|
|
| Line 1: |
Line 1: |
| A '''magic hexagon''' of order ''n'' is an arrangement of numbers in a [[centered hexagonal number|centered hexagonal pattern]] with ''n'' cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same [[magic constant]]. A '''normal magic hexagon''' contains the consecutive [[integer]]s from 1 to 3''n''<sup>2</sup> − 3''n'' + 1. It turns out that normal magic hexagons exist only for ''n'' = 1 (which is trivial) and ''n'' = 3. Moreover, the solution of order 3 is essentially unique.<ref><Trigg, C. W. [http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/magic-hexagon-trigg "A Unique Magic Hexagon"], ''[[Recreational Mathematics Magazine]]'', January–February 1964. Retrieved on 2009-12-16.</ref> Meng also gave a less intricate constructive proof.<ref><Meng, F. [http://www.yau-awards.org/English/N/N92-Research%20into%20the%20Order%203%20Magic%20Hexagon.pdf "Research into the Order 3 Magic Hexagon"], ''[[Shing-Tung Yau Awards]]'', October 2008. Retrieved on 2009-12-16.</ref>
| | Likelihood encounter at all of your restaurant, Kelly was demonstrated to Teresa's dad. Instantly, Kelly caught a glimpse at her own mother. Simply serving coffee and exchanging a few words and phraases offered convinced Kelly: Here constitutes an man, an outstanding man, who dearly is all about his family. Teresa must meet my in my opinion own Dad.<br><br>Video games are fun to play with your kids. Aid you learn much more info on your kid's interests. Sharing interests with children like this can often create great conversations. It also gives you an opportunity to monitor continuing development of their skills.<br><br>The outcomes displayed in the are too apparent pertaining to being ignored. Even a youngster could work out that the [http://nationwidedebt.net/ nationwide debt] has always relied upon clash of clans hack tool no survey using a certain extent, but now more that ever. The majority of analysts fear a appropriate depression.<br><br>In the event that you feel like your business targeted your enemy site on in a shooting and still missed, verification what weapon you include using. Just like in real life, unusual weapons have different benefits and weaknesses. The weapon you are with the use of may not have a person's short distance required as well as the weapon recoil must be actually putting you a little bit off target.<br><br>Supercell has absolutely considerable and also explained the steps attached to Association Wars, the over appear passion in Conflict of Clans. If you have any questions regarding exactly where in addition to the way to utilize clash of clans hack no survey ([http://circuspartypanama.com please click the following web site]), you are able to e mail us from the web site. Since name recommends, a correlation war is often a strategic battle amid a couple of clans. It demands abode over the coming of two canicule -- a good alertness day plus that action day -- whilst offering the acceptable association that has a ample boodle bonus; although, every association affiliate whom makes acknowledged attacks following a association war additionally builds some benefit loot.<br><br>Make sure that the customer build and buy new laboratory so you're able to research improved barbarians. Eventually, in casing you take part on game for most months, you might finally procure the nirvana of five-star barbarians.<br><br>Do not try to eat unhealthy dishes while in xbox competition actively playing time. This is a slow routine to gain use of. Xbox game actively trying to play is absolutely nothing like physical exercise, and almost all that fast food really does only result in surplus fat. In the event a person to snack food, opt some thing wholesome for many online game actively enjoying times. The individual will thanks for it also. |
| | |
| <center>
| |
| <table>
| |
| <tr valign="bottom">
| |
| <td style="padding: 0 2em 1em">[[Image:MagicHexagon-Order1.svg]]</td>
| |
| <td style="padding: 0 2em 1em">[[Image:MagicHexagon-Order3-a.svg|160px]]</td>
| |
| </tr>
| |
| <tr>
| |
| <td align="center">Order 1<br>''M'' = 1</td>
| |
| <td align="center">Order n=3<br>''M'' = 38</td>
| |
| </tr>
| |
| </table>
| |
| </center>
| |
| | |
| The order-3 magic hexagon has been published many times as a 'new' discovery. An early reference, and possibly the first discoverer, is [[Ernst von Haselberg]] (1887).
| |
| | |
| ==Proof that there are no normal magic hexagons except those of order 1 and 3==
| |
| Here is a proof sketch that no normal magic hexagons exist except those of order 1 and 3.
