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Received: from MIT.EDU (PACIFIC-CARRIER-ANNEX.MIT.EDU [18.69.0.28]) by bloom-picayune.MIT.EDU (8.6.13/2.3JIK) with SMTP id PAA05233; Sat, 20 Apr 1996 15:10:04 -0400 Received: from [199.164.164.1] by MIT.EDU with SMTP id AA15729; Sat, 20 Apr 96 14:11:37 EDT Received: by questrel.questrel.com (940816.SGI.8.6.9/940406.SGI) for news-answers-request@mit.edu id LAA25195; Sat, 20 Apr 1996 11:11:57 -0700 Newsgroups: rec.puzzles,news.answers,rec.answers Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!spool.mu.edu!howland.reston.ans.net!europa.eng.gtefsd.com!uunet!questrel!chris From: chris@questrel.questrel.com (Chris Cole) Subject: rec.puzzles Archive (competition), part 07 of 35 Message-Id: <puzzles/archive/competition/part2_745653851@questrel.com> Followup-To: rec.puzzles Summary: This is part of an archive of questions and answers that may be of interest to puzzle enthusiasts. Part 1 contains the index to the archive. Read the rec.puzzles FAQ for more information. Sender: chris@questrel.questrel.com (Chris Cole) Reply-To: archive-comment@questrel.questrel.com Organization: Questrel, Inc. References: <puzzles/archive/Instructions_745653851@questrel.com> Date: Wed, 18 Aug 1993 06:04:47 GMT Approved: news-answers-request@MIT.Edu Expires: Thu, 1 Sep 1994 06:04:11 GMT Lines: 1524 Xref: senator-bedfellow.mit.edu rec.puzzles:24993 news.answers:11513 rec.answers:1913 Apparently-To: news-answers-request@mit.edu Archive-name: puzzles/archive/competition/part2 Last-modified: 17 Aug 1993 Version: 4 ==> competition/games/cube.p <== What are some games involving cubes? ==> competition/games/cube.s <== Johan Myrberger's list of 3x3x3 cube puzzles (version 930222) Comments, corrections and contributions are welcome! MAIL: myrberger@e.kth.se Snailmail: Johan Myrberger Hokens gata 8 B S-116 46 STOCKHOLM SWEDEN A: Block puzzles A.1 The Soma Cube ______ ______ ______ ______ |\ \ |\ \ |\ \ |\ \ | \_____\ | \_____\ | \_____\ | \_____\ | | |____ _____| | | | | |____ | | |____ |\| | \ |\ \| | |\| | \ |\| | \ | *_____|_____\ | \_____*_____| | *_____|_____\ | *_____|_____\ | |\ \ | | |\ \ | | | |\ \ | | | | \| \_____\ | \| \_____\ | \| | \_____\ \| | | * | |___| * | |___| *_____| | | *_____|_____| \| | \| | \| | *_____| *_____| *_____| ______ ______ ____________ |\ \ |\ \ |\ \ \ | \_____\ | \_____\ | \_____\_____\ | | |__________ _____| | |____ _____| | | | |\| | \ \ |\ \| | \ |\ \| | | | *_____|_____\_____\ | \_____*_____|_____\ | \_____*_____|_____| | | | | | | | | | | | | | | \| | | | \| | | | \| | | *_____|_____|_____| *_____|_____|_____| *_____|_____| A.2 Half Hour Puzzle ______ ______ ______ |\ \ |\ \ |\ \ | \_____\ | \_____\ | \_____\ | | |__________ _____| | |____ | | |__________ |\| | \ \ |\ \| | \ |\| | \ \ | *_____|_____\_____\ | \_____*_____|_____\ | *_____|_____\_____\ | | | | | | | | | | | | |\ \ | \| | | | \| | | | \| | \_____\ | *_____|_____|_____| *_____|_____|_____| *_____| | |___| \| | *_____| ______ ______ ______ |\ \ |\ \ |\ \ | \_____\ | \_____\ | \_____\ _____| | | _____| | | | | | |\ \| | |\ \| | |\| | | \_____*_____| | \_____*_____|______ ___|_*_____|______ | |\ \ | | | |\ \ \ |\ \ \ \ \| \_____\ | \| | \_____\_____\ | \_____\_____\_____\ * | |___| *_____| | | | | | | | | \| | \| | | \| | | | *_____| *_____|_____| *_____|_____|_____| A.3 Steinhaus's dissected cube ______ ______ ______ |\ \ |\ \ |\ \ | \_____\ | \_____\ | \_____\ | | |__________ _____| | | | | |____ |\| | \ \ |\ \| | |\| | \ | *_____|_____\_____\ | \_____*_____| | *_____|_____\ | | | | | | |\ \ | | | |\ \ \| | | | \| \_____\ | \| | \_____\ *_____|_____|_____| * | |___| *_____| | | \| | \| | *_____| *_____| ____________ ______ ______ |\ \ \ |\ \ |\ \ | \_____\_____\ | \_____\ | \_____\ | | | | | | | ___________| | | \| | | |\| | |\ \ \| | *_____|_____|______ _________|_*_____| | \_____\_____*_____| \ |\ \ \ |\ \ \ \ | | |\ \ | \| \_____\_____\ | \_____\_____\_____\ \| | \_____\ | * | | | | | | | | *_____| | |___| \| | | \| | | | \| | *_____|_____| *_____|_____|_____| *_____| A.4 ______ |\ \ | \_____\ | | |____ Nine of these make a 3x3x3 cube. |\| | \ | *_____|_____\ | | | | \| | | *_____|_____| A.5 ______ ____________ |\ \ |\ \ \ | \_____\ | \_____\_____\ ____________ | | |____ | | | | |\ \ \ |\| | \ |\| | | | \_____\_____\ | *_____|_____\ | *_____|_____| | | | | | | | | | | | | \| | | \| | | \| | | *_____|_____| *_____|_____| *_____|_____| ______ ______ |\ \ |\ \ | \_____\ | \_____\ ______ ______ | | |____ | | |__________ |\ \ |\ \ |\| | \ |\| | \ \ | \_____\ | \_____\ | *_____|_____\ | *_____|_____\_____\ | | |___| | | | | | |____ | | | | | |\| | \| | |\| | | \ |\| | | | | *_____|_____*_____| | *_____|_____|_____\ | *_____|_____|_____| | | | | | | | | | | | | | | | \| | | | \| | | | \| | | | *_____|_____|_____| *_____|_____|_____| *_____|_____|_____| A.6 ______ ______ ______ ______ |\ \ |\ \ |\ \ |\ \ | \_____\ | \_____\ | \_____\ | \_____\ | | |____ _____| | | | | |____ | | |____ |\| | \ |\ \| | |\| | \ |\| | \ | *_____|_____\ | \_____*_____| | *_____|_____\ | *_____|_____\ | |\ \ | | |\ \ | | | |\ \ | | | | \| \_____\ | \| \_____\ | \| | \_____\ \| | | * | |___| * | |___| *_____| | | *_____|_____| \| | \| | \| | *_____| *_____| *_____| ______ ____________ ____________ |\ \ |\ \ \ |\ \ \ | \_____\ | \_____\_____\ | \_____\_____\ _____| | |____ _____| | | | _____| | | | |\ \| | \ |\ \| | | |\ \| | | | \_____*_____|_____\ | \_____*_____|_____| | \_____*_____|_____| | | | | | | | | | | | | | \| | | | \| | | \| | | *_____|_____|_____| *_____|_____| *_____|_____| A.7 ____________ |\ \ \ | \_____\_____\ | | | | |\| | | Six of these and three unit cubes make a 3x3x3 cube. | *_____|_____| | | | | \| | | *_____|_____| A.8 Oskar's ____________ ______ |\ \ \ |\ \ | \_____\_____\ | \_____\ _____| | | | | | |__________ __________________ |\ \| | | |\| | \ \ |\ \ \ \ | \_____*_____|_____| x 5 | *_____|_____\_____\ | *_____\_____\_____\ | | | | | | | | | | | | | | \| | | \| | | | \| | | | *_____|_____| *_____|_____|_____| *_____|_____|_____| A.9 Trikub ____________ ______ ______ |\ \ \ |\ \ |\ \ | \_____\_____\ | \_____\ | \_____\ | | | | | | |__________ _____| | |____ |\| | | |\| | \ \ |\ \| | \ | *_____|_____| | *_____|_____\_____\ | \_____*_____|_____\ | | | | | | | | | | | | | | \| | | \| | | | \| | | | *_____|_____| *_____|_____|_____| *_____|_____|_____| ______ ______ ____________ |\ \ |\ \ |\ \ \ | \_____\ | \_____\ | \_____\_____\ | | |____ | | |____ _____| | | | |\| | \ |\| | \ |\ \| | | | *_____|_____\ | *_____|_____\ | \_____*_____|_____| | |\ \ | | | |\ \ | | | | \| \_____\ | \| | \_____\ \| | | * | |___| *_____| | | *_____|_____| \| | \| | *_____| *_____| and three single cubes in a different colour. The object is to build 3x3x3 cubes with the three single cubes in various positions. E.g: * * * as center * * * as edge * *(3) as * *(2) as * S * * * * *(2)* space *(2)* center * * * * * S (1)* * diagonal (2)* * diagonal (The other two variations with the single cubes in a row are impossible) A.10 ______ ______ ______ |\ \ |\ \ |\ \ | \_____\ | \_____\ | \_____\ _____| | | | | |____ | | |____ |\ \| | |\| | \ |\| | \ | \_____*_____| | *_____|_____\ ___|_*_____|_____\ | |\ \ | | | |\ \ |\ \ \ | \| \_____\ | \| | \_____\ | \_____\_____\ | * | |___| *_____| | | | | | |___| \| | \| | \| | | *_____| *_____| *_____|_____| ______ ______ ______ |\ \ |\ \ |\ \ | \_____\ | \_____\ | \_____\ | | |__________ _____| | |____ | | |____ |\| | \ \ |\ \| | \ |\| | \ | *_____|_____\_____\ | \_____*_____|_____\ | *_____|_____\______ | |\ \ | | | | | | | | | |\ \ \ \| \_____\ | | \| | | | \| | \_____\_____\ * | |___|_____| *_____|_____|_____| *_____| | | | \| | \| | | *_____| *_____|_____| B: Coloured blocks puzzles B.1 Kolor Kraze Thirteen pieces. Each subcube is coloured with one of nine colours as shown below. The object is to form a cube with nine colours on each face. ______ |\ \ | \_____\ | | | ______ ______ ______ ______ ______ ______ |\| 1 | |\ \ |\ \ |\ \ |\ \ |\ \ |\ \ | *_____| | \_____\ | \_____\ | \_____\ | \_____\ | \_____\ | \_____\ | | | | | | | | | | | | | | | | | | | | | |\| 2 | |\| 2 | |\| 2 | |\| 4 | |\| 4 | |\| 7 | |\| 9 | | *_____| | *_____| | *_____| | *_____| | *_____| | *_____| | *_____| | | | | | | | | | | | | | | | | | | | | | \| 3 | \| 3 | \| 1 | \| 1 | \| 5 | \| 5 | \| 5 | *_____| *_____| *_____| *_____| *_____| *_____| *_____| ______ ______ ______ ______ ______ ______ |\ \ |\ \ |\ \ |\ \ |\ \ |\ \ | \_____\ | \_____\ | \_____\ | \_____\ | \_____\ | \_____\ | | | | | | | | | | | | | | | | | | |\| 9 | |\| 9 | |\| 3 | |\| 6 | |\| 6 | |\| 6 | | *_____| | *_____| | *_____| | *_____| | *_____| | *_____| | | | | | | | | | | | | | | | | | | \| 7 | \| 8 | \| 8 | \| 8 | \| 7 | \| 4 | *_____| *_____| *_____| *_____| *_____| *_____| B.2 Given nine red, nine blue and nine yellow cubes. Form a 3x3x3 cube in which all three colours appears in each of the 27 orthogonal rows. B.3 Given nine red, nine blue and nine yellow cubes. Form a 3x3x3 cube so that every row of three (the 27 orthogonal rows, the 18 diagonal rows on the nine square cross-sections and the 4 space diagonals) contains neither three cubes of like colour nor three of three different colours. B.4 Nine pieces, three of each type. Each subcube is coloured with one of three colours as shown below. The object is to build a 3x3x3 cube in which all three colours appears in each of the 27 orthogonal rows. (As in B.2) ______ ______ ______ |\ \ |\ \ |\ \ | \_____\ | \_____\ | \_____\ | | |____ | | |____ | | |____ |\| A | \ x 3 |\| B | \ x 3 |\| A | \ x 3 | *_____|_____\ | *_____|_____\ | *_____|_____\ | | | | | | | | | | | | \| B | C | \| A | C | \| C | B | *_____|_____| *_____|_____| *_____|_____| C: Strings of cubes C.1 Pululahua's dice 27 cubes are joined by an elastic thread through the centers of the cubes as shown below. The object is to fold the structure to a 3x3x3 cube. ____________________________________ |\ \ \ \ \ \ \ | \_____\_____\_____\_____\_____\_____\ | | | | | | | | |\| :77|77777|77: | :77|77777|77: | | *__:__|_____|__:__|__:__|_____|__:__| | | : |___| | : | : |___| | : | |\| : | \| 777|777 | \| : | | *__:__|_____*_____|_____|_____*__:__| | | : | | |___| | | : |____ \| 777|77777|77: | \| :77|777 | \ *_____|_____|__:__|_____*__:__|_____|_____\ | | : | | : | | | |\| : | + | 777|77777|77: | | *__:__|__:__|_____|_____|__:__| | | : | : | | | : | \| + | : | :77|77777|777 | *_____|__:__|__:__|_____|_____| | | : | : | \| 777|777 | *_____|_____| C.1.X The C.1 puzzle type exploited by Glenn A. Iba (quoted) INTRODUCTION "Chain Cube" Puzzles consist of 27 unit cubies with a string running sequentially through them. The string always enters and exits a cubie through the center of a face. The typical cubie has one entry and one exit (the ends of the chain only have 1, since the string terminates there). There are two ways for the string to pass through any single cubie: 1. The string enters and exits non-adjacent faces (i.e. passes straight through the cubie) 2. It enters and exits through adjacent faces (i.e. makes a "right angle" turn through the cubie) Thus a chain is defined by its sequence of straight steps and right angle turns. Reversing the sequence (of course) specifies the same chain since the chain can be "read" starting from either end. Before making a turn, it is possible to "pivot" the next cubie to be placed, so there are (in general) 4 choices of how to make a "Turn" in 3-space. The object is to fold the chain into a 3x3x3 cube. It is possible to prove that any solvable sequence must have at least 2 straight steps. [The smallest odd-dimensioned box that can be packed by