Movie Magic Squares: Volume 1
Minimum Requirements: Adobe Reader 9. A single copy of this curriculum may be used by one teacher in his or her own classroom. Please purchase one copy for each teacher. More info. Help your students discover the power of pattern recognition in problem solving. Problems are drawn from athletics, board games, and magic squares, as well as from science, geometry, probability, and number theory.
Adaptable to a wide range of ability and experience levels. Includes solutions, suggestions for use, and optional challenges for advanced students. The free sample below contains the table of contents, and a free activity from this book. Note: All files are pdf documents requiring Adobe Reader. Contact Us: 1. Welcome New User!
Discounts expires at end of day, November 1, ! Downloads Don't Download! A Pattern Discovery Approach to Problem Each is a modern adaptation based on an old puzzle design. I purchased some from In and Out Gifts which seems defunct. Each puzzle consists of a set of cards with various design fragments and cutouts. Stack the cards to achieve specific patterns, such as uniform color front and back, or unbroken paths of given colors from point to point.
Transposer 6 Bonbons Genesis Kaboozle Tiffany Tower of London Struzzle Along the same lines as the Toyo Glass puzzles, combined with the weave concept, Strip Tease requires you to create a 5x5 weave using 10 clear strips having various arrangements of quarter-squares so that all solid squares result. Trixxy designed by Dror Green - superpose four cards having transparent and opaque colored sections in order to produce a solid column of each of the four colors. This is the "2D 3D Burr" - stack the transparencies so that the image of a 3-piece burr appears. Cover Your Tracks - Thinkfun Four pieces and a set of challenge cards - for each card, pack the four pieces into the tray so that the bootprints on the card are all covered.
Four transparent overlays, each with a 6-unit enclosure. On each problem background, fence off unlike things. Frustables by Gameophiles Unlimited Six sets of six cards.
Each set of six cards has one colored puzzle on the front and another color on the back - twelve puzzles in all. A card has a given color, and may have some black area.
The objective given the six cards belonging to a given puzzle is to pile the cards such that all the black areas are covered but no colored area is covered. These are difficult puzzles! Ariel Laden's Kookoo Puzzles - Funny Fliers Four six-card puzzles, and one large puzzle using the backs of all 24 cards. Position and interleave the cards of a set to form a complete picture.
This principle is very similar to that of the puzzle Frustables by Gameophiles Unlimited. Four well done puzzle challenges and two expansion packs from Smart Games. I admit I had trouble with the very first problem! Six square cards, each almost cut into quarters and each bearing a portion of four different images. Interlock and overlap the cards to create each image, in turn. Make a large hexagon from 18 transparent trapezoidal tiles, while matching edge colors. A refined version of Pavel's exchange puzzle from IPP Nine transparent pieces, and a frame in the shape of a magnifying glass.
Arrange the pieces in three layers inside the frame. Find two distinct overall arrangements to determine "two things given to Snow White. Pathwords invented by Derrick Niederman For each of 40 challenges of graduated difficulty, fit a subset of the supplied transparent pieces to completely cover the field of letters such that each piece covers one word that runs backwards or forwards.
Each tile has either black segments, cutout segments through which whatever is behind the tile, including the black base will show, or blank area. On the Line - issued by Brainwright A set of graduated overlay-pattern-assembly challenges. Arrange four identical transparencies to form a given shape.
Star Shuffle 2 Stack the four disks so that a star appears in all twelve holes. Each dot is one of seven colors. Form a 7x7 mat by joining strips back-to-back cross-wise, such that where two strips cross the dots on both sides are the same color. From the patent description - the Weave-O-Gram "comprises a framework and a plurality" patent lawyers love that term "of flexible bands some extending in one direction and the others at right angles thereto, the various bands being interwoven On each band there is provided a number of sections of a picture Each band will be provided with the sections of a number of different pictures so that a number of different complete pictures may be made up.
The bands move pretty freely, occasionally catching on small rough edges or tears. It is best to use both hands to move a band, slowly, from both edges of the frame simultaneously. As you might expect, Weave-O-Gram is not too challenging, however the graphics are nice and this is a great implementation of one of those ideas that seems obvious once you've seen it done.
Perry's device employs linear strips rather than looped bands. It is pretty much a copy of Shamah's idea - Frankl's improvements are to "effect economies of manufacture and assembly and to provide for ease of manipulation and attractiveness. Paul MN. I learned that the company Century as of March purchased by Allcraft Marine makes powerboats, and was founded in Milwaukee in In , F.
Since boat production orders were seasonal - heavy from March through July and slack during the rest of the year, one of Hewitt's initiatives was to offset the downtimes by producing other, stable products such as toys. Century produced a line of educational toys under the brand Edcraft , including Weave-O-Gram.
Shamah also patented in the UK a locking mechanism for a money box. This problem first appeared in the Berliner Schachzeitung , where only two solutions were provided c. Singmaster Source2. Bezzel was born in in Herrnberchtheim, one of six brothers and three sisters; he was a math teacher and lawyer, and died young at age 47 in , probably of cancer. The full solution to the problem did not appear until published in the Leipziger Illustrierte Zeitung 15 No.