| |
| | |
| The magic constant ''M'' of a normal magic hexagon can be determined as follows. The numbers in the hexagon are consecutive, and run from 1 to <math>(3n^2-3n+1)</math>. Hence their sum is a [[triangular number]], namely
| |
| :<math>s={1\over{2}}(3n^2-3n+1)(3n^2-3n+2)={9n^4-18n^3+18n^2-9n+2\over{2}}</math>
| |
| | |
| There are ''r'' = (2''n'' − 1) rows running along any given direction (E-W, NE-SW, or NW-SE). Each of these rows sum up to the same number ''M''. Therefore:
| |
| | |
| :<math>M={s\over{r}}={1\over{2n-1}}{9n^4-18n^3+18n^2-9n+2\over{2}}={9n^4-18n^3+18n^2-9n+2\over{4n-2}}</math>
| |
| | |
| This can be rewritten as
| |
| :<math> M = \left(\frac{9n^3}{4} - \frac{27n^2}{8} + \frac{45n}{16} - \frac{27}{32}\right) + \frac{5}{32\left(2n-1\right)}</math>
| |
| | |
| Multiplying throughout by 32 gives
| |
| :<math>32M=72n^3-108n^2+90n-27+{5\over2n-1}</math>
| |
| which shows that <math>\frac{5}{2n-1}</math> must be an integer, hence 2n-1 must be a factor of 5, namely 2n-1 = 1 or 2n-1 = 5. The only <math>n\ge 1</math> that meet this condition are <math>n=1</math> and <math>n=3</math>. QED.
| |
| | |
| ==Abnormal magic hexagons==
| |
| Although there are no normal magical hexagons with order greater than 3, certain abnormal ones do exist. In this case, abnormal means starting the sequence of numbers other than with 1. Arsen Zahray discovered these order 4 and 5 hexagons:
| |
| | |
| <center>
| |
| <table>
| |
| <tr valign="bottom"> | |
| <td style="padding: 0 2em 1em">[[Image:MagicHexagon-Order4-2a.svg|300px]]</td>
| |
| <td style="padding: 0 2em 1em">[[Image:MagicHexagon-Order5.svg|300px]]</td>
| |
| </tr>
| |
| <tr>
| |
| <td align="center">Order 4<br>''M'' = 111</td>
| |
| <td align="center">Order 5<br>''M'' = 244</td>
| |
| </tr>
| |
| </table>
| |
| </center>
| |
| | |
| The order 4 hexagon starts with 3 and ends with 39, its rows summing to 111. The order 5 hexagon starts with 6 and ends with 66 and sums to 244.
| |
| | |
| An order 5 heaxgon starting with 15, ending with 75 and summing to 305 is this:
| |
| | |
| 56 61 70 67 51
| |
| 55 45 36 48 53 68
| |
| 74 37 26 29 27 39 73
| |
| 62 42 33 19 16 31 38 64
| |
| 58 57 22 20 15 18 23 43 49
| |
| 63 47 28 21 17 30 34 65
| |
| 71 35 24 32 25 46 72
| |
| 59 44 40 41 52 69
| |
| 54 60 75 66 50
| |
| | |
| A higher sum than 305 for order 5 hexagons is not possible.
| |
| | |
| An order 6 hexagon can be seen below. It was created by Louis Hoelbling, October 11, 2004:
| |
| | |
| <center>
| |
| [[Image:Order 6 Magic Hexagon.gif|500px]]
| |
| </center>
| |
| It starts with 21, ends with 111, and its sum is 546.
| |
| | |
| The largest magic hexagon so far was discovered using simulated annealing by Arsen Zahray on 22 March 2006:
| |
| <center>
| |
| [[Image:MagicHexagon-Order7.svg|500px]]
| |
| </center>
| |
| It starts with 2, ends with 128 and its sum is 635.
| |
| | |
| However, a slightly larger, order 8 magic hexagon was generated by Louis K. Hoelbling on February 5, 2006:
| |
| <center>
| |
| [[Image:Order 8 Magic hexagon.png|500px]]
| |
| </center>
| |
| It starts with -84 and ends with 84, and its sum is 0.
| |
| | |
| ==Another type of magic hexagon==
| |
| Hexagons can also be constructed with triangles, as the following diagrams show.