a chain of all Turns and no Straights is 3x5x7. Not a 3x3x3 puzzle, but an interesting challenge. The 5x5x5 can be done too, but its not the smallest in volume]. With the aid of a computer search program I've produced a catalog of the number of solutions for all (solvable) sequences. Here is an example sequence that has a unique solution (up to reflections and rotations): (2 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1) the notation is a kind of "run length" coding, where the chain takes the given number of steps in a straight line, then make a right-angle turn. Equivalently, replace 1 by Turn, 2 by Straight followed by a Turn. The above sequence was actually a physical puzzle I saw at Roy's house, so I recorded the sequence, and verified (by hand and computer) that the solution is unique. There are always 26 steps in a chain, so the "sum" of the 1's and 2's in a pattern will always be 26. For purposes of taxonomizing, the "level" of a string pattern is taken to be the number of 2's occuring in its specification. COUNTS OF SOLVABLE AND UNIQUELY SOLVABLE STRING PATTERNS (recall that Level refers to the number of 2's in the chain spec) Level Solvable Uniquely Patterns Solvable 0 0 0 1 0 0 2 24 0 3 235 15 4 1037 144 5 2563 589 6 3444 1053 7 2674 1078 8 1159 556 9 303 187 10 46 34 11 2 2 12 0 0 13 0 0 _______ ______ Total Patterns: 11487 3658 SOME SAMPLE UNIQUELY SOLVABLE CHAINS In the following the format is: ( #solutions palindrome? #solutions-by-start-type chain-pattern-as string ) where #solutions is the total number of solutions up to reflections and rotations palindrome? is T or NIL according to whether or not the chain is a palindrome #solutions by-start-type lists the 3 separate counts for the number of solutions starting the chain of in the 3 distinct possible ways. chain-pattern-as-string is simply the chain sequence My intuition is that the lower level chains are harder to solve, because there are fewer straight steps, and staight steps are generally more constraining. Another way to view this, is that there are more choices of pivoting for turns because there are more turns in the chains at lower levels. Here are the uniquely solvable chains for level 3: (note that non-palindrome chains only appear once -- I picked a "canonical" ordering) ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; Level 3 ( 3 straight steps) -- 15 uniquely solvable patterns ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; (1 NIL (1 0 0) "21121111112111111111111") (1 NIL (1 0 0) "21121111111111111121111") (1 NIL (1 0 0) "21111112112111111111111") (1 NIL (1 0 0) "21111111211111111111112") (1 NIL (1 0 0) "12121111111111112111111") (1 NIL (1 0 0) "11211211112111111111111") (1 NIL (1 0 0) "11112121111111211111111") (1 NIL (1 0 0) "11112112112111111111111") (1 NIL (1 0 0) "11112112111111211111111") (1 NIL (1 0 0) "11112111121121111111111") (1 NIL (1 0 0) "11112111111211211111111") (1 NIL (1 0 0) "11112111111112121111111") (1 NIL (1 0 0) "11111121122111111111111") (1 NIL (1 0 0) "11111112122111111111111") (1 NIL (1 0 0) "11111111221121111111111") C.2 Magic Interlocking Cube (Glenn A. Iba quoted) This chain problem is marketed as "Magic Interlocking Cube -- the Ultimate Cube Puzzle". It has colored cubies, each cubie having 6 distinctly colored faces (Red, Orange, Yellow, Green, Blue, and White). The object is to fold the chain into a 3x3x3 cube with each face being all one color (like a solved Rubik's cube). The string for the chain is actually a flexible rubber band, and enters a cubie through a (straight) slot that cuts across 3 faces, and exits through another slot that cuts the other 3 faces. Here is a rough attempt to picture a cubie: (the x's mark the slots cut for the rubber band to enter/exit) __________ / /| xxxxxxxxxxx | / / x | /_________/ x | | | x | | | | | | / | x | / | x | / | x |/ -----x----- Laid out flat the cubie faces would look like this: _________ | | | | | x | | x | |____x____|_________ _________ _________ | x | | | | | x | | | | | x | x x x x x x x x x x x | | x | | | | |____x____|_________|_________|_________| | x | | x | | x | | | |_________| The structure of the slots gives 3 choices of entry face, and 3 choices of exit face for each cube. It's complicated to specify the topology and coloring but here goes: Imagine the chain stretched out in a straight line from left to right. Let the rubber band go straight through each cubie, entering and exiting in the "middle" of each slot. It turns out that the cubies are colored so that opposite faces are always colored by the following pairs: Red-Orange Yellow-White Green-Blue So I will specify only the Top, Front, and Left colors of each cubie in the chain. Then I'll specify the slot structure. Color sequences from left to right (colors are R,O,Y,G,B,and W): Top: RRRRRRRRRRRRRRRRRRRRRRRRRRR Front: YYYYYYYYYYYYWWWYYYYYYYYYYYY Left: BBBBBGBBBGGGGGGGGGBBGGGGBBB For the slots, all the full cuts are hidden, so only the "half-slots" appear. Here is the sequence of "half slots" for the Top (Red) faces: (again left to right) Slots: ><><><><<><><<<><><>>>>><>> Where > = slot goes to left < = slot goes to right To be clearer, > (Left): _______ | | | | xxxxx | | | |_______| < (Right): _______ | | | | | xxxxx | | |_______| Knowing one slot of a cubie determines all the other slots. I don't remember whether the solution is unique. In fact I'm not certain whether I actually ever solved it. I think I did, but I don't have a clear recollection. D: Blocks with pins D.1 Holzwurm (Torsten Sillke quoted) Inventer: Dieter Matthes Distribution: Pyramo-Spiele-Puzzle Silvia Heinz Sendbuehl 1 D-8351 Bernried tel: +49-9905-1613 Pieces: 9 tricubes Each piece has one hole (H) which goes through the entire cube. The following puctures show the tricubes from above. The faces where you see a hole