puzzles of the mechanical type. I'm still missing the Pyraminx Star (basically a Pyraminx with more tips on it), Pyraminx Ball, and the Puck. If anyone out here is a fellow collector I'd like to hear from you. If you have a cube variant which you think is rare, or an idea for a cube variant we could swap notes. I'm in the middle of compiling an exhaustive library for computer simulations of puzzles. This includes simulations of all Uwe Meffert's puzzles which he prototyped but _never_ produced. In fact, I'm in the middle of working on a Pyraminx Hexagon solver. What? Never heard of it? Meffert did a lot of other puzzles which never were made. I invented some new "scramble by rotation" puzzles myself. My favourite creation is the Twisty Torus. It is a torus puzzle with segments (which slide around 360 degrees) with multiple rings around the circumference. The computer puzzle simulation library I'm forming will be described in depth in DOTC #4 (The Domain of the Cube Newsletter). So if you have any interesting computer puzzle programs please email me and tell me all about them! Also to the people interested in obtaining a subscription to DOTC, who are outside of Canada (which it seems is just about all of you!) please don't send U.S. or non-Canadian stamps (yeah, I know I said Self-Addressed Stamped Envelope before). Instead send me an international money order in Canadian funds for $6. I'll send you the first 4 issues (issue #4 is almost finished). Mark Longridge Address: 259 Thornton Rd N, Oshawa Ontario Canada, L1J 6T2 Email: mark.longridge@canrem.com One other thing, the six bucks is not for me to make any money. This is only to cover the cost of producing it and mailing it. I'm just trying to spread the word about DOTC and to encourage other mechanical puzzle lovers to share their ideas, books, programs and puzzles. Most of the programs I've written and/or collected are shareware for C64, Amiga and IBM. I have source for all my programs (all in C or Basic) and I am thinking of providing a disk with the 4th issue of DOTC. If the response is favourable I will continue to provide disks with DOTC. -- Mark Longridge <mark.longridge@canrem.com> writes: It may interest people to know that in the latest issue of "Cubism For Fun" % (# 28 that I just received yesterday) there is an article by Herbert Kociemba from Darmstadt. He describes a program that solves the cube. He states that until now he has found no configuration that required more than 21 turns to solve. He gives a 20 move manoeuvre to get at the "all edges flipped/ all corners twisted" position: DF^2U'B^2R^2B^2R^2LB'D'FD^2FB^2UF'RLU^2F' or in Varga's parlance: dofitabiribirilobadafodifobitofarolotifa Other things #28 contains are an analysis of Square 1, an article about triangular tilings by Martin Gardner, and a number of articles about other puzzles. -- % CFF is a newsletter published by the Dutch Cubusts Club NKC. Secretary: Anneke Treep Postbus 8295 6710 AG Ede The Netherlands Membership fee for 1992 is DFL 20 (about$ 11). -- -- dik t. winter <dik@cwi.nl> References: Mathematical Papers ------------------- Rubik's Revenge: The Group Theoretical Solution Mogens Esrom Larsen A.M.M. Vol. 92 No. 6, June-July 1985, pages 381-390 Rubik's Groups E. C. Turner & K. F. Gold, American Mathematical Monthly, vol. 92 No. 9 November 1985, pp. 617-629. Cubelike Puzzles - What Are They and How Do You Solve Them? J.A. Eidswick A.M.M. Vol. 93 No. 3 March 1986, pp 157-176 The Slice Group in Rubik's Cube, David Hecker, Ranan Banerji Mathematics Magazine, Vol. 58 No. 4 Sept 1985 The Group of the Hungarian Magic Cube Chris Rowley Proceedings of the First Western Austrialian Conference on Algebra, 1982 Books ----- Rubik's Cubic Compendium Erno Rubik, Tamas Varga, et al, Edited by David Singmaster Oxford University Press, 1987 Enslow Publishers Inc ==> competition/games/rubiks/rubiks.magic.p <== How do you solve Rubik's Magic? ==> competition/games/rubiks/rubiks.magic.s <== The solution is in a 3x3 grid with a corner missing. +---+---+---+ +---+---+---+---+ | 3 | 5 | 7 | | 1 | 3 | 5 | 7 | +---+---+---+ +---+---+---+---+ | 1 | 6 | 8 | | 2 | 4 | 6 | 8 | +---+---+---+ +---+---+---+---+ | 2 | 4 | Original Shape +---+---+ To get the 2x4 "standard" shape into this shape, follow this: 1. Lie it flat in front of you (4 going across). 2. Flip the pair (1,2) up and over on top of (3,4). 3. Flip the ONE square (2) up and over (1). [Note: if step 3 won't go, start over, but flip the entire original shape over (exposing the back).] 4. Flip the pair (2,4) up and over on top of (5,6). 5. Flip the pair (1,2) up and toward you on top of (blank,4). 6. Flip the ONE square (2) up and left on top of (1). 