A proof that there are only 12 unique solutions was published in the Philosophical Magazine by English mathematician and renowned pottery collector James Whitbread Lee Glaisher , pictured at right. The text of PM is available online. The problem is generalizable to the N Queens Problem , using n queens on an nxn chessboard.
There is no known formula for computing the number of solutions given n. You can see a chart of findings for various n up to 26 at www. Solution counts for the first few n are given below N 1 2 3 4 5 6 7 8 9 10 11 12 Unique 1 0 0 1 2 1 6 12 46 92 Total 1 0 0 2 10 4 40 92 The problem is discussed by Edouard Lucas , inventor of the Towers of Hanoi puzzle, in his book L'Arithmetique Amusante available online , in which he gives the nice table of the 12 basic solutions shown below. Over the years there have been many instantiations of this problem posed as mechanical puzzles.
Arrange the four squares such that in the resulting 8x8 grid, no two holes appear in the same row, column, or diagonal. The tiles may be flipped over and rotated. I have had this puzzle for a long time and it remains one of my favorites despite its simplicity.
If you examine the 12 unique solutions to the 8 Queens puzzle and divide each into quadrants, you'll find that there are only six types of quadrants that each contain two queens. In the diagram at right I have labeled each quadrant type, regardless of rotation or reflection, assigning a letter A through F. Two of the solutions, labeled VI and VIII by Lucas, have a single queen in two quadrants and three queens in the other two - I haven't labeled those quadrant types since they cannot be used in the Brain Drain puzzle.
This table summarizes the use of the six quadrants in each of the relevant 10 solutions. Only one set - those from solution V, has no duplicate of a quadrant piece type, and has a unique solution.
Are you sure?
This is the Brain Drain set! See X. See IV. The Frustr8tor From my friends at Puzzlemaster. The front side shows an 8x8 grid. The back side has 8 tracks corresponding to the 8 columns of the grid. Along each track, each of the 8 row positions is marked by a number from 1 to 28 - some appear three times, some two. One red and one green tab ride in each track - green at the top and red below.
To try a puzzle, choose a number and set a red tab at every position marked by that number. Then, using the green tabs in the remaining six columns, fill in the grid according to the usual rules - a dot appears in the grid on the front at the position where a tab is set. This vintage French boxed version called " Jeu des Manifestants " uses 8 tiles, and is available in two versions with either triangular or rectangular tiles. For another version using battleships, see U. Patent - Reibstein Lots O Spots by Peterson - "an L.
Gordon creation. When the tile are arranged in a 4x4 grid, the quadrants define 8 rows and 8 columns. The spots come in three colors - red, purple, and green - and are distributed so that there are 8 of each color. In the hardest of four challenges, you must arrange the tiles in a 4x4 grid such that all rows, columns, main diagonals, and all short diagonals contain no more than one spot of each color. This amounts to solving the 8 queens problem simultaneously for 3 colors of queens.
The Schpotz puzzle by Peterson Games. Arrange the nine tiles in a 3x3 grid such that every row, column, and main diagonal contains exactly three spots. The objective is to arrange the eight pegs in the 8x8 grid so that no more than one peg is in any row, column, or diagonal.
In addition, at least one peg must be in each of the five differently shaded areas - one of the areas is a single position at the lower right corner. The cover shows a motorcycle gang - I think they're supposed to be "out of line. An 8x8 board, with inner nested 4x4 and 6x6 areas marked off. There seem to have been at least two different sets of instructions issued with this puzzle. The first two challenges cited on the first instructions, on the 4x4 and 6x6 grids, are easy since you can use more than 4 and 6 colors respectively.
However, the 8x8 challenge amounts to superimposing 8 solutions of the 8 queens problem, and is impossible! This puzzle is interesting because it requires superposition , and also because it seems to be one of those rare instances where prize money is offered for an impossible challenge. Think about the two main diagonals - all 16 spaces must be filled, but some reflection should convince you that each of the 8 colors must contribute exactly two pegs somewhere on the main diagonals - one on each.
If a solution failed to contribute, that would mean two spaces on one diagonal would have to be filled by another color - which would violate the rules. That means that the only usable solutions from the 12 are the six that have a queen on each of the main diagonals. But , each of those six solutions also has a queen on one of the penultimate corner squares. Since every possible rotation and reflection of any of those six solutions will also have a queen on a penultimate corner square, and there are only four such squares to go around, we cannot superimpose more than four solutions before we have a conflict!
I did more research into this puzzle and found that the superposition problem has been discussed by Martin Gardner in his book The Unexpected Hanging and Other Mathematical Diversions. Martin writes that " When the order of the board is not divisible by 2 or 3, it is possible to superimpose n solutions that completely fill all the cells.
Thus on the 5x5 one can place 25 queens - 5 each of 5 colors - such that no queen attacks another of the same color. In the Chapter 16 addendum, however, a reader points out a reference to a proof of the impossibility of the 8 color superposition, given by Thorold Gosset in the Messenger of Mathematics Vol. The MoM is available online. Gosset's proof and mine are similar, but I think mine is more elegant :- A solution for 6 superpositions on the 8x8 is given by Lucas in his book.
