Magic Hexagon: 3 19 16 17 7 2 12 18 1 5 4 10 11 6 8 13 9 14 15 An arrangement of close-packed hexagons containing the numbers 1, 2, ..., H, where H = 3n(n-1)+1 is the nth hex number, such that the numbers along each straight line add up to the same sum. In the above magic hexagon, each line (those of lengths 3, 4, and 5) adds up to 38. This is the only magic hexagon of the counting numbers for any size hexagon, as proved by Trigg (Gardner 1984, p. 24). It was discovered several times independently: Ernst von Haselberg, Stralsund, 1887 [Haselberg], [Bauch], [Hemme], W. Radcliffe, Isle of Man, U.K. 1895 [Hemme], [Heinz], [Tapson] H. Lulli [Hendricks], [Heinz], Martin Kühl, Hannover, Germany 1940 [Gardner], [Honsberger], Clifford W. Adams, who worked on the problem from 1910 to 1957 [Gardner], [Honsberger]. hex numbers: Let n be the number of points along the edge of the hexagon. Let r be the number of rows in the hexagon than r = 2n - 1. Example n=3, r=5, H=19. a a a b a a a b b o c c b b c c b c c (1) H = 3n(n-1) + 1 = n^3 - (n-1)^3 (2) H = (3r^2 + 1)/4 Figurate proof of (2): 4H = 3r^2 + 1 a1 a1 a1 a2 a2 a2 a3 a3 a3 a4 a4 a4 b1 a1 a1 a1 b2 a2 a2 a2 b3 a3 a3 a3 b4 a4 a4 a4 b1 b1 o1 c1 c1 b2 b2 o2 c2 c2 b3 b3 o3 c3 c3 b4 b4 o4 c4 c4 b1 b1 c1 c1 b2 b2 c2 c2 b3 b3 c3 c3 b4 b4 c4 c4 b1 c1 c1 b2 c2 c2 b3 c3 c3 b4 c4 c4 = a4 a4 a3 a3 a3 b4 b4 b3 b3 b3 c4 c4 c3 c3 c3 a4 a4 a3 a3 a3 b4 b4 b3 b3 b3 c4 c4 c3 c3 c3 a4 a4 o1 a2 a2 b4 b4 o2 b2 b2 c4 c4 o3 c2 c2 o4 a1 a1 a1 a2 a2 b1 b1 b1 b2 b2 c1 c1 c1 c2 c2 a1 a1 a1 a2 a2 b1 b1 b1 b2 b2 c1 c1 c1 c2 c2 Magic constant: The sum S = 1 + 2 + 3 + ... + H is calculated by double counting 1 2 3 ... H-1 H | S H H-1 H-2 ... 2 1 | S -----------------------------------+---- H+1 H+1 H+1 ... H+1 H+1 | 2S S = H(H+1)/2 = (3n(n-1) + 1)(3n(n-1) + 2)/2 = (3r^2 + 1)(3r^2 + 5)/32 = ( 9r^4 + 18r^2 + 5 ) / 32. The magic constant is M = S / r = ( 9r^3 + 18r + 5/r ) / 32. If the magic constant is integral than 32 M is integral and that is r must divide 5. The divisors of 5 are {-5, -1, 1, 5}. Checking shows that r=1 and r=5 are valid. This simple form of the magic constant is first noted in [Berlekamp, Conway, Guy]. Magic constant for consecutive integers a, a+1, a+2 ... a+H-1: The sum is S = H(H+2a-1)/2 = (3r^2 + 1)(3r^2 + 8a - 3)/32 The magic constant is M = S / r = ( 9r^3 + 24ar - 6r + (8a-3)/r ) / 32. We see that there is no magic hexagon (except for n=1) if a=0. Magic Hexagon of order 3 Let the vertex elements be a_i, the mid-side elements be b_i, the central element be d, and the other elements be c_i, i=1, 2, 3, 4, 5, 6. Also, let Sum a_i = A, Sum b_i = B, and Sum c_i = C. a1 b1 a2 b6 c1 c2 b2 a6 c6 d c3 a3 b5 c5 c4 b3 a5 b4 a4 (I) Summing the rows of 3 elements, 2A + B = 6M. (II) Summing the rows of 4 elements, 2B + 2C = 