are marked with 'H'. If you see a hole at the top then there is a hole at the bottom too. Each peace has a worm (W) one one face. You have to match the holes and the worms. As a worm fills a hole completely, you can not put two worms at both ends of the hole of the same cube. __H__ _____ _____ | | | | | | | | | |W | | |_____|_____ |_____|_____ |_____|_____ | | | | | | | | | | | |W | | |H | H | |W |_____|_____| |_____|_____| |_____|_____| __H__ _____ _____ | | | | | | | | | | | W | |_____|_____ |_____|_____ |_____|_____ | | | | | | | | | | | | | W | H | | | H | |_____|_____| |_____|_____| |_____|_____| W __W__ _____ _____ | | | | | | | | H| |H | | |_____|_____ |_____|_____ |_____|_____ | | | | | | | | | | | H | | | | H| | W | |_____|_____| |_____|_____| |_____|_____| W Aim: build a 3*3*3 cube without a worm looking outside. take note, it is no matching problem, as | | worm> H| |H <worm | | is not allowed. E: Other E.1 Rubik's cube E.2 Magic cube Make a magic cube with the numbers 1 - 27. E.3 ==> geometry/coloring/cheese.cube.p <== A cube of cheese is divided into 27 subcubes. A mouse starts at one corner and eats through every subcube. Can it finish in the middle? ==> geometry/coloring/cheese.cube.s <== Give the subcubes a checkerboard-like coloring so that no two adjacent subcubes have the same color. If the corner subcubes are black, the cube will have 14 black subcubes and 13 white ones. The mouse always alternates colors and so must end in a black subcube. But the center subcube is white, so the mouse can't end there. E.4 Cut the 3*3*3 cube into single cubes. At each slice you can rearrange the blocks. Can you do it with fewer than 6 cuts? ==> competition/games/go-moku.p <== For a game of k in a row on an n x n board, for what values of k and n is there a win? Is (the largest such) k eventually constant or does it increase with n? ==> competition/games/go-moku.s <== Berlekamp, Conway, and Guy's _Winning_Ways_ reports proof that the maximum k is between 4 and 7 inclusive, and it appears to be 5 or 6. They report: . 4-in-a-row is a draw on a 5x5 board (C. Y. Lee), but not on a 4x30 board (C. Lustenberger). . N-in-a-row is shown to be a draw on a NxN board for N>4, using a general pairing technique devised by A. W. Hales and R. I. Jewett. . 9-in-a-row is a draw even on an infinite board, a 1954 result of H. O. Pollak and C. E. Shannon. . More recently, the pseudonymous group T. G. L. Zetters showed that 8-in-a-row is a draw on an infinite board, and have made some progress on showing infinite 7-in-a-row to be a draw. Go-moku is 5-in-a-row played on a 19x19 go board. It is apparently a win for the first player, and so the Japanese have introduced several 'handicaps' for the first player (e.g., he must win with _exactly_ five: 6-in-a-row doesn't count), but apparently the game is still a win for the first player. None of these apparent results have been proven. ==> competition/games/hi-q.p <== What is the quickest solution of the game Hi-Q (also called Solitaire)? For those of you who aren't sure what the game looks like: 32 movable pegs ("+") are arranged on the following board such that only the middle position is empty ("-"). Just to be complete: the board consists of only these 33 positions. 1 2 3 4 5 6 7 1 + + + 2 + + + 3 + + + + + + + 4 + + + - + + + 5 + + + + + + + 6 + + + 7 + + + A piece moves on this board by jumping over one of its immediate neighbor (horizontally or vertically) into an empty space opposite. The peg that was jumped over, is hit and removed from the board. A move can contain multiple hits if you use the same peg to make the hits. You have to end with one peg exactly in the middle position (44). ==> competition/games/hi-q.s <== 1: 46*44 2: 65*45 3: 57*55 4: 54*56 5: 52*54 6: 73*53 7: 43*63 8: 75*73*53 9: 35*55 10: 15*35 11: 23*43*63*65*45*25 12: 37*57*55*53 13: 31*33 14: 34*32 15: 51*31*33 16: 13*15*35 17: 36*34*32*52*54*34 18: 24*44 Found by Ernest Bergholt in 1912 and was proved to be minimal by John Beasley in 1964. References The Ins and Outs of Peg Solitaire John D Beasley Oxford U press, 1985 ISBN 0-19-853203-2 Winning Ways, Vol. 2, Ch. 23 Berlekamp, E.R. Academic Press, 1982 ISBN 01-12-091102-7 ==> competition/games/jeopardy.p <== What are the highest, lowest, and most different scores contestants can achieve during a single game of Jeopardy? ==> competition/games/jeopardy.s <== highest: $283,200.00, lowest: -$29,000.00, biggest difference: $281,600.00 (1) Our theoretical contestant has an itchy trigger finger, and rings in with an answer before either of his/her opponents. (2) The daily doubles (1 in the Jeopardy! round, 2 in the Double Jeopardy! round) all appear under an answer in the $100 or $200 rows. (3) All answers given by our contestant are (will be?) correct. Therefore: Round 1 (Jeopardy!): Max. score per category: $1500. For 6 categories - $100 for the DD, that's $8900. Our hero bets the farm and wins - score: $17,800. Round 2 (Double Jeopardy!): Max. score per category: $3000. Assume that the DDs are found last, in order. For 6 categories - $400 for both DDs, that's $17,600. Added to his/her winnings in Round 1, that's $35,400. After the 1st DD, where the whole thing is wagered, the contestant's score is $70,800. Then the whole amount is wagered again, yielding a total of $141,600. Round 3 (Final Jeopardy!): Our (very greedy! :) hero now bets the whole thing, to see just how much s/he can actually win. Assuming that his/her answer is right, the final amount would be $283,200. But the contestant can only take home $100,000; the rest is donated to charity. To calculate the lowest possible socre: -1500 x 6 = -9000 + 100 = -8900. On the Daily Double that appears in the 100 slot, you bet the maximum allowed, 500, and lose. So after the first round, you are at -9400. -3000 x 6 = -18000 + 400 = -17600 On the two Daily Doubles in the 200 slots, bet the maximum allowed, 1000. So after the second round you are at -9400 + -19600 = -29000. This is the lowest score you can achieve in Jeopardy before the Final Jeopardy round. The caveat here is that you *must* be the person sitting in the left-most seat (either a returning champion or the luckiest of the three people who come in after a five-time champion "retires") at the beginning of the game, because otherwise you will not have control of the board when the first Daily Double comes along. The scenario for the maximum difference is the same as the highest score, except that on every question that isn't a daily double, the worst contestant rings in ahead of the best one, and makes a wrong guess, after which the best contestant rings in and gets it right. However, since contestants with negative scores are disqualified before Final Jeopardy!, it is arguable that the negative score ceases to exist at that point. This also applies to zero scores. In that case, someone else would have to qualify for Final Jeopardy! for the maximum difference to exist, taking one $100 or $200 question away from the best player. In that case the best player would score 8*$200 lower, so the maximum difference would be $281,600.00. ==> competition/games/nim.p <== Place 10 piles of 10 $1 bills in a row. A valid move is to reduce the last i>0 piles by the same amount j>0 for some i and j; a pile reduced to nothing is considered to have been removed. The loser is the player who picks up the last dollar, and they must forfeit half of what they picked up to the winner. 1) Who is the winner in Waldo Nim, the first or the second player? 2) How much more money than the loser can the winner obtain with best play on both parties? ==> competition/games/nim.s <== For the particular game described we only need to consider positions for which the following condition holds for each pile: (number of bills in pile k) + k >= (number of piles) + 1 A GOOD position is defined as one in which this condition holds, with equality applying only to one pile P, and all piles following P having the same number of bills as P. ( So the initial position is GOOD, the special pile being the first. ) I now claim that if I leave you a GOOD position, and you make any move, I can move back to a GOOD position. Suppose there are n piles and the special pile is numbered (n-p+1) (so that the last p piles each contain p bills). (1) You take p bills from p or more piles; (a) If p = n, you have just taken the last bill and lost. (b) Otherwise I reduce pile (n-p) (which is now the last) to 1 bill. (2) You take p bills from r(<p) piles; I take r bills from (p-r) piles. (3) You take q(<p) bills from p or more piles; I take (p-q) bills from q piles. (4) You take q(<p) bills from r(<p) piles; (a) q+r>p; I take (p-q) bills from (q+r-p) piles (b) q+r<=p; I take (p-q) bills from (q+r) piles Verifying that each of the resulting positions is GOOD is tedious but straightforward. It is left as an exercise for the reader. -- RobH ==> competition/games/online/online.scrabble.p <== How can I play Scrabble online on the Internet? ==> competition/games/online/online.scrabble.s <== Announcing ScrabbleMOO, a server running at 134.53.14.110, port 7777 (nextsrv.cas.muohio.edu 7777). The server software is version 1.7.0 of the LambdaMOO server code. To reach it, you can use "telnet 134.53.14.110 7777", and sign on. You will have a unique name and password on the server, and directions are provided in the opening screen on how to accomplish signing on. The first time, you will need to type "create YourName YourPassword", and each time thereafter, "connect YourName YourPassword". There are currently 5 Scrabble boards set up, with global individual high score and game-cumulative high score lists. Games can be saved, and restored at a later time. There are complete command instructions at each board (via the command "instructions"); usage is simple and intuitive. There are commands to undo turns, exchange tiles, and pass, and there are a variety of options available to change the way the board and rack are displayed. We do not yet have a dictionary for challenges installed on-line, and that is coming very soon. I am seriously contemplating using the OSPD.shar wordlist that Ross Beresford listed in a recent Usenet article. It seems to have the full wordlist from the 1st edition of the OSPD, plus longer words from some other source. I have personal wordlists updating the OSPD to the 2nd edition, for words up to 4 letters long, and will have the longer words in the near future. Usage of a certain dictionary for challenges is not enforced, and really can't be. Many of the regular players there have their personal copy of the OSPD. It's all informal, and for fun. Players agree what dictionary to use on a game-by-game basis, though the OSPD is encouraged. There are even commands to enable kibitzing, if watching rather than playing is what you're into. Come by and try it out. We have all skill levels of players, and we welcome more! ==> competition/games/online/unlimited.adventures.p <== Where can I find information about unlimited adventures? ==> competition/games/online/unlimited.adventures.s <== ccosun.caltech.edu -- pub/adnd/inbound/UA wuarchive.wustl.edu -- pub/msdos_uploads/games/UA ==> competition/games/othello.p <== How good are computers at Othello? ==> competition/games/othello.s <== ("Othello" is a registered trademark of the Anjar Company Inc.) As of 1992, the best Othello programs may have reached or surpassed the best human players [2][3]. As early as 1980 Jonathon Cerf, then World Othello Champion, remarked: "In my opinion the top programs [...] are now equal (if not superior) to the best human players." [1] However, Othello's game theoretic value, unlike checkers, will likely remain unknown for quite some time. Barring some unforeseen shortcut or bankroll, a perfect Othello playing program would need to search in the neighborhood of 50 plies. Today, even a general 30 ply search to end the game, i.e. 30 remaining empty squares, is beyond reach. Furthermore, the game of Othello does not lend itself to endgame database techniques that have proven so effective in checkers, and in certain chess endgames. Progress of the best Othello computer programs: 1980 "Iago" (by Rosenbloom) [2] 1990 "Bill 3.0" (by Lee and Mahajan) [3] uses: 1. sophisticated searching and timing algorithms, e.g. iterative deepening, hash/killer tables, zero-window search. 