Of note is that only four of the superimposed six require a penultimate square - the other two do not - but not all main diagonals are filled. Now, can you find the superposition for 5 colors on a 5x5? This puzzle is called Orchard and is offered by the Australian company Dr.
Wood in their Mind Challenge series - it is discontinued but I found a mint copy. It poses another interesting superposition problem. You are given a board having an 8x8 grid of sockets, with a 2x2 house obstruction along the center of one side. You are also given 40 trees, 10 of each of 4 types. You are to plant the trees in the grid such that each set of 10 forms 5 rows with 4 in each row. This type of problem is discussed by Prof.
David Singmaster in his Sources in Recreational Mathematics in section 6. AO, Configuration Problems. Singmaster defines a notation to describe the various related flavors of this type of problem: a,b,c meaning arrange a points in b rows of c each. Singmaster does not point out the earliest appearance of this puzzle, but cites many examples in the puzzle literature. The Orchard puzzle calls for a superposition on an 8x8 grid, with the obstruction, of 4x 10,5,4.
Singmaster cites Dudeney's book Amusements in Mathematics available online for an example of a 10,5,4 problem, and notes that Dudeney describes six basic solutions. Dudeney states that there are only six fundamental solutions to the problem of arranging 10 points in 5 rows of 4 each though each of the six patterns can be infinitely distorted depending on the overall underlying grid size , and he shows diagrams with names. Only the Dart or the Funnel could be used on the 8x8 grid of the Orchard puzzle. When four copies are superimposed, it turns out the points conflict with the house obstruction, so the Funnel shape cannot be used in the solution to the Orchard puzzle.
The Dart shape can be stretched and moved around in the three configurations shown. The a and b versions also ultimately cause a conflict with the house. This is the solution to the Dr. Wood Orchard puzzle! The Pin and Dot puzzle - "Insert six pins, each in a separate dot, so that no two pins shall be on the same line. I don't have this - shown for reference. The six-queens puzzle has only one unique solution. Since that solution is degree symmetric, it has only four rotations and reflections.
I don't have these - shown for reference. Jeu des Sentinelles "A police chief represented by the red piece, with 7 sentinels represented by the white pieces, must position himself and his sentinels so that no man can see any of the other men along any straight vertical, horizontal, or diagonal line. L'Intraitable - "The Intractable" Given a 6x6 grid and 24 tokens - six each of four colors, arrange the tokens on the grid so that no horizontal, vertical, or diagonal line contains more than one token of the same color.
This vintage French boxed puzzle is an instance of a superposition of four six-queens puzzles. Both of these vintage French boxed puzzles are instances of, using Singmaster's notation, a 12,6,4 configuration problem. There is a second solution see The Sociable , or Hoffmann , but it does not fit on the 5x7 grid. Note that some a,b,c configuration solutions rely on the trick of stacking more than one token at a given point.
The Five-Queens puzzle has only two unique solutions. The asymmetric solution has eight rotations and reflections. The symmetric solution has only two, giving a total of A selection of five can be superimposed. Thanks very much, Dave! Arrange the five layers so that the five numbers appearing in each of the 16 radial columns total With the base plate and four additional movable disks, there are 65, combinations to try! This "SafeCracker" type of puzzle was the subject of U.
Davidson in But, see below for versions with earlier dates The Davidson patent seems to have been applied to the " Great Burglar Puzzle " which requires columns to total I obtained the example shown here. You can see the reference to the patent near the center. Here is an image of another copy showing an envelope the puzzle evidently came in. I don't have this. Bobrick, New York. Unfortunately I could not find a patent pertaining to the puzzle.
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While the USPTO website does allow searching for applications as opposed to granted patents , the database only contains records from onwards. The USPTO allows one to search for granted patents from through but only by issue date, patent number, and current U. Vandyke It was manufactured by the Geiger Bros. They can be found under classification section 1. I don't know what is on the back of the Leinbach puzzle another sum-to type or the " Safe Combination Puzzle " make all 12 columns sum to 55 , but their fronts display no dates. The Hecker's puzzle says "Pat. The Steinfeld puzzle says "Pat.
June 20th, " - with a bit of research I found that the patent in question is US issued to William H. Reiff of Philadelphia on June 20, That makes this the oldest version as far as I know. Dave at Creative Crafthouse has brought back the puzzle and issued the Word Wheel. Thanks for the copy, Dave!
Another type of "combination" puzzle relies on a large combinatorial space to be explored, in which one must find particular combinations. Includes four single-sided cards, each containing four variations on one quadrant of a caricature of a head - more card sets were offered. It was patented Feb. The derivative Changeable Charlie was produced in the 's copyright and onwards.
There was also a Changeable Charlie's Aunt. Sometimes the additional restriction of disallowing a repeated symbol along either main diagonal is also added. Also see a nice article by Elaine Young. Leonard Euler studied Latin Squares in the late eighteenth century, and research into them has continued, not simply because of their use as puzzles, but more for their application to experimental designs and cryptography.