6M. (III) Summing the rows of 5 elements, A + C + 3d = 3M. (IV) Summing all of the elements, A + B + C + d = 5M. As the sum of all rows is three times the sum of all elements this system of equations is indeterminate. A = A = (6M - B)/2 = (3M + C)/2 = 2M - d. B = 6M - 2A = B = 3M - C = 2M + 2d. C = 2A - 3M = 3M - B = C = M - 2d. d = 2M - A = ( B - 2M)/2 = ( M - C)/2 = d. Special properties of the magic hexagon of order 3: Connecting the elements x and y with x+y=19 we get many parallel edges. 3 19 16 3-- 0 --3 17 7 2 12 2-- 7===2 --7 18 1 5 4 10 1---1 5 4 9 11 6 8 13 8-- 6=|=8 --6 9 14 15 9 5 4 magic hexagon labels min(x,19-x) Connecting the elements x and x+1 was first analyzed by Martin Gardner. He only horizontal or vertical symmetric edges are shown. 18 11 9 18 11---9 17 1 | 6 |14 17 | 1 6 14 3 7-| 5 --8 |15 3-| 7===5 | 8 15 19 2 | 4 |13 19 | 2 4 13 16 12 10 16 12--10 connecting x,x+1 connecting x,x+2 Magic Hexagon order 3 for subsets of 0..20: Constant Solutions 35 12 36 18 37 35 38 27 39 18 40 19 41 14 Magic Hexagon Ring: 3 3 3 3 3 2 2 2 3 3 2 1 1 2 3 3 2 1 0 1 2 3 3 2 1 1 2 3 3 2 2 2 3 3 3 3 3 Magic Hexagon for arithmetic progressions: Progression Magic Constant Solutions 6, 7,...,24 3*19 = 57 0 1, 2,...,19 2*19 = 38 1 -4,-3,...,14 1*19 = 19 36 -9,-8,..., 9 0*19 = 0 26 = 12 + 2*7 The impossible a,a+1,...,a+H-1 third order magic hexagon for a>=6: Eliminate M from B = 2M + 2d and C = M - 2d gives B = 2C + 6d Now B <= 18+17+16+15+14+13+6a = 93+6a but 2C+6d >= 2(1+2+3+4+5+6)+18a = 42+18a. As 93+6a >= B = 2C + 6d >= 42+18a we have 17/4 >= a. Therefore with a>=5 the hexagon is impossible. List of Solutions for 1, 2,...,19 with magic constant 38: 3 19 16 17 7 2 12 18 1 5 4 10 11 6 8 13 9 14 15 List of Solutions for -4,-3,...,14 with magic constant 19: -4 14 9 10 7 3 -1 13 -3 -2 0 11 1 4 12 2 5 8 6 -4 14 9 13 7 0 -1 10 -3 -2 3 11 1 4 12 2 8 5 6 -4 11 12 14 7 1 -3 9 -1 -2 3 10 2 0 13 4 8 6 5 -3 13 9 14 6 1 -2 8 -4 3 0 12 4 -1 11 5 7 10 2 -2 13 8 14 9 -3 -1 7 -4 0 4 12 1 3 10 5 11 6 2 -2 11 10 12 6 5 -4 9 -1 -3 1 13 3 0 14 2 7 8 4 -2 10 11 14 0 3 2 7 -4 9 1 6 13 -3 4 5 -1 12 8 -2 8 13 9 14 -3 -1 12 -4 0 4 7 1 3 5 10 6 11 2 -2 7 14 10 13 -1 -3 11 -4 0 4 8 3 1 6 9 5 12 2 -1 13 7 8 9 4 -2 12 -4 -3 0 14 1 5 11 2 6 10 3 -1 13 7 12 9 0 -2 8 -4 -3 4 14 1 5 11 2 10 6 3 -1 11 9 8 10 -3 4 12 -4 7 -2 6 2 1 3 13 5 14 0 -1 8 12 11 0 6 2 9 -3 7 1 5 14 -2 3 4 -4 13 10 -1 8 12 7 10 6 -4 13 -3 -2 0 11 4 1 9 5 2 14 3 -1 6 14 9 4 8 -2 11 -3 1 3 7 12 0 5 2 -4 13 10 1 6 12 11 3 0 5 7 -4 10 4 2 14 -1 -3 9 -2 13 8 2 5 12 9 13 1 -4 8 -3 0 3 11 4 -1 6 10 7 14 -2 2 5 12 9 4 10 -4 8 -3 0 3 11 13 -1 6 1 -2 14 7 3 5 11 6 13 4 -4 10 -1 -3 1 12 2 0 8 9 7 14 -2 4 6 9 8 14 -1 -2 7 -3 0 3 12 2 1 5 11 10 13 -4 4 5 10 8 13 1 -3 7 -2 0 2 12 3 -1 6 11 9 14 -4 5 2 12 13 11 -3 -2 1 -4 -1 14 9 10 3 0 6 8 7 4 7 8 4 3 14 0 2 9 -1 -3 1 13 -2 6 5 10 12 11 -4 7 4 8 14 9 -3 -1 -2 -4 0 13 12 10 3 1 5 11 6 2 7 3 9 8 14 -1 -2 4 -3 0 6 12 5 1 2 11 10 13 -4 7 2 10 8 13 1 -3 4 -2 0 5 12 6 -1 3 11 9 14 -4 7 1 11 4 5 12 -2 8 -1 -4 6 10 14 3 2 0 -3 13 9 8 -1 12 7 10 6 -4 4 -3 -2 9 11 13 1 0 5 2 14 3 10 6 3 11 5 1 2 -2 -4 7 4 14 12 -1 0 8 9 13 -3 10 6 3 11 9 -3 2 -2 -4 -1 12 14 8 7 0 4 13 5 1 11 6 2 10 3 1 5 -2 -4 9 4 12 14 0 -3 8 7 13 -1 11 6 2 10 5 1 3 -2 -4 7 4 14 12 0 -1 8 9 13 -3 11 5 3 12 6 -1 2 -4 -2 4 7 14 10 0 1 8 13 9 -3 11 1 7 8 9 4 -2 0 -4 -3 12 14 13 5 -1 2 6 10 3 12 5 2 8 6 1 4 -1 -3 7 3 13 11 0 -2 10 9 14 -4 12 5 2 11 6 -1 3 -4 -2 4 7 14 10 1 0 8 13 9 -3 List of Solutions for -9,-8,..., 9 with magic constant 0: -9 1 8 6 5 -4 -7 3 -8 -3 9 -1 2 4 0 -6 -5 -2 7 -8 9 -1 3 6 -2 -7 5 -9 0 -4 8 -6 2 7 -3 1 4 -5 -8 7 1 3 4 2 -9 5 -7 0 -6 8 -4 -2 9 -3 -1 6 -5 -8 -1 9 2 5 -4 -3 6 -7 0 7 -6 3 4 -5 -2 -9 1 8 -8 -1 9 7 4 -5 -6 1 -9 3 8 -3 6 0 -4 -2 -7 2 5 -7 8 -1 3 1 2 -6 4 -9 6 -8 7 0 -3 5 -2 -4 9 -5 -7 6 1 2 8 -2 -8 5 -5 -6 -1 7 -9 3 9 -3 4 0 -4 -7 -1 8 5 4 -3 -6 2 -9 0 9 -2 6 3 -4 -5 -8 1 7 -6 -1 7 8 -3 4 -9 -2 -5 0 5 2 9 -4 3 -8 -7 1 6 -5 1 4 8 -2 3 -9 -3 -8 6 0 5 9 -7 2 -4 -6 7 -1 -5 -2 7 1 8 -6 -3 4 -9 0 9 -4 3 6 -8 -1 -7 2 5 -5 -4 9 4 6 -3 -7 1 -9 2 8 -2 7 0 -6 -1 -8 5 3 -4 -5 9 1 7 -2 -6 3 -8 0 8 -3 6 2 -7 -1 -9 5 4 -3 -5 8 1 6 0 -7 2 -8 -2 9 -1 7 3 -6 -4 -9 4 5 2 -7 5 3 6 0 -9 -5 -8 1 8 4 9 -2 -6 -1 -4 7 -3 3 -7 4 1 6 2 -9 -4 -8 0 7 5 9 -1 -6 -2 -5 8 -3 3 -7 4 2 6 1 -9 -5 -8 0 8 5 9 -1 -6 -2 -4 7 -3 3 -7 4 1 6 2 -9 -4 -8 -1 8 5 9 0 -6 -3 -5 7 -2 3 -8 5 1 6 2 -9 -4 -7 0 7 4 9 -2 -6 -1 -5 8 -3 3 -8 5 2 6 1 -9 -5 -7 0 8 4 9 -2 -6 -1 -4 7 -3 4 -6 2 3 5 1 -9 -7 -8 0 8 7 9 -1 -5 -3 -2 6 -4 5 -4 -1 3 6 -2 -7 -8 -9 0 9 8 7 2 -6 -3 1 4 -5 5 -6 1 3 4 2 -9 -8 -7 0 7 8 9 -2 -4 -3 -1 6 -5 5 -8 3 -1 2 8 -9 -4 -3 -6 7 6 9 0 -2 -7 -5 4 1 6 -4 -2 3 5 -1 -7 -9 -8 0 8 9 7 1 -5 -3 2 4 -6 6 -7 1 2 3 4 -9 -8 -5 0 5 8 9 -4 -3 -2 -1 7 -6 List of Solutions for 0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9 with magic constant 18: 1 9 8 9 3 1 5 8 0 3 2 5 6 2 4 6 4 7 7 1 8 9 9 4 2 3 8 0 1 3 6 6 2 5 5 4 7 7 1 8 9 8 4 3 3 9 1 0 2 6 5 2 6 5 4 7 7 2 8 8 7 4 4 3 9 0 1 1 7 6 2 5 5 3 9 6 3 8 7 9 3 4 2 6 0 1 2 9 7 1 6 4 5 8 5 5 7 6 9 5 1 3 4 0 2 3 9 6 1 4 7 8 8 2 7 6 5 5 7 2 4 6 1 0 2 9 4 3 3 8 8 9 1 9 4 5 7 5 2 4 2 1 0 6 9 8 3 1 6 8 7 3 List of Solutions for 0, 1, 2,...,19 without 15 with magic constant 35: 2 14 19 16 8 1 10 17 0 3 9 6 13 7 4 11 5 12 18 6 12 17 18 4 3 10 11 0 9 7 8 19 1 2 13 5 16 14 7 19 9 16 10 1 8 12 0 3 2 18 6 5 11 13 17 14 4 11 8 16 10 17 2 6 14 1 0 7 13 9 5 3 18 12 19 4 12 6 17 16 10 5 4 7 0 3 11 14 19 1 2 13 9 18 8 List of Solutions for 0, 1, 2,...,19 without 10 with magic constant 36: 1 19 16 18 7 5 6 17 2 0 3 14 8 4 15 9 11 12 