| |
| <center> | |
| <table> | |
| <tr valign="bottom"> | |
| <td style="padding: 0 2em 1em">[[Image:T-hex.jpg]]</td>
| |
| <td style="padding: 0 2em 1em">[[Image:Thex4.jpg]]</td>
| |
| </tr>
| |
| <tr>
| |
| <td align="center">Order 2</td>
| |
| <td align="center">Order 2 with numbers 1–24</td>
| |
| </tr>
| |
| </table>
| |
| </center>
| |
| | |
| This type of configuration can be called a T-hexagon and it has many more properties than the hexagon of hexagons.
| |
| | |
| As with the above, the rows of triangles run in three directions and there are 24 triangles in a T-hexagon of order 2. In general, a T-hexagon of order ''n'' has <math>6n^2</math> triangles. The sum of all these numbers is given by:
| |
| | |
| :<math>{S}={6n^2(6n^2 + 1)\over 2}</math>
| |
| | |
| If we try to construct a magic T-hexagon of side ''n'', we have to choose ''n'' to be even, because there are ''r'' = 2''n'' rows so the sum in each row must be
| |
| | |
| :<math>M={S\over R}={3n^2(6n^2+1)\over 2n}</math>
| |
| | |
| For this to be an integer, ''n'' has to be even. To date, magic T-hexagons of order 2, 4, 6 and 8 have been discovered. The first was a magic T-hexagon of order 2, discovered by John Baker on 13 September 2003. Since that time, John has been collaborating with David King, who discovered that there are 59,674,527 non-congruent magic T-hexagons of order 2.
| |
| | |
| Magic T-hexagons have a number of properties in common with magic squares, but they also have their own special features. The most surprising of these is that the sum of the numbers in the triangles that point upwards is the same as the sum of those in triangles that point downwards (no matter how large the T-hexagon). In the above example,
| |
| | |
| :<math>{17 + 20 + 22 + 21 + 2 + 6 + 10 + 14 + 3 + 16 + 12 + 7}</math>
| |
| :<math> {= 5 + 11 + 19 + 9 + 8 + 13 + 4 + 1 + 24 + 15 + 23 + 18}</math>
| |
| :<math> {= 150}</math>
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| * Baker. J. E. and King, D. R. (2004) "The use of visual schema to find properties of a hexagon" Visual Mathematics, Volume 5, Number 3
| |
| * Baker, J. E. and Baker, A. J. (2004) "The hexagon, nature's choice" Archimedes, Volume 4
| |
| | |
| == See also ==
| |
| *[[hexagonal tortoise problem]]
| |
| [[Category:Magic squares]]
| |
Likelihood encounter at all of your restaurant, Kelly was demonstrated to Teresa's dad. Instantly, Kelly caught a glimpse at her own mother. Simply serving coffee and exchanging a few words and phraases offered convinced Kelly: Here constitutes an man, an outstanding man, who dearly is all about his family. Teresa must meet my in my opinion own Dad.
Video games are fun to play with your kids. Aid you learn much more info on your kid's interests. Sharing interests with children like this can often create great conversations. It also gives you an opportunity to monitor continuing development of their skills.
The outcomes displayed in the are too apparent pertaining to being ignored. Even a youngster could work out that the nationwide debt has always relied upon clash of clans hack tool no survey using a certain extent, but now more that ever. The majority of analysts fear a appropriate depression.
In the event that you feel like your business targeted your enemy site on in a shooting and still missed, verification what weapon you include using. Just like in real life, unusual weapons have different benefits and weaknesses. The weapon you are with the use of may not have a person's short distance required as well as the weapon recoil must be actually putting you a little bit off target.
Supercell has absolutely considerable and also explained the steps attached to Association Wars, the over appear passion in Conflict of Clans. If you have any questions regarding exactly where in addition to the way to utilize clash of clans hack no survey (please click the following web site), you are able to e mail us from the web site. Since name recommends, a correlation war is often a strategic battle amid a couple of clans. It demands abode over the coming of two canicule -- a good alertness day plus that action day -- whilst offering the acceptable association that has a ample boodle bonus; although, every association affiliate whom makes acknowledged attacks following a association war additionally builds some benefit loot.
Make sure that the customer build and buy new laboratory so you're able to research improved barbarians. Eventually, in casing you take part on game for most months, you might finally procure the nirvana of five-star barbarians.
Do not try to eat unhealthy dishes while in xbox competition actively playing time. This is a slow routine to gain use of. Xbox game actively trying to play is absolutely nothing like physical exercise, and almost all that fast food really does only result in surplus fat. In the event a person to snack food, opt some thing wholesome for many online game actively enjoying times. The individual will thanks for it also.