The enumeration of Latin Squares has not been easy - figures up to order 10 are summarized in the table below. A reduced or standard Latin Square is one where the symbols in the first row and the first column are in lexicographical order. In , in Discrete Mathematics , J. Shao and W. Wei published a formula for the number of Latin Squares of any order. It is non-trivial to specify. Arrange the pieces 7 each of a Hippo, Lion, Elephant, Rhino, Giraffe, Zebra, and Tree so that only one of each appears in each row and column.
No Date. It comes as a single 5x5 plastic sheet, scored with grooves along which you are to break apart the pieces. Each square is one of five colors - yellow, red, blue, green, or white. The unbroken sheet shows the solution - this is a Latin Square puzzle and in the grid, no color occurs more than once in each row or column. There are nine pieces - two are 1x2, seven are 1x3. Bird's Puzzle , by Chad Valley. The Bird's Puzzle is very similar to the More Madness puzzle. There are nine pieces, two 1x2 and seven 1x3, colored with five colors - yellow, red, blue, green, and a Bird's logo - to be arranged into a 5x5 grid such that no color appears more than once in each row and column.
A vintage French boxed puzzle called Les An order-5 Latin Square. Testa Cross Colour Same as Bird's. A vintage cardboard advertising puzzle, "Say Cheese Louder. I have highlighted the scored lines separating the pieces. Form a 9x9 Latin Square using the pieces made from colored woods. Utopia issued by Popular Playthings invented by Sjaak Griffioen Sixteen "buildings" - four each of four different heights.
Place on the 4x4 grid according to rules and hints given on 50 challenge cards divided into 25 Phase 1 and 25 Phase 2. I enjoyed Phase 1 but Phase 2 seems overly confusing. NOTE: this may seem to belong in the Graeco-Latin section, but notice that all buildings of the same height are the same color, so height is really the only differentiator. Setko Scramble Arrange the pegs so that no row, column, or main diagonal contains a duplicate letter, and so that "SETKO" is not spelled in any line. Graeco-Latin Squares A Graeco-Latin or Greco-Latin Square also known as an Euler Square is constructed by superimposing two Latin Squares having the same order but different sets of symbols usually designated by using Latin letters for one of the squares' symbols and Greek letters for the other, hence the name Greco-Latin , such that each combination of symbols one from each Latin square occurs only once in the superposition.
Euler demonstrated methods for constructing Graeco-Latin Squares when N is odd or a multiple of 4. The Thirty-six Officers Problem goes as follows: arrange six regiments of six officers each of six different ranks into a 6x6 square so that no regiment or rank is repeated in any row or column. Note that there are no diagonal restrictions.
Rob Beezer shows a nice colorful order 10 square on his web page. Since in such a superposition, the Latin Squares used cannot both be standard, a Greco-Latin Square in standard form is one where the first Latin Square is standard, and the second has only its first row in lexicographical order. They must be arranged in a 4x4 grid such that no two with the same feature appear in any row, column, or main diagonal.
Sometimes it is prohibited to have a repeated symbol among the four corners of the square, or among the four central cells see " Play Thinks" There is only one order 4 Graeco-Latin Square in reduced form, but it does not meet these additional constraints. This is a Graeco-Latin Square puzzle. Place the blocks in the box so that no two of the same number nor of the same color are in any of the 10 horizontal, vertical, or diagonal lines.
Remove one of the 4's, then, by sliding them about, arrange them in horizontal rows, each of a different color, and in the order of 1,2,3,4. The fourth or bottom row should be 1,2,3. After completing the second, slide them about again to arrange them as in second, but in a vertical position. Brain Strain An advertising puzzle consists of sixteen small playing cards - the Jack, Queen, King, and Ace in each suit.
As with any Graeco-Latin square puzzle, the objective is to arrange the pieces in a square grid so that neither of the two kinds of feature in this case, face value and suit appears more than once in each row, column, or main diagonal. This puzzle was first proposed by Jacques Ozanam. Quintessence Copyright by gametime, Inc. The objective is to arrange them in a 4x4 grid such that rows, columns, and main diagonals contain cars of all different models and colors, and the sum of license numbers is 34 in all rows and columns and the four corners, the four centers, and each group of four in a corner!
The 36 Cube - Thinkfun Presented as a 6x6 Graeco-Latin Square - in this case, one feature is the height of the tower, the other is its color. The usual rules would seem to apply. However, remember the proof that no Graeco-Latin Squares exist for order 6? Some info from Thinkfun's press release: According to Nick Baxter, "Its friendly look fools you into thinking it's easy, but 36 Cube takes perseverance and practical smarts to solve. Its solution lies in an 'a-ha' moment, an insight equally discoverable by almost anyone willing to spend time with it.
Hmmm - I wonder which supposition? You can play online - trying Graeco-Latin squares up to order 5. To try level 6 you've got to buy the puzzle. Thinkfun also sponsored a contest - I was one of the winners! Thanks, Dave! We found a picture of the old puzzle in Slocum and Botermans' book, New Book of Puzzles and were fascinated by it. It has a bit of a Sukoku edge to it, and it's really hard. The object is to arrange all the pieces in the base such that there is no color repeated on any row, column, or any diagonal.