7. Flip the pair (2,4) up and toward you. Your puzzle won't be completely solved, but this is how to get the shape. Notice that 3,5,6,7,8 don't move. ==> competition/games/scrabble.p <== What are some exceptional Scrabble Brand Crossword Game (TM) games? ==> competition/games/scrabble.s <== The shortest Scrabble game: The Scrabble Players News, Vol. XI No. 49, June 1983, contributed by Kyle Corbin of Raleigh, NC: [J] J U S S O X [X]U which can be done in 4 moves, JUS, SOX, [J]US, and [X]U. In SPN Vol. XI, No. 52, December 1983, Alan Frank presented what he claimed is the shortest game where no blanks are used, also four moves: C WUD CUKES DEY S This was followed in SPN, Vol. XII No. 54, April 1984, by Terry Davis of Glasgow, KY: V V O[X] [X]U, which is three moves. He noted that the use of two blanks prevents such plays as VOLVOX. Unfortunately, it doesn't prevent SONOVOX. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Record for the highest Scrabble score in a single turn (in a legal position): According to the Scrabble Players Newspaper (since renamed to Scrabble Players News) issue 44, p13, the highest score for one turn yet discovered, using the Official Scrabble Players Dictionary, 1st ed. (the 2nd edition is now in use in club and tournament play) and the Websters 9th New Collegiate Dictionary, was the following: d i s e q u i l i b r a t e D . . . . . . . e . . . . . . e . . . . . . . e . . . . . o m r a d i o a u t o g r a p(h)Y . . . . . . . . . . . w a s T . . . . . . . . . . b e . . h . . . . . . . . . . a . . g o . . . c o n j u n c t i v a L . . . . . . . . . . . . . n o . . . . . . . f i n i k i n G . . . . . . . a . . . (l) e i . . . . . . . d . s p e l t Z . . . . . . w e . . . . . . e . . . . . . r . . . . . . o r m e t h o x y f l u r a n e S for 1682 points. According to the May 1986 issue of GAMES, the highest known score achievable in one turn is 1,962 points. The word is BENZOXYCAMPHORS formed across the three triple-word scores on the bottom of the board. Apparently it was discovered by Darryl Francis, Ron Jerome, and Jeff Grant. As for other Scrabble trivia, the highest-scoring first move based on the Official Scrabble Players Dictionary is 120 points, with the words JUKEBOX, QUIZZED, SQUEEZE, or ZYMURGY. If Funk & Wagnall's New Standard Dictionary is used then ZYXOMMA, worth 130 points, can be formed. The highest-scoring game, based on Webster's Second and Third and on the Oxford English Dictionary, was devised by Ron Jerome and Ralph Beaman and totalled 4,142 points for the two players. The highest-scoring words in the game were BENZOXYCAMPHORS, VELVETEEN, and JACKPUDDINGHOOD. The following example of a SCRABBLE game produced a score of 2448 for one player and 1175 for the final word. It is taken from _Beyond Language_ (1967) by Dmitri Borgman (pp. 217-218). He credits this solution to Mrs. Josefa H. Byrne of San Francisco and implies that all words can be found in _Webster's Second Edition_. The two large words (multiplied by 27 as they span 3 triple word scores) are ZOOPSYCHOLOGIST (a psychologist who treats animals rather than humans) and PREJUDICATENESS (the condition or state of being decided beforehand). The asterisks (*) represent the blank tiles. (Please excuse any typo's). Board Player1 Player2 Z O O P S Y C H O L O G I S T ABILITY 76 ERI, YE 9 O N H A U R O W MAN, MI 10 EN 2 * R I B R O V E I FEN, FUN 14 MANIA 7 L T I K E G TABU 12 RIB 6 O L NEXT 11 AM 4 G I AX 9 END 6 I T IT, TIKE 10 LURE 6 * Y E LEND, LOGIC*AL 79 OO*LOGICAL 8 A R FUND, JUD 27 ATE, MA 7 L E N D M I ROVE 14 LO 2 E A Q DARE, DE 13 ES, ES, RE 6 W A X F E N U RE, ROW 14 IRE, IS, SO 7 E T A B U I A DARED, QUAD 22 ON 4 E N A M D A R E D WAX, WEE 27 WIG 9 P R E J U D I C A T E N E S S CHIT, HA 14 ON 2 PREJUDICATENESS, AN, MANIAC, QUADS, WEEP 911 OOP 8 ZOOPSYCHOLOGIST, HABILITY, TWIG, ZOOLOGICAL 1175 -------------------------------------- Total: 2438 93 F, N, V, T in loser's hand: +10 -10 -------------------------------------- Final Score: 2448 83 --------------------------------------------------------------------------- It is possible to form the following 14 7-letter OSPD words from the non-blank tiles: HUMANLY FATUOUS AMAZING EERIEST ROOFING TOILERS QUIXOTE JEWELRY CAPABLE PREVIEW BIDDERS HACKING OVATION DONATED ==> competition/games/set.p <== What is the size of the largest collection of cards from which NO "set" can be selected ? ==> competition/games/set.s <== I can get 20: 1ROL 1GDL 1GDM 1GOM 1GSL 1PDH 1PDM 1POL 1POM 2RSL 2PDM 3ROL 3ROM 3RSL 3RSH 3GDM 3GOL 3GSL 3GSM 3POM This collection of 20 is a local maximum. The small C progam shown below was used to check for all possible extensions to a collection of 21. Of course this leaves open the question whether there exists a completely different collection of 21 from which no "set" can be selected. -- Gene Miller ------- C Program enclosed ------- #define N 21 static int data[N][4]= { 1, 1, 2, 1, /* 00 */ 1, 2, 1, 1, /* 01 */ 1, 2, 1, 2, /* 02 */ 1, 2, 2, 2, /* 03 */ 1, 2, 3, 1, /* 04 */ 1, 3, 1, 3, /* 05 */ 1, 3, 1, 2, /* 06 */ 1, 3, 2, 1, /* 07 */ 1, 3, 2, 2, /* 08 */ 2, 1, 3, 1, /* 09 */ 2, 3, 1, 2, /* 10 */ 3, 1, 2, 1, /* 11 */ 3, 1, 2, 2, /* 12 */ 3, 1, 3, 1, /* 13 */ 3, 1, 3, 3, /* 14 */ 3, 2, 1, 2, /* 15 */ 3, 2, 2, 1, /* 16 */ 3, 2, 3, 1, /* 17 */ 3, 2, 3, 2, /* 18 */ 3, 3, 2, 2, /* 19 */ 0, 0, 0, 0 /* 20 */ /* leave space for Nth combo */ }; main() { int x, y, z, w; for (x=1; x<=3; x++) /* check all combos */ for (y=1; y<=3; y++) for (z=1; z<=3; z++) for (w=1; w<=3; w++) printf ("%d %d %d %d -> sets=%d\n", x, y, z, w, check (x, y, z, w)); } int check(x,y,z,w) int x, y, z, w; { int i,j,k,m; int errors, sets; for (i=0; i<N; i++) /* check for pre-existing combos */ if (x==data[i][0] && y==data[i][1] && z==data[i][2] && w==data[i][3] ) { return -1; /* discard pre-existing*/ } data[N-1][0] = x; /* make this the Nth combo */ data[N-1][1] = y; data[N-1][2] = z; data[N-1][3] = w; sets = 0; /* start counting sets */ for (i=0; i<N; i++) /* look for sets */ for (j=i+1; j<N; j++) for (k=j+1; k<N; k++) { errors = 0; for (m=0; m<4; m++) { if (data[i][m] == data[j][m]) { if (data[k][m] != data[i][m]) errors++; if (data[k][m] != data[j][m]) errors++; } else { if (data[k][m] == data[i][m]) errors++; if (data[k][m] == data[j][m]) errors++; } } if (errors == 0) /* no errors means is a set */ sets++; /* increment number of sets */ } return sets; } -- I did some more experimenting. With the enclosed C program, I looked at many randomly generated collections. In an earlier version of this program I got one collection of 20 from a series of 100 trials. The rest were collections ranging in size from 16 to 19. Unfortunately, in an attempt to make this program more readable and more general, I changed the algorithm slightly and I haven't been able to achieve 20 since then. I can't remember enough about my changes to be able to get back to the previous version. In the most recent 1000 trials all of the maximaml collections range in size from 16 to 18. I think that this experiment has shed very little light on what is the global maximum, since the search space is many orders of magnitude larger than what can be tried in a reasonable amount of time through random searching. I assume that Mr. Ring found his collection of 20 by hand. This indicates that an intelligent search may be more fruitful than a purely random one. ------------------ Program enclosed ------------- int n; int data[81][4]; main() { int i; for (i=0; i<1000; i++) { /* Do 1000 independent trials */ printf ("Trial %d:\n", i); try(); } } try() { int i; int x, y, z, w; n = 0; /* set collection size to zero */ for (i=0; i<100; i++) { /* try 100 random combos */ x = 1 + rand()%3; y = 1 + rand()%3; z = 1 + rand()%3; w = 1 + rand()%3; check (x, y, z, w); } for (x=1; x<=3; x++) /* check all combos */ for (y=1; y<=3; y++) for (z=1; z<=3; z++) for (w=1; w<=3; w++) check (x, y, z, w); printf (" collection size=%d\n", n); } int check(x, y, z, w) /* check whether a new combo can be added */ int x, y, z, w; { int i,j,k,m; int errors, sets; for (i=0; i<n; i++) /* check for pre-existing combos */ if (x==data[i][0] && y==data[i][1] && z==data[i][2] && w==data[i][3] ) { return -1; /* discard pre-existing*/ } data[n][0] = x; /* make this the nth combo */ data[n][1] = y; data[n][2] = z; data[n][3] = w; sets = 0; /* start counting sets */ for (i=0; i<=n; i++) /* look for sets */ for (j=i+1; j<=n; j++) for (k=j+1; k<=n; k++) { errors = 0; for (m=0; m<4; m++) { if (data[i][m] == data[j][m]) { if (data[k][m] != data[i][m]) errors++; if (data[k][m] != data[j][m]) errors++; } else { if (data[k][m] == data[i][m]) errors++; if (data[k][m] == data[j][m]) errors++; } } if (errors == 0) /* no errors means is a set */ sets++; /* increment number of sets */ } if (sets == 0) { n++; /* increment collection size */ printf ("%d %d %d %d\n", x, y, z, w); } return sets; } ------------------ end of enclosed program ------------- -- Gene -- Gene Miller Multimedia Communications NYNEX Science & Technology Phone: 914 644 2834 500 Westchester Avenue Fax: 914 997 2997, 914 644 2260 White Plains, NY 10604 Email: gene@nynexst.com ==> competition/games/soma.p <== What is the solution to Soma Cubes? ==> competition/games/soma.s <== The soma cube is dissected in excruciating detail in volume 2 of "Winning Ways" by Conway, Berlekamp and Guy, in the same chapter as the excruciatingly detailed dissection of Rubik's Cube. ==> competition/games/square-1.p <== Does anyone have any hints on how to solve the Square-1 puzzle? ==> competition/games/square-1.s <== SHAPES 1. There are 29 different shapes for a side, counting reflections: 1 with 6 corners, 0 edges 3 with 5 corners, 2 edges 10 with 4 corners, 4 edges 10 with 3 corners, 6 edges 5 with 2 corners, 8 edges 2. Naturally, a surplus of corners on one side must be compensated by a deficit of corners on the other side. Thus there are 1*5 + 3*10 + C(10,2) = 5 + 30 + 55 = 90 distinct combinations of shapes, not counting the middle layer. 3. You can reach two squares from any other shape in at most 7 transforms, where a transform consists of (1) optionally twisting the top, (2) optionally twisting the bottom, and (3) flipping. 