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Ordering Numbers Homework Year 1984

A magic hexagon of order is an arrangement of close-packed hexagons containing the numbers 1, 2, ..., , where is the th hex number such that the numbers along each straight line add up to the same sum. (Here, the hex numbers are i.e., 1, 7, 19, 37, 61, 91, 127, ...; OEIS A003215). In the above magic hexagon of order , each line (those of lengths 3, 4, and 5) adds up to 38.

It was discovered independently by Ernst von Haselberg in 1887 (Bauch 1990, Hemme 1990), W. Radcliffe in 1895 (Tapson 1987, Hemme 1990, Heinz), H. Lulli (Hendricks, Heinz), Martin Kühl in 1940 (Gardner 1963, 1984; Honsberger 1973), Clifford W. Adams, who worked on the problem from 1910 to 1957 (Gardner 1963, 1984; Honsberger 1973), and Vickers (1958; Trigg 1964).

This problem and the solution have a long history. Adams came across the problem in 1910. He worked on the problem by trial and error and after many years arrived at the solution which he transmitted to M. Gardner, Gardner sent Adams' magic hexagon to Charles W. Trigg, who by mathematical analysis found that it was unique disregarding rotations and reflections (Gardner 1984, p. 24). Adams' result and Trigg's work were written up by Gardner (1963). Trigg (1964) did further research and summarized known results and the history of the problem.

the first few of which are 1, 28/3, 38, 703/7, 1891/9, 4186/11, ... (OEIS A097361 and A097362), which requires to be an integer for a solution to exist. But this is an integer for only (the trivial case of a single hexagon) and Adams's (Gardner 1984, p. 24).

SEE ALSO:Hex Number, Hexagon, Magic Graph, Magic Hexagram, Magic Square, Pascal's Theorem, Talisman HexagonREFERENCES:

Abraham, K. Philadelphia Evening Bulletin. July 19, 1963, p. 18 and July 30, 1963.

Bauch, H. F. "Zum magischen Sechseck von Ernst v. Haselberg." Wissenschaft und Fortschritt40, 240-242 and 4th page of dustjacket, 1990.

Bauch, H. F. "Magische Figuren in Parketten." Math. Semesterber.38, 99-115, 1991.

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.

Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982.

Gardner, M. "Permutations and Paradoxes in Combinatorial Mathematics." Sci. Amer.209, 112-119, Aug. 1963.

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 22-24, 1984.

Gardner, M. "Hexes and Stars." Ch. 2 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 15-24, 1988.

Heinz, H. D. "More Magic Squares." http://www.magic-squares.net/moremsqrs.htm.

Hemme, H. "Das magische Sechseck." Bild der Wissenschaft, 164-166, Oct. 1988. Reprinted as "Das magische Sechseck." §1.6 in Das Beste aus dem Mathematischen Kabinett (Ed. T. Devendran). Stuttgart, Germany: Deutsche Verlag-Anstalt, pp. 36-41, 1990.

Hemme, H. "Das magische Sechseck." Problem 88 in Mathematik zum Frühstück. Göttingen, Germany: Vandenhoeck & Ruprecht, p. 44, 1990.

Hendricks, J. "A Magic Square Course." p. 7.

Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 69-76, 1973.

Kschischang, F. R. "The Magic Hexagon." Sept. 2000. http://www.comm.toronto.edu/~frank/hexagon/proof.html.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 100-101, 1979.

Pickover, C. A. "The Magic Hexagon." §139 in The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press, pp. 325-340, 2002.

Radcliffe, W. "Magic Hexagon." 1895. http://www.johnrausch.com/PuzzleWorld/puz/magic_hexagon.htm.

Sloane, N. J. A. Sequences A003215/M4362, A097361, and A097362 in "The On-Line Encyclopedia of Integer Sequences."

Tapson, F. "The Magic Hexagon: An Historical Note." Math. Gaz.71, 217-229, Oct. 1987.

Trigg, C. W. "A Unique Magic Hexagon." Recr. Math. Mag.46, 40-43, Jan./Feb. 1964. http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/magic-hexagon-trigg.

Trigg, C. W. "P824: A Well-Known Magic Hexagon." Math. Mag.45, 100, 1972.

Trigg, C. W. "Solution to Problem P824." Math. Mag.46, 44-45, 1973.

Vickers, T. "Magic Hexagon." Math. Gaz.42, 291, Dec. 1958.

von Haselberg, E. "Problem and Solution of the Unique Magic Hexagon of Order 3." Manuscript, 1887.

von Haselberg, E. "Aufgabe." §795 in Zeitschrift für mathematische und naturwissenschaftlichen Unterricht19, 429, 1888.

von Haselberg, E. "Auflösung." §801 in Zeitschrift für mathematische und naturwissenschaftlichen Unterricht20, 263-264, 1889.

Referenced on Wolfram|Alpha: Magic HexagonCITE THIS AS:

Weisstein, Eric W. "Magic Hexagon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MagicHexagon.html

Roman Numerals

Ancient Romans used a special method of showing numbers

Examples: They wrote V instead of 5
And wrote IX instead of 9

Read on to learn about Roman Numerals or go straight to the Roman Numeral Conversion Tool.

The Roman Symbols

Romans Numerals are based on the following symbols:

1

5

10

50

100

500

1000

I

V

X

L

C

D

M

Basic Combinations

Which can be combined like this:

1

2

3

4

5

6

7

8

9

I

II

III

IV

V

VI

VII

VIII

IX

 

10

20

30

40

50

60

70

80

90

X

XX

XXX

XL

L

LX

LXX

LXXX

XC

 

100

200

300

400

500

600

700

800

900

C

CC

CCC

CD

D

DC

DCC

DCCC

CM

Forming Numbers - The Rules

When a symbol appears after a larger (or equal) symbol it is added

  • Example: VI = V + I = 5 + 1 = 6
  • Example: LXX = L + X + X = 50 + 10 + 10 = 70

But if the symbol appears before a larger symbol it is subtracted

  • Example: IV = V − I = 5 − 1 = 4
  • Example: IX = X − I = 10 − 1 = 9

To Remember: After Larger is Added

Don't use the same symbol more than three times in a row (but IIII is sometimes used for 4, particularly on clocks)

How to Convert to Roman Numerals

Break the number into Thousands, Hundreds, Tens and Ones, and write down each in turn.

Example: Convert 1984 to Roman Numerals.

Break 1984 into 1000, 900, 80 and 4, then do each conversion

  • 1000 = M
  • 900 = CM
  • 80 = LXXX
  • 4 = IV

1000 + 900 + 80 + 4 = 1984, so 1984 = MCMLXXXIV

How To Remember

Think "MeDiCaL XaVIer".
It has the roman numerals in descending order from 1000 to 1.

I, for one, like Roman numerals!

Really Big Numbers

Numbers greater than 1,000 are formed by placing a dash over the symbol, meaning "times 1,000", but these are not commonly used:

5,000

10,000

50,000

100,000

500,000

1,000,000

V

X

L

C

D

M

Conversion Tool

You can convert to/from Roman Numerals here:

 

 

Decimal Numbers

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