Also, no number can be repeated on any row, column or any diagonal. Also, each row, column and major diagonal must add to You might try to tackle each of the requirements separately before you take on the full challenge of making them all happen at once! If all diagonals main as well as partial also sum to the magic constant, the square is a pandiagonal or panmagic square. If replacing each number by its square also results in a magic square, the square is bimagic. If the sequence of numbers used in a square of order n is from 1 to n 2 , it's known as a normal magic square.
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Supposedly the order 3 magic square was invented in China between and B. There are of order 4, and over million of order 5. In the order 3 normal square, all rows, columns, and the main diagonals total An order 4 Magic Square appears in Albrecht Durer's famous engraving called Melencolia I : 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1. Each row, column, and main diagonal sum to 34, as do the four corners and the four central cells.
Fascinating 15 - Crestline Crestline put out this order-3 magic square - what could be simpler to manufacture, eh? Thinking Man's 34 - Skor Mor. Also 34 Skidoo by Reiss, Sixteen wooden tiles printed with the numbers one through sixteen. Arrange them in a 4x4 grid such that every row and column and main diagonal totals Mystifying 65 - Crestline An order-5 magic square from Crestline. Its magic constant is It was discovered in by William Radcliffe.
In Charles Trigg published a proof that this is the only magic hexagon of any size save the trivial single hexagon. Patent - Castanis and Hi-Q Enigma by Ideal are two puzzles where one must fit a group of pieces onto a surface bearing a pattern. Gridlock by Gabriel is a travel version of Hi-Q Enigma. The pieces match various portions of the pattern, and the proper "covering" must be found so that all the pieces will be accomodated and the surface completely covered.
This will work with just patterns although both of these puzzles employ an embossed surface with corresponding holes in the pieces. Of the two, Catch is more logically straightforward. It includes a set of domino-like pieces. Here is my solution "diary" for the 2nd side puzzle of Catch Please refer to the diagram at left. This shows the 6x7 board - the number inside each square indicates the pattern on the board.
The numbers outside the board label the x horizontal and y vertical coordinate axes. I will refer to locations on the board via a pair of x and y coordinate pairs. The tiles consist of the 21 "dominoes" representing all ways of pairing the numbers 1 through 6. Getting started here is not as easy as for the puzzle on the first side, since here there are multiple possible locations for every tile.
Choose to place it at the latter location. Ergo we have made an error and must backtrack. Now consider the 6 at location 5,0. Next consider the 3 at location 0,6. Now look at the 6 at location 1,6. Now consider the 6 at location 5,4. Next consider the 5 at location 4,3. There are two ways of placing them and either is OK. Consider the 1 at location 4,1. Now look at the 2 at location 2,1. Next consider the 4 at 1,2. Lastly, consider the 3 at 5,6. Again, two arrangements are possible and either is OK. You can make and play this type of puzzle with just some paper and a set of dominoes.
Wonder Workshops had this inexpensive version of a domino puzzle I don't have it. An arrangement of pips is presented on a card. Using a set of dominoes, cover the card by matching pairs. Thanks, Mom! BTW, this puzzle obsession is all your fault! Instant Insanity was invented by Franz O. Frank Armbruster , a California computer programmer, and marketed by Parker Brothers during It sold about 12 million copies.
There are many puzzles in what I call the "Instant Insanity Family. Four colors are used, and each of the cubes' faces are colored with one of the four colors. The objective is to find a linear arrangement of the cubes a rectangular prism such that all four long sides show each color only once. The faces on the ends and in between cubes don't matter.
There are 41, distinct ways to line up the four cubes, so exploring them all by hand will take a while! The number of ways to color a cube, allowing repeated colors, with at most k colors is given by Polya's Enumeration Formula , which follows from Burnside's Lemma. Sloane's integer sequence A : 10, 57, , , , , , , So for four-color type puzzles such as Instant Insanity, there are possible cubes to choose from when creating the puzzle.
The trick is to find a set that guarantees a solution, and preferably only one solution! The cubes are typically diagrammed by "unfolding" them and representing them as a cross:. Note that when coloring a cube, once you've colored two pairs of opposing faces, then given colors for the last pair of opposing faces, you define a pair of mirror image cubes by assigning the colors to those two remaining faces in one way or the other.
For purposes of this type of puzzle, mirror image cubes are identical and distinctions between them are immaterial to the solution. Ivars Peterson's MathTrek article of August 9, , "Averting Instant Insanity," is devoted to graphical solution techniques pertaining to this type of puzzle. Smith, writing as "F. For the graphical solution technique, the set of cubes is represented as a single undirected pseudograph. The term "graph" usually formally excludes more than one edge connecting any two nodes, and self loops.
The term "pseudograph" allows both, and we need both for this type of puzzle. Each of the four colors is a node. Each cube results in three labeled edges on the graph for twelve edges altogether - an edge connecting two nodes or a node with itself when those two colors appear on opposite faces of the cube, labeled with an identifier for that cube. Given this graph, the solution to the puzzle can be determined, and all puzzles of this type with isomorphic graphs are in essence identical.