4. Each transform toggles the middle layer between Square and Kite, so you may need 8 transforms to reach a perfect cube. 5. The shapes with 4 corners and 4 edges on each side fall into four mutually separated classes. Side shapes can be assigned values: 0: Square, Mushroom, and Shield; 1: Left Fist and Left Paw; 2: Scallop, Kite, and Barrel; 3. Right Fist and Right Paw. The top and bottom's sum or difference, depending on how you look at them, is a constant. Notice that the side shapes with bilateral symmetry are those with even values. 6. To change this constant, and in particular to make it zero, you must attain a position that does not have 4 corners and 4 edges on each side. Almost any such position will do, but returning to 4 corners and 4 edges with the right constant is left to your ingenuity. 7. If the top and bottom are Squares but the middle is a Kite, just flip with the top and bottom 30deg out of phase and you will get a cube. COLORS 1. I do not know the most efficient way to restore the colors. What follows is my own suboptimal method. All flips keep the yellow stripe steady and flip the blue stripe. 2. You can permute the corners without changing the edges, so first get the edges right, then the corners. 3. This transformation sends the right top edge to the bottom and the left bottom edge to the top, leaving the other edges on the same side as they started: Twist top 30deg cl, flip, twist top 30deg ccl, twist bottom 150deg cl, flip, twist bottom 30deg cl, twist top 120deg cl, flip, twist top 30deg ccl, twist bottom 150deg cl, flip, twist bottom 30deg cl. Cl and ccl are defined looking directly at the face. With this transformation you can eventually get all the white edges on top. 4. Check the parity of the edge sequence on each side. If either is wrong, you need to fix it. Sorry -- I don't know how! (See any standard reference on combinatorics for an explanation of parity.) 5. The following transformation cyclically permutes ccl all the top edges but the right one and cl all the bottom edges but the left one. Apply the transformation in 3., and turn the whole cube 180deg. Repeat. This is a useful transformation, though not a cure-all. 6. Varying the transformation in 3. with other twists will produce other results. 7. The following transformation changes a cube into a Comet and Star: Flip to get Kite and Kite. Twist top and bottom cl 90deg and flip to get Barrel and Barrel. Twist top cl 30 and bottom cl 60 and flip to get Scallop and Scallop. Twist top cl 60 and bottom cl 120 and flip to get Comet and Star. The virtue of the Star is that it contains only corners, so that you can permute the corners without altering the edges. 8. To reach a Lemon and Star instead, replace the final bottom cl 120 with a bottom cl 60. In both these transformation the Star is on the bottom. 9. The following transformation cyclically permutes all but the bottom left rear. It sends the top left front to the bottom, and the bottom left front to the top. Go to Comet and Star. Twist star cl 60. Go to Lemon and Star -- you need not return all the way to the cube, but do it if you're unsure of yourself by following 7 backwards. Twist star cl 60. Return to cube by following 8 backwards. With this transformation you should be able to get all the white corners on top. 10. Check the parity of the corner sequences on both sides. If the bottom parity is wrong, here's how to fix it: Go to Lemon and Star. The colors on the Star will run WWGWWG. Twist it 180 and return to cube. 11. If the top parity is wrong, do the same thing, except that when you go from Scallop and Scallop to Lemon and Star, twist the top and bottom ccl instead of cl. The colors on the Star should now run GGWGGW. 12. Once the parity is right on both sides, the basic method is to go to Comet and Star, twist the star 120 cl (it will be WGWGWG), return to cube, twist one or both sides, go to Comet and Star, undo the star twist, return to cube, undo the side twists. With no side twists, this does nothing. If you twist the top, you will permute the top corners. If you twist the bottom, you will permute the bottom corners. Eventually you will get both the top and the bottom right. Don't forget to undo the side twists -- you need to have the edges in the right places. Happy twisting.... -- Col. G. L. Sicherman gls@windmill.att.COM ==> competition/games/think.and.jump.p <== THINK & JUMP: FIRST THINK, THEN JUMP UNTIL YOU ARE LEFT WITH ONE PEG! O - O O - O / \ / \ / \ / \ O---O---O---O---O BOARD DESCRIPTION: To the right is a model of \ / \ / \ / \ / the Think & Jump board. The O---O---O---O---O---O O's represent holes which / \ / \ / \ / \ / \ / \ contain pegs. O---O---O---O---O---O---O \ / \ / \ / \ / \ / \ / O---O---O---O---O---O DIRECTIONS: To play this brain teaser, you begin / \ / \ / \ / \ by removing the center peg. Then, O---O---O---O---O moving any direction in the grid, \ / \ / \ / \ / jump over one peg at a time, O - O O - O removing the jumped peg - until only one peg is left. It's harder then it looks. But it's more fun than you can imagine. SKILL CHART: 10 pegs left - getting better 5 pegs left - true talent 1 peg left - you're a genius Manufactured by Pressman Toy Corporation, NY, NY. ==> competition/games/think.and.jump.s <== Three-color the board in the