A Bouquet for Gardner.

A Bouquet for Gardner. Our tribute bouquet starts with several PELARGONIUMS: all but twoof which are red, all but two of which are yellow, and all but two ofwhich are green. The reader should be able to determine from thisinformation just how many flowers are in our main bouquet. The answer tothis riddle which is an adaptation of one of Martin Gardner'scharming Snarkteasers (#52, in [G73]) will be given later. Meanwhile, there is some mathematics to be explained about ourPELARGONIUMS and the hybrids we obtain from them. We start with one ofGardner's earliest articles "The Five Platonic Solids"(Chapter 1 in [G61]). There he proves that the tetrahedron, thehexahedron (a cube), the octahedron, the dodecahedron, and theicosahedron are the only possible regular convex solids (Fig. 1). [FIGURE 1 OMITTED] Most of our flowers will grow from these five Platonic seeds. Inthe article Gardner points out that "the cube and octahedron are'duals' in the sense that if the centers of all pairs ofadjacent faces on one are connected by straight lines, the lines formthe edges of the other. The dodecahedron and icosahedron are duallyrelated in the same way. The tetrahedron is its own dual". We willcombine planar graphs of the solids with their duals in a special way toobtain our PELARGONIUMS. Such planar graphs are called Schlegetdiagrams. This is a model that is, as Gardner explains (p. 23 in [G83])".[??] simply the distorted diagram of the solid, with its backface stretched to become the figure's outside border." InFigure 2 we draw with solid, curved lines the Schlegel diagrams of atetrahedron, an octagon and an icosahedron. Superimposed with dashedlines are the Schlegel diagrams of the respective duals, thetetrahedron, hexahedron, and the dodecahedron. The three outer dashedlines in each diagram are to be regarded as meeting in the back face. [FIGURE 2 OMITTED] After labeling the nodes as ROSE, ORCHID, and PELARGONIUMS, thedrawings are extended to the completed flowers where the new nodesbecome the intersections of the former dashed and solid arcs and inheritthe two-letter labels of the endpoints of the former nodes. A study ofFigure 3 will make this clear. Thus three flowers answers our openingriddle. [FIGURE 3 OMITTED] It is possible to form hybrids by judicious relabeling of theflowers. For instance, Figure 4 illustrates the new flowers GERANIUM andPELARGONIUMS formed on ORCHID. It may be possible to obtain otherhybrids using as labels WILD ROSE, VIOLET, MARIGOLD, MYRTLE, etc.Remember any labeling should yield bona fide dictionary entries on thenew names of the parts. [FIGURE 4 OMITTED] In addition to the nodes our flowers contain 3-, 4- and 5-cyclepetals corresponding to the former regular faces of the Platonic solids.All of the petals inherit labels from their boundaries and all labelsare words which are main entries in most unabridged dictionaries oratlases. (We used the Merriam-Webster New International, 3rd ed.) Wewill be able to use these labels for certain puzzles and games we havein mind on the flower graphs. The number of flower parts are: Nodes 3-cycles 4-cycles 5-cycles ROSE 6 8 0 0 ORCHID 12 8 6 0PELARGONIUMS 30 20 0 12 Recall that the outside of each flower is also a cycle. There is a famous puzzle invented in the 1850s by the Irishmathematician Sir William Rowan Hamilton that was originally played on asolid dodecahedron that can be played on our green PELARGONIUMS grid.Gardner first described this puzzle in "The Icosian Game and theTower of Hanoi" (Chapter 6 in [G59]). "... the basic puzzle isas follows. Start at any corner of the solid (Hamilton labeled each corner withthe name of a large city), then by traveling along the edges make acomplete 'trip around the world', visiting each vertex onceand only once, and return to the starting corner." Today this iscalled finding a Hamiltonian circuit. The 20 3-cycles of PELARGONIUMSare the "vertices" that must be visited in a circuit on ourgraph. They are each joined by a two-letter node on the flower. It isalso possible to write the 20 three-letter words on tiles and try toarrange them in a chain so that abutting tiles have two letters incommon. If the chain closes, you have solved the puzzle. Gardner writes "On a dodecahedron with unmarked vertices thereare only two Harniltonian circuits that are different in form, one amirror image of the other. But if the corners are labeled, and weconsider each route 'different' if it passes through the 20vertices in a different order, there are 30 separate circuits, notcounting reverse runs of these same sequences. Similar Hamiltonian pathscan be found on the other four Platonic solids". One of the 30solutions is given at the end of this article. Other informative Gardnerarticles about Hamiltonian circuits include "Graph Theory: (Chapter10 in [G71]), "'Knights of the Square Table" (Chapter 14in [G77], and "'Uncrossed