Note that the Graph Isomorphism Problem is a well-known area of research. On the right is the graph for Instant Insanity I numbered the cubes arbitrarily. Here is my copy of the solution letter sent out by Parker Brothers on request:. Schossow of Detroit. It was marketed as the Katzenjammer Puzzle. Gottechalk Patentee Chicago, Ill.
Patent No. Slocum's book shows the layout of each of the four cubes, as does the patent, pictured here. In the physical puzzle, each block is a different color - orange, pink, green, and yellow. I have a complete example and another example with one block missing. I am missing the fourth block from the top in the illustration in the book, which I believe is the yellow one. Katzenjammer brought this little box of blocks to his wife, and said to her:- 'Katerina, you will notice that on the top row of these blocks there is a diamond, a heart, a spade, and a club.
Now take the blocks out of the box and place them together so that all four sides will have one spot of each kind in a row. It comes easy, Katerina,' he said. I've colored the Schossow nodes to show the correspondence. I numbered the Schossow cubes according to their top-down order in the patent image. I've analyzed over 20 puzzles in this family, and most of them copy Schossow's design. O'Beirne had already remarked that many versions of this puzzle known to him were isomorphic - including the Katzenjammer, Great Tantalizer, and Symington's.
The symbols might be replaced or re-mapped, but the essential pattern of the set of cubes is isomorphic. For example, see patent - Silkman Silkman's cubes are virtually identical - just the symbols have been re-mapped, and two cubes are mirror images of Schossow's, but this makes no difference to the solution.
So, how does one establish that two such puzzles are in fact isomorphic? Well, I read up on the Graph Isomorphism Problem mentioned previously, and found references to nauty by Brendan McKay, which is evidently the best publicly available software. However, it seems like overkill for my purposes. I've chosen DOT since the graphviz package is handy. II-type Instant Insanity - type puzzles, with 6 faces per cubic element, four cubes, and four colors, are 6,4,4 in my scheme.
Many approaches to graph isomorphism entail establishing some way to convert a graph to a canonical form , and showing that two graphs with the same canonical form are in fact isomorphic. Another useful concept is the invariant - some property of a graph, that all graphs isomorphic to it must share - e. Obviously, two invariants that apply here are four nodes and twelve edges.
One can also try to define a complete or sufficient invariant which completely decides isomorphism - the number of nodes and edges doesn't meet this need. Yet another useful concept is the signature or certificate of a graph - a way of assigning a coded ID to the graph as a whole, that plays the role of a sufficient invariant.
I've come up with what I believe is a good way of assigning certificates to II-type puzzles. First , encode the nodes using four hex digits 0-F based on their connectedness: The rightmost or least significant place counts the number of self-loops on the node. The other three places give the counts of connections to the other three nodes in no particular order , but sorted in increasing value from left to right. For twelve-edge graphs, we really only need up to digit 'C' and in practice most counts are only as high as 3.
This is convenient since each node gets a unique code. Next , encode the cubes themselves, each as a series of six node-codes derived previously - remember, an edge on the graph was defined by a pair of opposing faces on the cube. So an edge can be encoded with 8 digits by concatenating the codes for the two nodes it connects, sorted in lexicographic order the edges are undirected, after all.
And, a cube can be represented with 24 digits by concatenating the codes for the three edges it contributes to the graph, again sorted. These two sorting steps mean that we lose info to distinguish mirror image cubes, but as previously established, that doesn't matter to the solution! Finally , the whole puzzle's digit certificate is the four codes for the cubes, again sorted.
And here again, the sorting doesn't matter, since the order of the cubes doesn't matter to the solution. This is a vintage advertising premium called Symington's Puzzle. It contains four cardboard cubes, each with a different arrangement of four Symington's product advertisements on their faces: Soup, Custard Powder, Ideal Cream, and Gravy. Isomorphic to SK. Tantalizer - Shackman Isomorphic to SK. Here is another vintage cube matching puzzle called The FourAce Puzzle. The four wooden cubes are decorated with various arrangements of the four playing card suite symbols: hearts, diamonds, clubs, and spades.
The box says "Provisionally Protected" but does not identify the manufacturer or date of manufacture. According to Slocum, this puzzle was sold in Britain at Gamage's in An advertising version of the Tantalizer puzzle, from Bass. I am not sure what this puzzle is actually called, but on the bottom of the tray it says "Masudaya Made in Hong Kong" so I call them the Masudaya Cubes. This may be the same as Ideal's Face Four puzzle, which I also found. I have examples in red, white, and black.
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Nice Cubes Isomorphic to SK. The vintage Cubo Color Puzzle. Not isomorphic to SK. Idiot's Delight - Field Mfg. The Allies Flag Puzzle is another very old example of this family. This puzzle has five cubes, and each cube has some arrangement of five flags on its faces. This is the Masudaya Hexagon Mind Exerciser. It has six hexagonal pieces. The objective is to line up the six hexagons so that each of the six rows of six faces shows all six colors.