obvious way. The initial configuration has 12 of each color, and each jump changes the parity of all three colors. Thus, it is impossible to achieve any position where the colors do not have the same parity; in particular, (1,0,0). If you remove the requirement that the initially-empty cell must be at the center, the game becomes solvable. The demonstration is left as an exercise. Karl Heuer rutgers!harvard!ima!haddock!karl karl@haddock.ima.isc.com Here is one way of reducing Think & Jump to two pegs. Long simplifies Balsley's scintillating snowflake solution: 1 U-S A - B C - D 2 H-U / \ / \ / \ / \ 3 V-T E---F---G---H---I 4 S-H \ / \ / \ / \ / 5 D-M J---K---L---M---N---O 6 F-S / \ / \ / \ / \ / \ / \ 7 Q-F P---Q---R---S---T---U---V 8 A-L \ / \ / \ / \ / \ / \ / 9 S-Q W---X---Y---Z---a---b 10 P-R / \ / \ / \ / \ 11 Z-N c---d---e---f---g 12 Y-K \ / \ / \ / \ / 13 h-Y h - i j - k 14 k-Z The board should now be in the snowflake pattern, i.e. look like o - * * - o / \ / \ / \ / \ *---o---*---o---* \ / \ / \ / \ / *---*---*---*---*---* / \ / \ / \ / \ / \ / \ o---o---o---o---o---o---o \ / \ / \ / \ / \ / \ / *---*---*---*---*---* / \ / \ / \ / \ *---o---*---o---* \ / \ / \ / \ / o - * * - o where o is empty and * is a peg. The top and bottom can now be reduced to single pegs individually. For example, we could continue 15 g-T 16 Y-a 17 i-Z 18 T-e 19 j-Y 20 b-Z 21 c-R 22 Z-X 23 W-Y 24 R-e which finishes the bottom. The top can be done in a similar manner. -- Chris Long ==> competition/games/tictactoe.p <== In random tic-tac-toe, what is the probability that the first mover wins? ==> competition/games/tictactoe.s <== Count cases. First assume that the game goes on even after a win. (Later figure out who won if each player gets a row of three.) Then there are 9!/5!4! possible final boards, of which 8*6!/2!4! - 2*6*4!/0!4! - 3*3*4!/0!4! - 1 = 98 have a row of three Xs. The first term is 8 rows times (6 choose 2) ways to put down the remaining 2 Xs. The second term is the number of ways X can have a diagonal row plus a horizontal or vertical row. The third term is the number of ways X can have a vertical and a horizontal row, and the 4th term is the number of ways X can have two diagonal rows. All the two-row configurations must be subtracted to avoid double-counting. There are 8*6!/1!5! = 48 ways O can get a row. There is no double- counting problem since only 4 Os are on the final board. There are 6*2*3!/2!1! = 36 ways that both players can have a row. (6 possible rows for X, each leaving 2 possible rows for O and (3 choose 2) ways to arrange the remaining row.) These cases need further consideration. There are 98 - 36 = 62 ways X can have a row but not O. There are 48 - 36 = 12 ways O can have a row but not X. There are 126 - 36 - 62 - 12 = 16 ways the game can be a tie. Now consider the 36 configurations in which each player has a row. Each such can be achieved in 5!4! = 2880 orders. There are 3*4!4! = 1728 ways that X's last move completes his row. In these cases O wins. There are 2*3*3!3! = 216 ways that Xs fourth move completes his row and Os row is already done in three moves. In these cases O also wins. Altogether, O wins 1728 + 216 = 1944 out of 2880 times in each of these 36 configurations. X wins the other 936 out of 2880. Altogether, the probability of X winning is ( 62 + 36*(936/2880) ) / 126. win: 737 / 1260 ( 0.5849206... ) lose: 121 / 420 ( 0.2880952... ) draw: 8 / 63 ( 0.1269841... ) The computer output below agress with this analysis. 1000000 games: won 584865, lost 288240, tied 126895 Instead, how about just methodically having the program play every possible game, tallying up who wins? Wonderful idea, especially since there are only 9! ~ 1/3 million possible games. Of course some are identical because they end in fewer than 8 moves. It is clear that these should be counted multiple times since they are more probable than games that go longer. The result: 362880 games: won 212256, lost 104544, tied 46080 #include <stdio.h> int board[9]; int N, move, won, lost, tied; int perm[9] = { 0, 1, 2, 3, 4, 5, 6, 7, 8 }; int rows[8][3] = { { 0, 1, 2 }, { 3, 4, 5 }, { 6, 7, 8 }, { 0, 3, 6 }, { 1, 4, 7 }, { 2, 5, 8 }, { 0, 4, 8 }, { 2, 4, 6 } }; main() { do { bzero((char *)board, sizeof board); for ( move=0; move<9; move++ ) { board[perm[move]] = (move&1) ? 4 : 1; if ( move >= 4 && over() ) break; } if ( move == 9 ) tied++; #ifdef DEBUG printf("%1d%1d%1d\n%1d%1d%1d w %d, l %d, t %d\n%1d%1d%1d\n\n", board[0], board[1], board[2], board[3], board[4], board[5], won, lost, tied, board[6], board[7], board[8]); #endif N++; } while ( nextperm(perm, 9) ); printf("%d games: won %d, lost %d, tied %d\n", N, won, lost, tied); exit(0); } int s; int *row; over() { for ( row=rows[0]; row<rows[8]; row+=3 ) { s = board[row[0]] + board[row[1]] + board[row[2]]; if ( s == 3 ) return ++won; if ( s == 12 ) return ++lost; } return 0; } nextperm(c, n) int c[], n; { int i = n-2, j=n-1, t; while ( i >= 0 && c[i] >= c[i+1] ) i--; if ( i < 0 ) return 0; while ( c[j] <= c[i] ) j--; t = c[i]; c[i] = c[j]; c[j] = t; i++; j = n-1; while ( i < j ) { t = c[i]; c[i] = c[j]; c[j] = t; i++; j--; } return 1; } ==> competition/tests/analogies/long.p <== 1. Host : Guest :: Cynophobia : ? 