13 2 18 16 15 6 8 7 19 0 3 1 13 12 4 11 9 5 17 14 3 16 17 15 6 8 7 18 0 5 1 12 14 2 9 11 4 19 13 11 12 13 9 17 2 8 16 1 0 4 15 6 7 5 18 14 19 3 15 9 12 13 14 4 5 8 2 0 7 19 11 3 6 16 17 18 1 17 8 11 12 14 4 6 7 1 0 9 19 13 5 3 15 16 18 2 List of Solutions for 0, 1, 2,...,19 without 5 with magic constant 37: 9 10 18 17 13 4 3 11 0 2 8 16 14 1 7 15 12 19 6 11 14 12 17 13 1 6 9 0 2 7 19 10 4 8 15 18 16 3 18 9 10 15 11 3 8 4 1 0 13 19 16 7 2 12 17 14 6 List of Solutions for subsets of 1, 2,...,21 with magic constant 39: 9 12 18 13 7 8 11 17 1 6 5 10 19 4 2 14 3 21 15 9 12 18 16 5 8 10 14 1 7 6 11 21 2 3 13 4 20 15 15 14 10 19 3 6 11 5 1 8 7 18 21 2 4 12 13 17 9 List of Solutions for subsets of 1, 2,...,21 with magic constant 40: 3 19 18 21 5 2 12 16 1 7 6 10 15 4 8 13 9 14 17 10 13 17 12 6 14 8 18 2 1 4 15 19 5 7 9 3 21 16 12 20 8 19 6 4 11 9 1 7 2 21 13 3 10 14 18 17 5 19 10 11 14 12 6 8 7 1 2 9 21 17 5 3 15 16 20 4 List of Solutions for 0, 1 with magic constant 1: 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 List of Solutions for 0, 1 with magic constant 2: 0 1 1 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 List of Solutions for 0, 1 with magic constant 3: 1 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 1 References: K. Abraham; Philadelphia Evening Bulletin. July 19, 1963, p. 18 and July 30, 1963. John Baker; Hexagonia, http://naturalmaths.com.au/hexagonia/ Section: Magic Hexagons Hans F. Bauch; Zum magischen Sechseck von Ernst v. Haselberg, wissenschaft und fortschritt 40:9 (1990) 240-242, und 4. Umschlagseite Hans F. Bauch; Magische Figuren in Parketten, Mathematische Semesterberichte 38:1 (1991) 99-115 Zbl 0749.05029 Regular tilings (triangle-tilings, square-tilings, hexagon-tilings) are considered. A generalization of magic squares in square-tilings is discussed for triangle-tilings and hexagon-tilings. Necessary and sufficient conditions for the existence of magic figures in regular tilings are proved. Beeler, M. et al. Item 49 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 18, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item49. Elwyn R. Berlekamp, John H. Conway, Richard K. Guy; Winning Ways, for your mathematical play, Vol 2, (Part 4: Diamond) Academic Press, London, 1982 (german: Gewinnen, Strategien für mathematische Spiele, Band 4: Solitairspiele, Vieweg Verlag, Braunschweig, 1985) Martin Gardner; Permutations and Paradoxes in Combinatorial Mathematics, Scientific American 209, 112-119, Aug. 1963. Martin Gardner; The Sixth Book of Mathematical Games from Scientific American, Chicago, IL: University of Chicago Press, 1984. Chapter 3: "Combinatorial Theory", pp. 22-24 Martin Gardner; Time Travel and Other Mathematical Bewilderments, New York: W. H. Freeman, 1988. Chapter 2: "Hexes and Stars", pp. 