2. lookup tables to encode positional evaluation knowledge. 3. Bayesian learning for the evaluation function. The average middle game search depth is 8 plies. Exhaustive endgame search within tournament-play time constraints, is usually possible with 13 to 15 empty squares remaining. "Bill 3.0" defeated Brian Rose, the highest rated American Othello player, by a score of 56-8. 1992 At the 4th AST Computer Olympiad [4][5], the top three programs were: Othel du Nord (France) Aida (The Netherlands) Jacp'Oth (France) References ---------- [1] Othello Quarterly 3(1) (1981) 12-16 [2] P.S. Rosenbloom, A World Championship-Level Othello Program, "Artificial Intelligence" 19 (1982) 279-320 [3] Kai-Fu Lee and Sanjoy Mahajan, The Development of a World Class Othello Program, "Artificial Intelligence" 43 (1990) 21-36 [4] D.M. Breuker and J. Gnodde, The AST 4th Computer Olympiad, "International Computer Chess Association Journal 15-3 (1992) 152-153 [5] Jos Uiterwijk, The AST 4th Conference on Computer Games, "International Computer Chess Association Journal 15-3 (1992) 158-161 Myron P. Souris EDS/Unigraphics St. Louis, Missouri souris@ug.eds.com ==> competition/games/pc/best.p <== What are the best PC games? ==> competition/games/pc/best.s <== Read "net pc games top 100" in newsgroup comp.sys.ibm.pc.games.announce. ==> competition/games/pc/reviews.p <== Are reviews of PC games available online? ==> competition/games/pc/reviews.s <== Presenting... the Game Bytes Issues Index! (Issues 1-8) Game Bytes has covered well over 100 games in the past several issues. Using this index, you can look up the particular games you're interested in, find out what issues of Game Bytes cover them, and download those issues. Also included is a list of the interviews GB has done to date - - the interviews from several issues ago still contain a lot of current material. The easiest way to use the games index is to employ the search command of your favorite word processor to find a distinctive string, such as "Ultima","Perfect", or "Lemmings". The list is alphabetized; series have been listed together rather than by individual subtitle. All issues of Game Bytes to date are available by anonymous FTP at ftp.ulowell.edu in the /msdos/Games/GameByte directory and are mirrored on other FTP sites as well. Contact Ross Erickson, ross@kaos.b11.ingr.com, if you need assistance acquiring Game Bytes or have other questions. Game Bytes Interview List, Issues 1 - 7, Chronological Order ----------------------------------------------------------------- Issue Person(s) Company Sample Games ----- --------- ------- ------------ 2 Richard Garriott Origin Ultima series 3 Chris Roberts Origin Wing Commander, Strike C. 4 ID Software team Apogee* Wolfenstein 3D, Commander Keen 5 Damon Slye Dynamix Red Baron, Aces of the Pacific 5 Scott Miller Apogee Wolf3D, C. Keen, Duke Nukem 6 Bob Bates (Part 1) Legend Spellcasting 101 7 Bob Bates (Part 2) "" "" 8 Looking Glass Tech Origin Underworld 1 and 2 * distributing/producing company Game Bytes Reviews Index, Issues 1 - 8, Alphabetical by Title --------------------------------------------------------------------- Title Review Preview Tips ----- ------ ------- ---- A-Train 3 A.T.A.C. 5 Aces of the Pacific 3 1 8 Action Stations! 8 Air Combat 5 Air Force Commander 8 Alien 3 (Sega Genesis) 7 Amazon 8 6 Axelay (Super Nintendo) 8 B-17 Flying Fortress 6 4 B.A.T. II: The Koshan Conspiracy 7 Battlecruiser 3000 A.D. 8 Birds of Prey 7 4 Carrier Strike 6 Carriers at War 6 Castle Wolfenstein 3-D 2 Challenge of the Five Realms 4 Chessmaster 3000 2 Civilization 5 Comanche: Maximum Overkill 6 Conflict: Korea 6 Conquered Kingdoms 7 Conquests of the Longbow 3 Contra 3: The Alien Wars (Super Nintendo) 5 Crisis in the Kremlin 6 D/Generation 2 Dark Sun: Shattered Lands 6 Darklands 7 3 7 Darkseed 5 Dune 3 Dungeon Master 7 Dynamix Football 3 Earl Weaver Baseball 2 4 Ecoquest: The Search for Cetus 2 5 Eric the Unready 8 Eye of the Beholder 2 1 Eye of the Beholder 3 8 F-117A Stealth Fighter 3 F-15 Strike Eagle III 5 Falcon 3.0 1 5,8 Falcon 3.0: Operation Flying Tiger 6 Flight Simulator 4.0 Scenery 8 Front Page Sports: Football 8 6 Galactix 6 Gateway 4 Global Conquest 3 Gods 6 Gravis Gamepad 4 Great Naval Battles 8 Greens! 2 Gunship 2000 2 Hardball 3 4,5 Hardball 3 Statistical Utilities 7 Harpoon 1.3 Designer Series / IOPG 6 Heaven and Earth 4 Heimdall 7 Hong Kong Mahjong 3 Indiana Jones and the Fate of Atlantis 5 Jack Nicklaus Golf: Signature Edition 2 Joe and Mac (SNES) 2 Johnny Castaway 8 King's Quest VI: Heir Today, Gone Tomorrow 6 Laura Bow 2: The Dagger of Amon Ra 4 3 Legends of Valor 8 Les Manley: Lost in L.A. 1 Links 386 Pro 5 1 Links Courses: Troon North 2 Loom -- CD-ROM version 5 Lord of the Rings II: The Two Towers 7 3 Lost Treasures of Infocom 5 Lure of the Temptress 8 Mantis: XF5700 Experimental Space Fighter 7 4 Martian Memorandum 5 Micro League Baseball 4 6 Might and Magic: Clouds of Xeen 8 Mike Ditka's Ultimate Football 6 Monkey Island 2: LeChuck's Revenge 5 NCAA Basketball (Super Nintendo) 8 NCAA: The Road to the Final Four 3 NFL Pro League 7 NHLPA Hockey '93 (Sega