2. Mountain : Plain :: Acrocephalic : ? 3. Lover : Believer :: Philofelist : ? 4. 4 : 6 :: Bumblebee : ? 5. 2 : 1 :: Major : ? 6. 1 : 24 :: Group : ? 7. 4 : 64 :: Crotchet : ? 8. Last : First :: Grave : ? 9. 7 : 9 :: Throne : ? 10. Pride : Hatred :: Beelzebub : ? 11. Dollar : Bond :: Grant : ? 12. Ek : Sankh :: 1 : ? ==> competition/tests/analogies/long.s <== 1. Lyssophobia Cynophobia is the fear of dogs; lyssophobia is the fear of rabies. As Rodney Adams pointed out, a word meaning the fear of fleas would be even better, but I've been unable to locate such a word (although surely one must exists). 2. Homalocephalic Acrocephalic is having a pointed head; homalocephalic is having a flat head. Rodney Adamas suggested "planoccipital", but on checking this apparently refers to having a flat back of the skull, so he only gets partial credit. 3. Galeanthropist A philofelist is a cat-lover (also commonly called an ailurophile); a galeanthropist is a person who believes they are a cat. 4. Blue Bird A Camp Fire Girl who is 4 is in the Bumblebee Club; a Campfire Girl who is 6 in the Blue Bird Club. I should have had "4 or 5" instead of "4" to remove ambiguity (e.g. Mark Brader suggested "triplane"). 5. Brigadier A 2-star general in the army is a major general; a 1-star general in the army is a brigadier general. 6. Field Army Army groupings; there are 24 "groups" in a "field army". 7. Hemidemisemiquaver A crotchet is a quarter-note; a hemidemisemiquaver is a sixty-fourth note. Rodney Adams and Mark Brader both got this. 8. Prestissimo In music prestissimo means extremely fast; grave means very slow to the point of solemnity. This question was poorly worded (I received both "womb" and "cradle" as answers). 9. Seraph In the ninefold hierarchy of angels, a throne ranks 7th and a seraph 9th (9th being most powerful). Rodney Adams got this one. 10. Sonneillon In Father Sebastien Machaelis's (one of the more famous exorcists) hierarchy of devils, Beelzebub is responsible for pride, Sonneillon for hatred. 11. Roosevelt Grant is on the $50 bill; Roosevelt on the $50 savings bond. 12. 10^14 Ek is 1 in Hindi; sankh is 10^14 (one hundred quadrillion). -- Chris Long, 265 Old York Rd., Bridgewater, NJ 08807-2618 ==> competition/tests/analogies/pomfrit.p <== 1. NATURAL: ARTIFICIAL :: ANKYLOSIS: ? 2. RESCUE FROM CHOKING ON FOOD, etc.: HEIMLICH :: ADJUSTING MIDDLE EAR PRESSURE: ? 3. LYING ON OATH: PERJURY :: INFLUENCING A JURY: ? 4. RECTANGLE: ELLIPSE :: MERCATOR: ? 5. CLOSED: CLEISTOGAMY :: OPEN: ? 6. FO'C'SLE: SYNCOPE :: TH'ARMY: ? 7. FILMS: OSCAR :: MYSTERY NOVELS: ? 8. QUANTITATIVE DATA: STATISTICS :: HUMAN SETTLEMENTS: ? 9. 7: 19 : : SEPTIMAL: ? 10. 3 TO 2: SESQUILATERAL :: 7 TO 5: ? 11. SICILY: JAPAN :: MAFIA: ? 12. CELEBRITIES: SYCOPHANTIC :: ANCESTORS: ? 13. 95: 98 :: VENITE: ? 14. MINCES: EYES :: PORKIES: ? 15. POSTAGE STAMPS: PHILATELIST: MATCHBOX LABELS: ? 16. MALE: FEMALE :: ARRENOTOKY: ? 17 TAILOR: DYER :: SARTORIAL: ? 18. HERMES: BACCHUS :: CADUCEUS: ? 19. 2823: 5331 :: ELEPHANT: ? 20. CENTRE OF GRAVITY: BARYCENTRIC :: ROTARY MOTION: ? 21. CALIFORNIA: EUREKA :: NEW YOKK: ? 22. MARRIAGE: DIGAMY :: COMING OF CHRIST: ? 23. 6: 5 :: PARR: ? 24. GROUP: INDIVIDUAL :: PHYLOGENESIS: ? 25. 12: 11 :: EPSOM: ? ==> competition/tests/analogies/pomfrit.s <== 1. ARTHRODESIS 2. VALSALVA 3. EMBRACERY 4. MOLLWEIDE 5. CHASMOGAMY 6. SYNAL(O)EPHA 7. EDGAR 8. EKISTICS 9. DECENNOVAL 10. SUPERBIQUINTAL 11. YAKUZA 12. FILIOPETISTIC 13. CANTATE 14. LIES 15. PHILLUMENIST 16. THELYTOKY 17. TINCTORIAL 18. THYRSUS 19. ANTIQUARIAN 20. TROCHILIC 21. EXCELSIOR (mottos) 22. PAROUSIA 23. HOWARD (wives of Henry VIII) 24. ONTOGENESIS 25. GLAUBER (salts) ==> competition/tests/analogies/quest.p <== 1. Mother: Maternal :: Stepmother: ? 2. Club: Axe :: Claviform: ? 3. Cook Food: Pressure Cooker :: Kill Germs: ? 4. Water: Air :: Hydraulic: ? 5. Prediction: Dirac :: Proof: ? 6. Raised: Sunken :: Cameo: ? 7. 1: 14 :: Pound: ? 8. Malay: Amok :: Eskimo Women: ? 9. Sexual Intercourse: A Virgin :: Bearing Children: ? 10. Jaundice, Vomiting, Hemorrhages: Syndrome :: Jaundice: ? 11. Guitar: Cello :: Segovia: ? 12. Bars: Leaves :: Eagle: ? 13. Roll: Aileron :: Yaw: ? 14. 100: Century :: 10,000: ? 15. Surface: Figure :: Mobius: ? 16. Logic: Philosophy :: To Know Without Conscious Reasoning: ? 17. Alive: Parasite :: Dead: ? 18. Sea: Land :: Strait: ? 19. Moses: Fluvial :: Noah: ? 20. Remnant: Whole :: Meteorite: ? 21. Opossum, Kangaroo, Wombat: Marsupial :: Salmon, Sturgeon, Shad: ? 22. Twain/Clemens: Allonym :: White House/President: ? 23. Sculptor: Judoka :: Fine: ? 24. Dependent: Independent :: Plankton: ? 25. Matthew, Mark, Luke, John: Gospels :: Joshua-Malachi: ? 26. Luminous Flux: Lumen :: Sound Absorption: ? 27. 2: 3 :: He: ? 28. Growth: Temperature :: Pituitary Gland: ? 29. Spider: Arachnoidism :: Snake: ? 30. Epigram: Anthology :: Foreign Passages: ? 