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Unlike a set of cubes, where on each cube two faces are not used in a solution, here all six faces of every hexagon will be used. This means that there must be in total six faces of each of the six colors. In my copy of the puzzle, all six hexagons have distinct color arrangements - i. I have found at least one solution - one can employ the graphical technique, but not in exactly the same way as for cubic puzzles - here, three mutually consistent sub-graphs are needed, and they are not independent.
A vintage Cylinder Ten by Masudaya. Ten rotating disks each with ten color segments. The English instructions say, "Arrange 10 different colors in a line to get all rows correct. The probability of getting the correct order is one in a billion. This puzzle will give the whole family hours of amusement. Note: it is packed in the factory with colors in the correct order i. However, I translated "Cylinder 10" into Japanese and did a search on Google Japan just to see what would turn up, and I found a "FrogPort" blog article from April talking about the release of this puzzle and how it is in a series with Masudaya's other puzzles Face 4, Hexagon Mind Exerciser, and Circus 7.
Here is a link to the blog via google translation to English. This is unlike the traditional Instant Insanity type where one of each color must be present. This allows the rotating disks to have repeated or missing colors. A "line" is a row of stickers across all ten disks. Go Crazy Embree Manufacturing Co. NJ Arrange the five disks so that alternate rows have five different colors, then three different colors.
The five disks are separate and may be removed from the case and re-ordered. The Buvos Golyok is a clever variant using balls enclosed in a tube. High Octane puzzles. These versions use octahedrons rather than cubes, and a clever mirrored base to allow one to see all faces. Thanks, Stan! The cubes are usually labeled with colors, but sometimes numbers are used Twenty Teazer Arrange the cubes so that each side totals Kathy's Kubes by R. Gee Arrange so dots total 10 on each long side. This type of puzzle can be arranged vertically, too The connectivity of this graph is similar to that of the Nice Cubes, though its edges seem differently labeled.
Are the puzzles in fact isomorphic? Explorations So, why have so many puzzles over the years simply copied the topology of the Schossow-Katjenjammer cubes? Is it so difficult to find a different set of four cubes that satisfies the constraints and yet has only one solution? I set out to try to answer that question - here are notes on my explorations and findings.
Recall that there are ways to color a cube using 4 colors. Given a "stable" of cubes from which to choose four, how many possible 4-cube sets are there? So, we've got over million sets to explore! Also, for a set of 4 cubes, there are 41, possible ways to arrange them - that's going to be a lot of work. But we can reduce this by eliminating from consideration all cubes that do not have at least one face of each color. It's a little arbitrary, and it means my conclusions are not universal over the entire universe of possible puzzles, but I think it's reasonable - most commercial puzzles comply.
Of the possible four-color cubes, there are 68 that use all four colors at least once each. Each cube can be represented by a four-node, 3-edge pseudograph, where the nodes are the four colors symbolized by the letters ABCD and each edge corresponds to a pair of opposing faces on the cube. The graphs are the same for mirror-image cubes, since the edges are undirected. The 68 cubes give rise to only 52 unique graphs. The colors identify the six groups. It should be easy to construct a cube given its little diagram and the key on the lower right.
You just have to decide on your assignment of actual colors to ABCD, and then be sure to remain consistent. Rick assigned numbers to these groups as follows, and noted that cubes in a group have the given number of possible effective orientations within a puzzle, ranging from 10 to A higher number of possible orientations might contribute to a harder puzzle. You can see some of the cube models I made - I affixed a small label to each cube, showing its ID number. One bucket does not contain enough color panels to make all 52 cubes simultaneously. I created a program to solve a 4-cube set, and used it within another program that explores 4-cube sets taken from the stable of 52 cubes above.
I did not allow repeated cubes in a set - so, here again, I have simplified the problem space, but again I believe in a reasonable manner. I also eliminated repeated sets that differed only by the order in which the same cubes were selected, but this entails no loss of generality. This resulted in , sets being tried - a run that easily completed overnight on my PC. Here is a chart of number of solutions, number of sets that have the given number of solutions, and the first set found that has that number of solutions. I have omitted solution counts where no set had that number of solutions.
A set is represented by its "signature" - an eight-digit number zeroes on the left, if dropped, should be inferred that is simply the four cube IDs from the chart, concatenated, always from lowest numerical ID on the left to highest on the right. There were many sets - almost half - that had no solution at all. Some sets have more than one solution, and the maximum number of solutions any set can have seems to be Most solvable sets have four solutions. There is only one set that has exactly 27 solutions, and 27 solutions is the only count where only one set has that count.
Interestingly, the cubes used for this set are the four from the yellow group in the chart. It is not hard to solve this set by hand. Based on a cursory check, it appears that none of the cubes in groups 5 tan or 6 blue are used at all in single-solution sets! I ran the single-solution sets through my program that computes a puzzle's certificate and checks if it is isomorphic to SK.
I found that 24 of the are iso. If all four cubes in a set are permuted in a consistent manner, one may arrive at another set - and the two sets are in some sense the same puzzle. Rick Eason says that by using Burnside's Lemma to eliminate color permutations, the , figure can be collapsed to 11, sets. If we further eliminate the 12 cubes not used in single-solution sets the tan and blue groups , we end up with only sets. I did not perform this collapsing for my runs, so my totals do not take into account color permutations.