Knight's Tours (p. 186 in[G79]). John H. Conway' s very interesting "ADodecahedron--Quintomino Puzzle" (p. 23 in [G83]) can be adapted toour flower. Gardner has often written about magic squares in his ScientificAmerican columns and we were especially interested in his report aboutRoom squares reprinted in "'The Csaszar Polyhedron"(Chapter 11 in [G88]). "A Room square is an arrangement of an evennumber of objects, n+l, in a square array of side n. Each cell is eitherempty or holds exactly two different objects. In addition, each objectappears exactly once in every row and column, and each (unordered) pairof objects must occur in exactly one cell." The Australian mathematician Thomas G. Room had called this concept"A New type of magic square" in 1955 but it was laterdiscovered that they had been in use before 1900 in scheduling bridgetournaments. We have discovered in our flowers a generalization of thesesquares. For instance using the yellow ORCHID we can form this 4x4square from the 12 nodes.4 RI HD CO3 CH ID OR2 OH IC DR1 DO CR HI a b c d This square is magic on the rows and columns in the sense that eachset of three entries transposes into the word ORCHID, the magicconstant. It is not a Room square since the taboo pairs IO, HR, and CDnever occur together. It is instructive to locate these pairs on theORCHID graph. A pleasant little puzzle is possible by preparing 12 tiles with thetwo-letter words on them and trying to reconstruct one of the 1152solutions to the puzzle that look different to the eye. Two persons canplay the puzzle as a game by drawing a tile in turn and placing it onthe grid so that no common letter occurs in any row or column. The lastplayer to be able to play wins. To play expertly one must heedcomplementary pairs of words consisting of taboo mates. That is, RI-OH,D-CO, IC-DO, HI-OR, CR-HD, and CH-DR. If any of these occur in the samerow or column it will be impossible to complete the trio of words inthat row or column. It can also be proved that where the blank squaresin the grid intersect, we must insert complementary words. For example,the two blanks at a2 and b4 intersect at a4 and b2 where the complementsRI and OH occur. This last property can be used in a magic trick. Let the subjectfind one of the possible solutions to the puzzle and then turn all thetiles face down. The subject turns over and exposes any tile and themagician can then call out another (the complement) and is able tolocate it in the grid. This is repeated until all tiles are exposed. Ifthe grid is on a board it may be carefully rotated before any tile isexposed and of course the trick will still work. Interchanging pairs ofrows or columns--including the blanks--can make the trick even moremysterious. The 12 words are the edges of a 3-dimensional cube and similarsquare grids are possible using the edges of an n-dimensional cube. Thisgeneralization, as far as we know, has never been explored. The 20 three-letter words in the green PELARGONIUMS flower yieldanother magic square. One solution with magic constant PELARGONIUMS is:MAR OIL SUN PEGPUG MRS ALE IONLIE PUN MOS RAG AGE PIN MOL SURSON RUG PIE LAM The taboo pairs are MP, AN, LU, RI, GO and ES which lead to thecomplementary pairs MAR-PIN, MRS-PIE, MOS-PEG, MOL-PUG, LAM-PUN,ALE-SUN, RAG-ION, AGE-SON, SUR-LIE and RUG-OIL. There are 28,800 solutions to this puzzle that will look differentto the eye. (Purists would not regard all of these as truly differentbecause of certain symmetries of the grid.) Each of the remarks aboutthe 4x4 ORCHID grid hold as well for this 5x5 PELARGONIUMS grid. These magic squares may have applications m tournament schedules.For instance suppose we have three two-person teams that are to playone-on-one games of four types, a, b, c, and d, over four days. If theteam members have initials IO, HR, and CD respectively, our 4x4 squaregives the pairings on each day. It is possible to extend our flower garden into higher dimensions.For example, there are six regular polytopes in four-dimensions that arethe analogues of the Platonic Solids. One is the hypercube (see Chapter13 [G01]) but perhaps the simplest is the polytope ASTER whose Schlegeldiagram appears in Figure 5. See also Gardner's"Tetrahedrons" (Chapter 19, [G71]). [FIGURE 5 OMITTED] ASTER, or the "regular simplex" as geometers call it, has5 nodes, 10 lines, 10 3-cycles and 5 4-cycles. Specifically, the partsare A-REST, S-TEAR, T-SEAR, E-STAR, R-SEAT, AS-RET, AT-ERS, EA-STR,RA-SET, ST-ARE, SE-RAT, SR-TEA, TE-RAS, RT-SEA, and RE-SAT. This floweris self-dual, like the tetrahedron it is similar to, and so a dualASTER, interchanging nodes with 4-cycles and lines with 3-cycles couldbe superimposed on the graph. When reproduced in two dimensions theresult would be a very "busy" flower! There are of course other solids of interest that are not regular.One such infinite class is the prisms, solids with polygonal bases