31. Pathogen: Thermometer :: Lethal Wave: ? 32. Russia: Balalaika :: India: ? 33. Involuntary: Sternutatory :: Voluntary: ? 34. Unusual Hunger: Bulimia :: Hunger for the Unusual: ? 35. Blind: Stag :: Tiresias: ? 36. River: Fluvial :: Rain: ? 37. Country: City :: Tariff: ? 38. $/Dollar: Logogram :: 3, 5, 14, 20/Cent: ? 39. Lung Capacity: Spirometer :: Arterial Pressure: ? 40. Gold: Ductile :: Ceramic: ? 41. 7: 8 :: Uranium: ? 42. Judaism: Messiah :: Islam: ? 43. Sight: Amaurosis :: Smell: ? 44. Oceans: Cousteau :: Close Encounters of the Third Kind: ? 45. Diamond/Kimberlite: Perimorph :: Fungus/Oak: ? 46. Compulsion to Pull One's Hair: Trichotillomania :: Imagine Oneself As a Beast: ? 47. Cross: Neutralism :: Hexagram: ? 48. Wing: Tail :: Fuselage: ? 49. Bell: Loud :: Speak: ? 50. Benevolence: Beg :: Philanthropist: ? 51. 10: Decimal :: 20: ? 52. Five-sided Polyhedron: Pentahedron :: ? Faces of Parallelepiped Bounded by a Square: ? 53. Motor: Helicopter :: Airflow: ? 54. Man: Ant :: Barter: ? 55. United States: Soviet Union :: Cubism: ? 56. State: Stipend :: Church: ? 57. Motorcycle: Bicycle :: Motordrome: ? 58. Transparent: Porous :: Obsidian: ? 59. pi*r^2*h: 1/3*pi*r^2*h :: Cylinder: ? ==> competition/tests/analogies/quest.s <== Annotated solutions. If there is more than one word that fits the analogy, we list the best word first. Goodness of fit considers many factors, such as parallel spelling, pronunciation or etymology. In general, a word that occurs in Merriam-Webster's Third New International Dictionary is superior to one that does not. If we are unsure of the answer, we mark it with a question mark. Most of these answers are drawn from Herbert M. Baus, _The Master Crossword Puzzle Dictionary_, Doubleday, New York, 1981. The notation in parentheses refers to the heading and subheading, if any, in Baus. 1. Mother: Maternal :: Stepmother: Novercal (STEPMOTHER, pert.) 2. Club: Axe :: Claviform: Dolabriform, Securiform (AXE, -shaped) "Claviform" is from Latin "clava" for "club"; "securiform" is from Latin "secura" for "axe"; "dolabriform" is from Latin "dolabra" for "to hit with an axe." Thus "securiform" has the more parallel etymology. However, only "dolabriform" occurs in Merriam-Webster's Third New International Dictionary. 3. Cook Food: Pressure Cooker :: Kill Germs: Autoclave (PRESSURE, cooker) 4. Water: Air :: Hydraulic: Pneumatic (AIR, pert.) 5. Prediction: Dirac :: Proof: Anderson (POSITRON, discoverer) 6. Raised: Sunken :: Cameo: Intaglio (GEM, carved) 7. 1: 14 :: Pound: Stone (ENGLAND, weight) 8. Malay: Amok :: Eskimo Women: Piblokto (ESKIMO, hysteria) 9. Sexual Intercourse: A Virgin :: Bearing Children: A Nullipara 10. Jaundice, Vomiting, Hemorrhages: Syndrome :: Jaundice: Symptom (EVIDENCE) 11. Guitar: Cello :: Segovia: Casals (SPAIN, cellist) 12. Bars: Leaves :: Eagle: Stars (INSIGNIA) 13. Roll: Aileron :: Yaw: Rudder (AIRCRAFT, part) 14. 100: Century :: 10,000: Myriad, Banzai? (NUMBER) "Century" usually refers to one hundred years, while "myriad" refers to 10,000 things, but "century" can also mean 100 things. "Banzai" is Japanese for 10,000 years. 15. Surface: Figure :: Mobius: Klein 16. Logic: Philosophy :: To Know Without Conscious Reasoning: Theosophy (MYSTICISM) There are many schools of philosophy that tout the possibility of knowledge without conscious reasoning (e.g., intuitionism). "Theosophy" is closest in form to the word "philosophy." 17. Alive: Parasite :: Dead: Saprophyte (SCAVENGER) 18. Sea: Land :: Strait: Isthmus (CONNECTION) 19. Moses: Fluvial :: Noah: Diluvial (FLOOD, pert.) 20. Remnant: Whole :: Meteorite: Meteoroid? (METEOR) A meteorite is the remains of a meteoroid after it has partially burned up in the atmosphere. The original meteoroid may have come from an asteroid, comet, dust cloud, dark matter, supernova, interstellar collision or other sources as yet unknown. 21. Opossum, Kangaroo, Wombat: Marsupial :: Salmon, Sturgeon, Shad: Andromous (SALMON) 22. Twain/Clemens: Allonym :: White House/President: Metonym (FIGURE, of speech) 23. Sculptor: Judoka :: Fine: Martial (SELF, -defense) 24. Dependent: Independent :: Plankton: Nekton (ANIMAL, free-swimming) 25. Matthew, Mark, Luke, John: Gospels :: Joshua-Malachi: Nebiim (HEBREW, bible books) 26. Luminous Flux: Lumen :: Sound Absorption: Sabin (SOUND, absorption unit) 27. 2: 3 :: He: Li (ELEMENT) 28. Growth: Temperature :: Pituitary Gland: Hypothalamus (BRAIN, part) 29. Spider: Arachnoidism :: Snake: Ophidism, Ophidiasis, Ophiotoxemia None of these words is in Webster's Third. 30. Epigram: Anthology :: Foreign Passages: Chrestomathy, Delectus (COLLECTION) These words are equally good answers. 31. Pathogen: Thermometer :: Lethal Wave: Dosimeter? (X-RAY, measurement) What does "lethal wave" refer to? If it is radiation, then a dosimeter measures the dose, not the effect, as does a thermometer. 32. Russia: Balalaika :: India: Sitar, Sarod (INDIA, musical instrument) Both are guitar-like instruments (lutes) native to India. 