In a brief paper entitled On "The Tantalizer" and "Instant Insanity" PDF online here , Frank Harary , recognized as one of the fathers of modern graph theory, notes that "Using standard methods of graphical enumeration [Harary and Palmer ], it is not difficult to develop a formula for the number of different 'Tantalizer' games with four cubes.
This can be done by associating a graph with each such game, and then counting graphs. If you know of one, please let me know! Here are the 24 signatures for the sets that share the SK certificate, 01 04 20 40 01 14 36 39 02 03 22 45 02 12 33 44 03 07 17 40 03 14 26 47 03 30 35 38 04 08 16 45 04 12 28 32 04 29 37 46 06 20 33 47 06 22 32 39 07 10 28 30 07 14 44 48 07 18 39 46 08 10 26 37 08 12 36 51 08 18 33 38 12 20 25 35 13 17 33 37 13 22 30 51 14 22 25 29 15 16 30 39 15 20 37 48 The set identified by O'Beirne in his Great Tantalizer article in Figure 1 is the first listed, SK consists of 1 cube from group 2 green , 1 cube from group 3 teal , and 2 from group 4 lavender.
Rick suggests that the most difficult puzzles will consist of only cubes from groups 1 yellow and 4 lavender , since those groups offer the most possible orientations 24 each. The single-solution sets I found group into just distinct certificates. I believe this reduction is also due to the permutations of the color assignments.
The certificate with the smallest number of sets having that certificate is shared by only six sets: is assigned to sets: , , , , , and In each set, a given pair of colors appears only once each on adjacent sides on all four cubes, with the other four sides of each cube having the other two colors. The graph of such a cube will have only one edge leading from each of the two colors, which will not share an edge. These sets are easy to solve - identify the special colors. Orient all cubes so that the sides with the special colors are on top and front.
Now it is a simple matter to turn three of them so that the solution is found. At the other end of the scale, there is one certificate that encompasses sets: The first example having that certificate is According to my analysis, there should be distinct single-solution puzzles, where every cube uses all four colors, and each set uses four distinct cubes.
You can download a text file "single-soln-sigs-and-certs. Those sets isomorphic to SK are noted. The tail of the file contains a summary of the unique certificates and the count of sets having a given certificate. Wellingtons Cube Puzzles An extensive set of additional puzzles in the Instant Insanity family were offered by three U. Many of them comprise four, six, or eight clear plastic cubes containing images on each side. In several cases the objectives are a departure from that of the Instant Insanity family.
The table below lists, in mostly alphabetical order I tried to keep sequels together , those I know of and shows images where I either have a copy or have been able to find pictures. I don't have them all and I will note the ones I don't have. Those I have are highlighted like this. Those I do not have are highlighted like this. I would like to acknowledge the following: The generous Liz Burrow , who gave me several of these puzzles, and Martin Watson, who was kind enough to bring them to me in Prague.
Colin James , a fellow collector, who has also sent me puzzles and helped catalogue the puzzles, supplying many photos of his collection. Bananas Bananas Onsworld Ltd. Unlike a standard II-type puzzle, the images have distinct orientations which must be respected to form the complete bananas.
The tips, stems, and middle segments must all be aligned properly. A little analysis reveals that the usual four-in-a-row arrangement of the cubes cannot satisfy the goal - but then, the objective does not stipulate that arrangement, does it? Bunkered Bunkered Wellingtons "Tee-off by placing the cubes together then rearrange them to show, simultaneously, TWO identical golf courses. Image from Jim Storer's collection. Here is a possible alternative package: Cubix 3D I don't have this, but see Rebus in the Pattern Blocks section below. Cuss Cuss Onsworld Ltd. No foreign words or proper names allowed.
A clue in case you get stuck and a solution are included. Double Cross Double Cross Wellingtons Fascinating Felines Fascinating Felines - Wellingtons Footsie Footsie Wellingtons "Arrange the four cubes in a square so that wherever the surfaces of two cubes join top, bottom, and sides pairs of feet are formed. Frantic Frantic Onsworld Ltd. Isomorphic to SK! Frantic alt. An alternative version of Frantic, with solid colors. Colin says it is the same graph. Also issued with black backgrounds. Frantic II Frantic II "Place the cubes together in a square so that wherever they meet, top, bottom, and sides, the coloured squares match.
Kenneth Miller. The letters must all point the same way. Try it - then baffle them at the 19th. Nuts Nuts Wellingtons "Place the four cubes in a row and simply rearrange them so that one of each kind of nut shows on each long side. The designs are: 10, Jack, Queen, King, Ace. Kenneth Miller Two puzzles based on the two versions of Rugby - for the Rugby League fan, place the cubes in a straight line so that each long side shows exactly 13 pieces of muddy lace.
For the Rugby Union fan, place the cubes in a straight line so that each long side shows exactly 15 pieces of muddy lace. Kenneth Miller Thanks to Rob Hegge for photos. Spellbound - alternative version - Onsworld Ltd. This and the other Tantalizer puzzles seem to be more dexterity than Insanity-type.