andtops with quadrilateral sides. As an example consider the followingflower. [ILLUSTRATION OMITTED] This flower is the Schelgel diagram of a hexagonal prism on whichwe ask the reader as a puzzle to place the 12 letters of PELARGONIUMS inthe nodes so that each of the six 4-cycles as well as the two 6-cyclesforming the base and top of the prism transpose into words. Our solutionwill be given later. The basic flower diagrams can be used for a variety of board games.One game that can be played on any of the three flowers starts byplacing tokens on all the nodes. Two players alternately remove one ormore tokens from any one of the cycles on the board. The player thatremoves the last token wins the game. This game is a nim type game thatsuperficially resembles David Gale's CHOMP. See Gardner'saccounts "Nim and Tac Tix" (Chapter 15 in [G59]) and"Sim, Chomp and Race Track" (Chapter 9 in [G86]) for details.On our flower boards, however, these games are second person wins.Perhaps the reader can discover the strategy before we give a hint onhow to play. A more challenging game, and one we can not predict the winner of,is Ten Men's Morris played on the green PELARGONIUMS flower. Eachof two players has ten distinctive tokens which they alternately placeon the nodes. When a player obtains a cycle of tokens, that player hasformed a Mill and may remove one of his or her opponent's tokens. Atoken that is part of a Mill cannot be removed. After all their tokenshave been placed, the players alternately move their tokens to empty,adjacent nodes trying to form Mills. The game continues until one playerloses by being reduced to two tokens. All of our games and puzzles may be played strictly as word gameswithout using the board at all. This usually makes them immensely harderto play. In "Jam, Hot and Other Games" (Chapter 16 in [G75])Gardner recounts a word version of ticktacktoe by Canadian mathematicianLeo Moser, who called it "Hot". Without the symmetry of theboard or grid as a guide, the games take on new life. Even our TenMen's Morris game can be played as a word game. The players have aword list composed from the 3- and 5-cycles of PELARGONIUMS and beginplay by drawing, in turn, ten tiles from a face-up bone pile of 30tiles. The tiles contain the two-letter words of the nodes. When someoneis able to form a word Mill with their tiles they take a tile from theiropponent's ten and place it back, face up, in the bone pile. Afterdrawing their ten tiles the players continue by exchanging one of thetiles in front of them with one from the bone pile that has a letter incommon, still hoping to form Mills. When a player is reduced to just twotiles, they have lost the game. Puzzling Pelargoniums. If there are n PELARGONIUMS, n - 2 of themare red, n - 2 of them are yellow, and n - 2 of them are green. Thus, n[greater than or equal to] (n - 2) + (n - 2) + (n - 2) = 3n - 6, or n =3. Implicit in the puzzle is that there are only three colors, implyingthat, in fact, n = 3. (Otherwise, technically, there could be just two PELARGONIUMS, ofsome other color.) Hamiltonian circuit puzzle. One solution isMAR-RAG-RUG-SUR-SUN-SON-ION-OIL-LIE-PIE-PIN-PUN-PUG-PEG-AGE-ALE-LAM-MOL-MOS-MRS. Hexagonal prism problem. The best set of words we found is GLAMOR,SUPINE, GAIN, LUNG, PLUM, POEM, ROSE, and AIRS. (See Figure 8.) [ILLUSTRATION OMITTED] Nim-like game. Our hint for the Nim-type game is to take advantageof the symmetry of the board, keeping in mind the complement of youropponent's play. For further insights on this kind of strategy, seeGardner's "The Game of Hex" [1, Chapter 8] and"Dodgem and Other Simple Games" [10, Chapter 12]. BIBLIOGRAPHY. All references are by Martin Gardner. [G59] The Sci. Am. Book of Mathematical Puzzles & Diversions.Simon & Schuster, 1959. [G61] The 2nd Sci. Am. Book of Mathematical Puzzles &Diversions. Simon & Schuster, 1961. [G71] Martin Gardner's Sixth Book of Mathematical Games fromSci, Am. W.H. Freeman, 1971. [G73] The Snark Puzzle Book. Simon & Schuster, 1973. [G75] Martin Gardner's Mathematical Carnival. Knopf, 1975. [G77] Mathematical Magic Show. Knopf, 1977. [G79] Mathematical Circus. Knopf, 1979. [G83] Wheels, Life and Other Mathematical Amusements. W.H. Freeman,1983. [G86] Knotted Doughnuts and Other Mathematical Entertainments. W.H.Freeman, 1986. [G88] Time Travel and Other Mathematical Bewilderments. W.H.Freeman, 1988. [G01] The Colossal Book of Mathematics. W. W. Norton, 2001. The co-authors, long-time collaborators, are avid collectors of allthings Gardner: books, articles, reviews, magic, puzzles, games, et. al. Jeremiah Farrell and Thomas Rodgers. "A Bouquet forGardner." In A Lifetime of Puzzles: A Collection of Puzzles inHonor of Martin Gardner's 90th Birthday, edited by Erik D. Demaine,Martin L. Demaine, and Tom Rodgers, pp. 253-263. Wellesley, MA: A KPeters, 2008. Rreprinted by permission of A K Peters, Ltd.(www.akpeters.com).