33. Involuntary: Sternutatory :: Voluntary: Expectorant? (SPIT) A better word would be an agent that tends to cause snorting or exsufflation, which is the voluntary, rapid expulsion of air from the lungs. 34. Unusual Hunger: Bulimia :: Hunger for the Unusual: Allotriophagy, Pica (HUNGER, unusual) These words are synonyms. 35. Blind: Stag :: Tiresias: Actaeon (STAG, changed to) 36. River: Fluvial :: Rain: Pluvial (RAIN, part.) 37. Country: City :: Tariff: Octroi (TAX, kind) 38. $/Dollar: Logogram :: 3, 5, 14, 20/Cent: Cryptogram (CODE) 39. Lung Capacity: Spirometer :: Arterial Pressure: Sphygmomanometer (PULSE, measurer) 40. Gold: Ductile :: Ceramic: Fictile (CLAY, made of) 41. 7: 8 :: Uranium: Neptunium (ELEMENT, chemical) 42. Judaism: Messiah :: Islam: Mahdi (MOHAMMEDAN, messiah) 43. Sight: Amaurosis :: Smell: Anosmia, Anosphresia (SMELL, loss) These words are synonyms. 44. Oceans: Cousteau :: Close Encounters of the Third Kind: Spielburg, Truffaut Steven Spielburg was the person most responsible for the movie; Francois Truffaut was a French person appearing in the movie. 45. Diamond/Kimberlite: Perimorph :: Fungus/Oak: Endophyte, Endoparasite (PARASITE, plant) An endoparasite is parasitic, while an endophyte may not be. Which answer is best depends upon the kind of fungus. 46. Compulsion to Pull One's Hair: Trichotillomania :: Imagine Oneself As a Beast: Zoanthropy, Lycanthropy Neither word is exactly right: "zoanthropy" means imagining oneself to be an animal, while "lycanthropy" means imagining oneself to be a wolf. 47. Cross: Neutralism :: Hexagram: Zionism (ISRAEL, doctrine) 48. Wing: Tail :: Fuselage: Empennage, Engines, Waist? (TAIL, kind) "Empennage" means the tail assemply of an aircraft, which is more a synonym for "tail" than "wing" is for "fuselage." The four primary forces on an airplane are: lift from the wings, negative lift from the tail, drag from the fuselage, and thrust from the engines. The narrow part at the end of the fuselage is called the "waist." 49. Bell: Loud :: Speak: Hear, Stentorian? The Sanskrit root of "bell" means "he talks" or "he speaks"; the Sanskrit root of "loud" means "he hears". A bell that makes a lot of noise is loud; a speaker who makes a lot of noise is stentorian. 50. Benevolence: Beg :: Philanthropist: Mendicant, Mendicate? If the analogy is attribute: attribute :: noun: noun, the answer is "mendicant"; if the analogy is noun: verb :: noun: verb the answer is "mendicate." 51. 10: Decimal :: 20: Vigesimal (TWENTY, pert.) 52. Five-sided Polyhedron: Pentahedron :: Faces of Parallelepiped Bounded by a Square: ? Does this mean a parallelepiped all of whose faces are bounded by a square (and what does "bounded" mean), or does it mean all six parallelograms that form the faces of a parallelepiped drawn in a plane inside of a square? 53. Motor: Helicopter :: Airflow: Autogiro (HELICOPTER) 54. Man: Ant :: Barter: Trophallaxis 55. United States: Soviet Union :: Cubism: ? (ART, style) If the emphasis is on opposition and collapse, there were several movements that opposed Cubism and that died out (e.g., Purism, Suprematism, Constructivism). If the emphasis is on freedom of perspective versus constraint, there were several movements that emphasized exact conformance with nature (e.g., Naturalism, Realism, Photo-Realism). If the emphasis is on dominating the art scene, the only movement that was contemporary with Cubism and of the same popularity as Cubism was Surrealism. A better answer would be an art movement named "Turkey-ism", since the Soviet Union offered to exchange missiles in Cuba for missiles in Turkey during the Cuban Missile Crisis. 56. State: Stipend :: Church: Prebend (STIPEND) 57. Motorcycle: Bicycle :: Motordrome: Velodrome (CYCLE, track) 58. Transparent: Porous :: Obsidian: Pumice (GLASS, volcanic) 59. pi*r^2*h: 1/3*pi*r^2*h :: Cylinder: Cone ==> competition/tests/math/putnam/putnam.1967.p <== In article <5840002@hpesoc1.HP.COM>, nicholso@hpesoc1.HP.COM (Ron Nicholson) writes: > Say that we have a hallway with n lockers, numbered from sequentialy > from 1 to n. The lockers have two possible states, open and closed. > Initially all the lockers are closed. The first kid who walks down the > hallway flips every locker to the opposite state, that is, opens them > all. The second kid flips the first locker door and every other locker > door to the opposite state, that is, closes them. The third kid flips > every third door, opening some, closing others. The forth kid does every > fourth door, etc. > > After n kid have passed down the hallway, which lockers are open, and > which are closed? B4. (a) Kids run down a row of lockers, flipping doors (closed doors are opened and opened doors are closed). The nth boy flips every nth lockers' door. If all the doors start out closed, which lockers will remain closed after infinitely many kids? ==> competition/tests/math/putnam/putnam.1967.s <== B4. (a) Only lockers whose numbers have an odd number of factors will remain closed, which are the squares. ## User Contributions: |

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