Challenge - find the magic octagon

Dakirel · 2014-11-09 08:55:40

After watching a fantastic video Numberfile made on magic hexagons (watch it HERE) I started doing my own research on if and how a magic octahedron could work. Using the same methods Numberphile used it turns out yes - you can! And, just like with magic hexagons, there appears to be just one special case other than the trivial n=1. That is.. [drum roll please]... n=5, resulting in a whooping 129 cells!

Encase you're unfamiliar with magic squares and magic hexagons, that means you have to fill the below grid with the numbers 1 to 129 so that all 13 rows, 13 columns, and 26 diagonals add up to 645. Yep, 645. This is for only the most dedicated mathematicians only. On the plus side, after extensive Googling, I don't think anyone has ever found the solution before... so if you find the solution, you'll be crowned to first person to solve the magic octagon, of all of time!... probably.

If you think you've got a solution use the layout below or an equivalent to fill in the blanks and repost as a comment on this thread. As to how to find one... if I knew that, I'd solve it myself. Though I'd highly recommend watching the Numberphile video I poster above and seeing how they found the solution to the magic hexagon first, and I'm sure this solution won't be far to find.

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Last edited by Dakirel (2014-11-09 09:14:59)

phrontister · 2014-11-13 04:40:55

Hi Dakirel,

Dakirel wrote:

This is for only the most dedicated mathematicians only.

Nevertheless I thought I might give it a go.

I haven't got far with it yet, but I've spotted a couple of things...

1. Missing diagonals:

You mentioned that there are 26, but I think there are 34: 17 left-pointing + 17 right-pointing. Maybe 26 being 2 × 13, and that there are 13 rows and 13 columns, played a part in that.

I suspect the 'missing' diagonals are those indicated by the red arrows in the image:

As each of the four groups (rows, columns, left-pointing diagonals and right-pointing diagonals) capture all cells, they should have the common total that is the sum of all of the cells in the octagon. That total is the triangular number 8385, from:

8385 also = 13 × 645, with 13 being the number of rows and columns, and 645 the magic number you gave.

2. Magic number too great:

I think the magic number has to be less than 645 to obey the rule that all 13 rows and 13 columns must equal the magic number, as the distribution of the 20 highest different numbers in the four 5-celled row and column groups = 2390:

However, 645 × 4 = 2580.

The magic number can't be greater than:

That greatest-possible number is reduced to 579 when taking into account the four 5-celled left- and right-pointing diagonals, with highest-possible values being located at the end cells where doubling-up occurs at intersections with rows and columns:

I might even hazard a guess and say that the magic number could be around the 559 mark.

EDIT:
Maybe I should explain...
559 is a very nice number that fits in well with some figuring:
1. Using any number placement (no matter how wild), it's the average of the sum of all 60 totals from the 13 rows, 13 columns and 34 diagonals. This tactic works for the hexagon from the video, which was an exciting and encouraging discovery. The fact that it didn't work for the magic squares I tried it on (3x3, 4x4 and 5x5) doesn't bother me at all, so I'll just ignore it. Lucky, otherwise it would spoil my theory!
2. 8385/15 = 559
(a) 8385 is the total of the octagon
(b) 15 is from 60/4
(i) 60 = the number of rows, columns and diagonals
(ii) 4 is some number to do with something or other (eg, there are 4 groups used for the '60' calculation)

That may not hold water, but you never know...

Last edited by phrontister (2017-02-25 22:29:23)

Dakirel · 2014-11-13 19:47:21

phrontister wrote:

Hi Dakirel,
Dakirel wrote:
This is for only the most dedicated mathematicians only.
Nevertheless I thought I might give it a go.
I haven't got far with it yet, but I've spotted a couple of things...
1. Missing diagonals:
You mentioned that there are 26, but I think there are 34: 17 left-pointing + 17 right-pointing. Maybe 26 being 2 × 13, and that there are 13 rows and 13 columns, played a part in that.
I suspect the 'missing' diagonals are those indicated by the red arrows in the image:
http://k003.kiwi6.com/hotlink/42h1263s9q/Magic_Octogon.png
As each of the four groups (rows, columns, left-pointing diagonals and right-pointing diagonals) capture all cells, they should have the common total that is the sum of all of the cells in the octagon. That total is the triangular number 8385, from:
8385 also = 13 × 645, with 13 being the number of rows and columns, and 645 the magic number you gave.
2. Magic number too great:
I think the magic number has to be less than 645 to obey the rule that all 13 rows and 13 columns must equal the magic number, as the distribution of the 20 highest different numbers in the four 5-celled row and column groups = 2390:
However, 645 × 4 = 2580.
The magic number can't be greater than:
That greatest-possible number is reduced to 579 when taking into account the four 5-celled left- and right-pointing diagonals, with highest-possible values being located at the end cells where doubling-up occurs at intersections with rows and columns:
I might even hazard a guess and say that the magic number could be around the 559 mark.
EDIT:
Maybe I should explain...
559 is a very nice number that fits in well with some figuring:
1. Using any number placement (no matter how wild), it's the average of the sum of all 60 totals from the 13 rows, 13 columns and 34 diagonals. This tactic works for the hexagon from the video, which was an exciting and encouraging discovery. The fact that it didn't work for the magic squares I tried it on (3x3, 4x4 and 5x5) doesn't bother me at all, so I'll just ignore it. Lucky, otherwise it would spoil my theory!
2. 8385/15 = 559
(a) 8385 is the total of the octagon
(b) 15 is from 60/4
(i) 60 = the number of rows, columns and diagonals
(ii) 4 is some number to do with something or other (eg, there are 4 groups used for the '60' calculation)
That may not hold water, but you never know...

Wow thanks for taking the time to respond! That method you used to calculate the maximum magic number is clever; I wish I thought of it! And I messed those extra diagonals too so thanks a lot for pointing them out for me.

Unfortunetly... these things make the magic octagon impossible. D:

First off (I'm not entirely versed on the mathematics on this so just trust Numberphile on this) that calculation can prove that if a solution exists, the magic number has to be 645... otherwise the maths doesn't hold up.

Second, and simpler to understand, we can prove from the diagonals that they can't share a common magic number with the straights. Using the straights we can see that the sum of every cell should be equal to 13 lots of the magic number. However using the diagonals we can count the same total as being 17 lots of the magic number... we can see a problem. The only way the total could be 13m and 17m at the same time is if m were 0, which it obviously isn't. Unfortunetly then we have to conclude that a solution would be impossible.

It's disappointing I'll admit, although at least we come to some kind of conclusion... it's a hell of a lot easier than correctly filling in all 129 cells would have been, that's for sure.

Last edited by Dakirel (2014-11-13 19:48:47)

phrontister · 2014-11-14 13:01:35

Hi Dakirel,

Second, and simpler to understand, we can prove from the diagonals that they can't share a common magic number with the straights...

Yes, that proof is really good and simple...and it applies to all sizes of octagons that have your cell configuration and rules.

Great pity about that, as I'd been looking forward to spending some months (minimum) of dedicated time trying to correctly fill the octagon! Most of that time would be taken up in working on formulas like those near the end of the video.

Last edited by phrontister (2014-11-14 13:03:47)

Math Is Fun Forum

#1 2014-11-09 08:55:40

Challenge - find the magic octagon

#2 2014-11-13 04:40:55

Re: Challenge - find the magic octagon

#3 2014-11-13 19:47:21

Re: Challenge - find the magic octagon

#4 2014-11-14 13:01:35

Re: Challenge - find the magic octagon

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