Genesis) 7 Nova 9 2 Oh No! More Lemmings 3 Out of This World 6 Pirates! Gold 2 Planet's Edge 3 Pools of Darkness 2 Powermonger 5 Prince of Persia 4 Prophecy of the Shadow 7 Pursue the Pennant 4.0 4 Quest for Glory I (VGA edition) 7 Quest for Glory III: The Wages of War 7 Rampart 4 Rampart (SNES) 7 RBI Baseball 4 (Sega Genesis) 7 Red Baron Mission Builder 8 4 Rex Nebular and the Cosmic Gender Bender 8 5 Risk for Windows 1 Robosport for Windows 8 Rules of Engagement 7 Secret Weapons of the Luftwaffe 4 Sega CD-ROM (Sega Genesis) 8 Sherlock Holmes, Consulting Detective Vol.I 7 Shining in the Darkness (Sega Genesis) 4 Siege 6 SimAnt 4 Solitaire's Journey 5 Sonic the Hedgehog 2 8 Space Megaforce (SNES) 7 Space Quest V: The Next Mutation 3 Speedball 2 5 Spellcasting 301: Spring Break 8 8 Spellcraft: Aspects of Valor 3 Splatterhouse 2 (Sega Genesis) 5 S.S.I. Goldbox summary 8 Star Control 2 8 Star Legions 6 Star Trek: 25th Anniversary 1 Street Fighter 2 8 Strike Commander 3 Stunt Island 8 7 Summer Challenge 8 5 Super Hi-Impact Football (Sega Genesis) 8 Super Star Wars (SNES) 7 Super Tetris 3 Take-a-Break Pinball 6 Tegel's Mercenaries 6 Terminator 2029: Cybergen 5 The 7th Guest 5 The Castle of Dr. Brain 5 The Incredible Machine 7 The Legend of Kyrandia 7 The Lost Admiral 6 The Magic Candle II: The Four and Forty 5 The Miracle 3 The Mystical Quest (SNES) 7 The Perfect General 3 Theatre of War 6 Thrustmaster 4 Thunderhawk 2 TimeQuest 2 Tony La Russa's Ultimate Baseball II 8 Turbo Science 7 Ultima 1, 2, and 3 (First Trilogy) 7 Ultima 7: Forge of Virtue 6 4 Ultima 7: The Black Gate 3 1 5,6 Ultima Underworld: The Stygian Abyss 3 7 Ultima Underworld 2: Labyrinth of Worlds 8 V for Victory: Utah Beach 7 Veil of Darkness 8 WaxWorks 7 Wayne Gretzky Hockey III 5 Wing Commander 2 1 Wing Commander 2: Special Operations 2 4 Winter Challenge 5 Wizardry 6: Bane of the Cosmic Forge 1 Wizardry 7: Crusaders of the Dark Savant 8 5 Wordtris 4 World Circuit 7 X-Wing: Star Wars Space Combat Simulator 7 ==> competition/games/pc/solutions.p <== What are the solutions to various popular PC games? ==> competition/games/pc/solutions.s <== Solutions, hints, etc. for many games exist at: pub/game_solutions directory on sun0.urz.uni-heidelberg.de pub/games/solutions directory on risc.ua.edu (130.160.4.7) pub/msdos/romulus directory on ftp.uwp.edu (131.210.1.4) ==> competition/games/poker.face.up.p <== In Face-Up Poker, two players each select five cards from a face-up deck, bet, discard and draw. Is there a winning strategy for this game? What if the players select cards alternately? ==> competition/games/poker.face.up.s <== If the first player draws four aces, the second player draws four kings. If the first player keeps the four aces on the draw, the second player draws a king-high straight flush, and if the first player pitches the aces to draw a straight flush, the second player can always make a higher straight flush. Instead, the winning strategy is for the first player to draw four tens. The second player cannot draw a royal flush, and in order to prevent the first player from getting one, the second player must draw at least one card higher than the ten from each suit, which means he can't do better than four-of-a-kind. Then the first player wins by drawing a straight flush from any suit. If the cards are dealt alternately as in real poker, the second player can always tie with proper strategy. The second player mirrors the first player's selections in rank and color. For example, if the first player picks up a red queen, the second player picks up a red queen. When they are done playing, their hands will be identical except one will have spades and hearts where the other has clubs and diamonds, and vice versa. Since suits aren't ranked in poker, the hands are tied. It is unknown if there is a winning strategy if the replacement cards are dealt together as in real poker, as opposed to alternately. ==> competition/games/risk.p <== What are the odds when tossing dice in Risk? ==> competition/games/risk.s <== Odds calculated with program by David Karr (karr@cs.cornell.edu): Attacker rolls 3 dice, defender rolls 2 dice: Attacker Defender Probability loses loses 0 2 2890/7776 = 0.3716563786 1 1 2611/7776 = 0.3357767490 2 0 2275/7776 = 0.2925668724 Attacker rolls 3 dice, defender rolls 1 dice: Attacker Defender Probability loses loses 0 1 855/1296 = 0.6597222222 1 0 441/1296 = 0.3402777778 Attacker rolls 2 dice, defender rolls 2 dice: Attacker Defender Probability loses loses 0 2 295/1296 = 0.2276234568 1 1 420/1296 = 0.3240740741 2 0 581/1296 = 0.4483024691 Attacker rolls 2 dice, defender rolls 1 dice: Attacker Defender Probability loses loses 0 1 125/216 = 0.5787037037 1 0 91/216 = 0.4212962963 Attacker rolls 1 dice, defender rolls 2 dice: Attacker Defender Probability loses loses 0 1 55/216 = 0.2546296296 1 0 161/216 = 0.7453703704 Attacker rolls 1 dice, defender rolls 1 dice: Attacker Defender Probability loses loses 0 1 15/36 = 0.4166666667 1 0 21/36 = 0.5833333333 ---------------------8<------snip here--------8<-------------------- /* * riskdice.c -- prints Risk dice odds * * This program calculates probabilities for one roll of the dice in Risk. * For each possible number of dice that the attacker might roll, for each * possible number of dice that the defender might roll, this program * lists all the possible outcomes (number of armies lost by attacker * and defender) and the probability of each outcome. * * Copyright 1993 by David A. Karr. */ #define MAX_ATTACK 3 /* max # of dice attacker may roll */ #define MAX_DEFEND 2 /* max # of dice defender may roll */ #define MAX_DICE MAX_ATTACK + MAX_DEFEND void main() { int a; /* number of dice rolled by attacker */ int d; /* number of dice rolled by defender */ void calc(); for (a = MAX_ATTACK; a > 0; --a) { for (d = MAX_DEFEND; d > 0; --d) { calc( a, d ); } } } void calc( a_dice, d_dice ) /* * Purpose: Print odds for the given numbers of dice rolled */ int a_dice; /* number of dice rolled by attacker */ int d_dice; /* number of dice rolled by defender */ { int num_dice; /* total number of dice rolled */ int num_armies; /* # armies that will be lost by both sides, total */ int kill_count[MAX_DEFEND + 1]; /* entry [i] counts # of times attacker loses i armies */ int roll[MAX_DICE + 1]; /* holds one roll of the dice */ int a_roll[MAX_ATTACK]; /* holds attacker's dice */ int d_roll[MAX_DEFEND]; /* holds defender's dice */ int n; /* cursor into the arrays */ int num_lost; /* # of armies lost by the attacker */ int cases; /* total # of events counted */ void dsort(); /* * The method is pure brute force. roll[] is set successively to * all possible rolls of the total number of dice; for each roll * the number of armies lost by the attacker (the outcome) is * computed and the event is counted. * Since all the counted events are equiprobable, the count of each * outcome merely needs to be scaled down by the total count to * obtain the probability of that outcome. */ /* The number of armies at stake is min(a_dice, d_dice) */ num_armies = a_dice < d_dice ? a_dice : d_dice; /* initialize event counters */ for (n = 0; n <= num_armies; ++n) kill_count[n] = 0; /* * The roll[] array is treated as a funny odometer whose wheels each * go from 1 to 6. Each roll of the dice appears in roll[0] through * roll[num_dice - 1], starting with all 1s and counting up to all 6s. * roll[num_dice] is used to detect when the other digits have * finished a complete cycle (it is tripped when they go back to 1s). */ num_dice = a_dice + d_dice; for (n = 0; n <= num_dice; ++n) roll[n] = 1; while (roll[num_dice] == 1) { /* examine a new possible roll of the dice */ /* * copy attacker's and defender's dice so as not to disturb * the "odometer" reading. */ for (n = 0; n < a_dice; ++n) a_roll[n] = roll[n]; for (n = 0; n < d_dice; ++n) d_roll[n] = roll[n + a_dice]; /* * sort attacker's and defender's dice, highest first. */ dsort(a_roll, a_dice); dsort(d_roll, d_dice); /* * compare attacker's and defender's dice, count attacker's loss */ num_lost = 0; for (n = 0; n < num_armies; ++n) if (d_roll[n] >= a_roll[n]) ++num_lost; ++kill_count[num_lost]; /* * Find next roll values (bump the "odometer" up one tick). */ n = 0; roll[0] += 1; while (roll[n] > 6) { /* place [n] rolled over */ roll[n] = 1; /* Carry 1 into the next place (which may in turn roll over) */ ++n; roll[n] += 1; } } cases = 0; for (n = 0; n <= num_armies; ++n) cases += kill_count[n]; printf( "Attacker rolls %d dice, defender rolls %d dice:\n\n", a_dice, d_dice ); printf( "Attacker Defender Probability\n" ); printf( " loses loses\n" ); for (n = 0; n <= num_armies; ++n) printf( "%5d %5d %5d/%d = %12.10lf\n", n, num_armies - n, kill_count[n], cases, ((double) kill_count[n]) / ((double) cases) ); printf( "\n\n" ); } void dsort( array, length ) /* * Sort the given array in descending order. */ int *array; int length; /* number of slots in the array */ { int level, n, tmp; /* Use bubble sort since the array will be tiny in this application */ for (level = 0; level < length - 1; ++level) { /* * Slots [0] through [level - 1] are already "stable." * Bubble up the value that belongs in the [level] slot. */ for (n = length - 1; n > level; --n) { if (array[n - 1] < array[n]) { /* swap them */ tmp = array[n - 1]; array[n - 1] = array[n]; array[n] = tmp; } } } } ==> competition/games/rubiks/rubiks.clock.p <== How do you quickly solve Rubik's clock? ==> competition/games/rubiks/rubiks.clock.s <== Solution to Rubik's Clock The solution to Rubik's Clock is very simple and the clock can be "worked" in 10-20 seconds once the solution is known. In this description of how to solve the clock I will describe the different clocks as if they were on a map (e.g. N,NE,E,SE,S,SW,W,NW); this leaves the middle clock which I will just call M. To work the Rubik's clock choose one side to start from; it does not matter from which side you start. Your initial goal will be to align the N,S,E,W and M clocks. Use the following algorithm to do this: [1] Start with all buttons in the OUT position. [2] Choose a N,S,E,W clock that does not already have the same time as M (i.e. not aligned with M). [3] Push in the closest two buttons to the clock you chose in [2]. [4] Using the knobs that are farest away from the clock you chose in [2] rotate the knob until M and the clock you chose are aligned. The time on the clocks at this point does not matter. [5] Go back to [1] until N,S,E,W and M are in alignment. [6] At this point N,S,E,W and M should all have the same time. Make sure all buttons are out and rotate any knob until N,S,E,W and M are pointing to 12 oclock. Now turn the puzzle over and repeat steps [1]-[6] for this side. DO NOT turn any knobs other than the ones described in [1]-[6]. If you have done this correctly then on both sides of the puzzle N,S,E,W and M will all be pointing to 12. Now to align NE,SE,SW,NW. To finish the puzzle you only need to work from one side. Choose a side and use the following algorithm to align the corners: [1] Start with all buttons OUT on the side you're working from. [2] Choose a corner that is not aligned. [3] Press the button closest to that corner in. [4] Using any knob except for that corner's knob rotate all the clocks until they are in line with the corner clock. (Here "all the clocks" means N,S,E,W,M and any other clock that you have already aligned) There is no need at this point to return the clocks to 12 although if it is less confusing you can. Remember to return all buttons to their up position before you do so. [5] Return to [1] until all clocks are aligned. [6] With all buttons up rotate all the clocks to 12. 