15-25 Martin Gardner; Correspondence: The history of the magic hexagon, The mathematical gazette 72 (June 1988) 133 Harvey D. Heinz; More Magic Squares, http://www.geocities.com/~harveyh/moremsqrs.htm Section: Magic Hexagons crediting W. Radcliffe, Isle of Man, U.K. with this discovery in 1895. Ernst von Haselberg; Manuscript 1887 problem and solution of the unique magic hexagon of order 3. Ernst von Haselberg; Section 795: Zeitschrift für mathematische und naturwissenschaftlichen Unterricht 19 (1888) 429 Aufgabe Section 801: Zeitschrift für mathematische und naturwissenschaftlichen Unterricht 20 (1889) 263-264 Auflösung John Hendricks; A magic Square Course, page 7, credits H. Lulli Heinrich Hemme; Das magische Sechseck, Bild der Wissenschaft (Oktober 1988) 164-166 Reprinted as Section 1.6: Das magische Sechseck, p36-41 Thiagar Devendran (editor) Das Beste aus dem Mathematischen Kabinett, Deutsche Verlag-Anstalt, Stuttgart 1990, ISBN 3-421-02758-7 Heinrich Hemme; Mathematik zum Frühstück, Vandenhoeck & Ruprecht, Göttingen, 1990 Problem 88: Das magische Sechseck, p44, 107-108 - magic hexagons of order 2. Ross Honsberger; Mathematical Gems I, The Dolciani Mathematical Expositions No. 1 The Mathematical Association of America, 1973 (german: Mathematische Edelsteine, Vieweg, 1981) Chapter 6: its combinatorics that counts! Section 6.2: Clifford W. Adams, pp. 69-76 David King; Hall of Hexagons, http://www.drking.plus.com/hexagons/index.html Section: Magic Hexagons Frank R. Kschischang; The Magic Hexagon, http://www.comm.toronto.edu/~frank/hexagon/ september 2000 - proof that the magic constant is integral for n=3 and 1 only. Joseph S. Madachy; Madachy's Mathematical Recreations. New York: Dover, pp. 100-101, 1979. (reprint: Mathematics on Vacation, 1966) Clifford A. Pickover; The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions Princeton University Press, 2002 - Magic hexagons, 139, 325-340 William Radcliffe; Magic Hexagon, 1895 http://www.johnrausch.com/PuzzleWorld/puz/magic_hexagon.htm Frank Tapson; The magic hexagon: an historical note, The mathematical gazette 71 (October 1987) 217-220 Charles W. Trigg; A Unique Magic Hexagon, Recreational Mathematics Magazine (January-February 1964) 40-43 http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/magic-hexagon-trigg Charles W. Trigg; P824: A Well-known Magic Hexagon, Mathematics Magazin 45 (March 1972) problem by Paul S. Lemke Mathematics Magazin 46 (Jan.-Feb. 1973) 44-45 solution by C. W. Trigg Tom Vickers; note 2799. Magic hexagon, The mathematical gazette 42 (December 1958) 291 shows the magic hexagon without comment Eric W. Weisstein; Magic Hexagon, From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MagicHexagon.html -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/