Jon Wharf

Jon Wharf	Introduces himself	November 8, 2003
Order-12a stars	Quantity for each top point value	November 8, 2003
Confirmation of orders 6-11	and some transformations	November 10, 2003
Some order-13 solutions	One each for patterns b, c, d, e.	November 10, 2003
Order 15-20 solutions	One for each pattern	November 10, 2003
Order-14	One solution for each of 5 patterns	November 11, 2003
Comparison of orders	And comments on transformations	November 16, 2003

The first email I received from Jon Wharf was on November 8, 2003. This was a coincidence because it was at that time I was also corresponding with another magic star investigator (Simon Whitechapel). And we are a rare breed!

Jon has confirmed my count of basic solutions for all patterns of orders 5 to 11. He has also found all solutions for the four patterns of order 12.
And he has made some interesting observations on transformations between patterns of a particular order.

Notice how fast his routine is compared with mine. See my times at 'Points, patterns, and total solutions' on my star definitions page.

I enjoyed your website very much. I rediscovered it after solving an "IBM-Ponder This" puzzle on an order-6 magic star. (Good old Google!)

Prior to discovering your site I had established that for the order -6 star and by extension for all other even type-a stars the alternate outer points must sum to the same value. This is why there are no 6-stars with the points having the lowest (or highest) 6 values, and this will be true for all type-a stars with #points = 2 mod 4, since the relevant triangular number is odd and so can't be split into two equal groups. (This will also apply to this size of star for other types which consist of an even number of independently traceable figures). I don't have a reason for the 8a star however - I may think about it a little more...

I reviewed your table of star numbers on your Definitions page, and took a clue from there to produce a much-improved algorithm for finding stars (hugely better for 'a' type). This means for example that finding all the 10a stars takes about 15 minutes, running on a setup which should be not much different in speed from yours. (My computer is 433Mhz but I'm using an Excel macro which is interpreted so should be slower).

You're welcome to the code of course; the central idea is that the evaluation of the different nodes is done in an order which helps to eliminate bad configurations as early as possible. More than half the code is concerned with setting up the evaluation order and associated housekeeping. There may still be opportunities to improve this.

Harvey
There are 396930 type 12a magic stars, with the following number of stars with the minimum (top) point:

top point #stars
1 207027
2 88456
3 45823
4 24378
5 13537
6 7660
7 4312
8 2328
9 1439
10 718
11 626
12 286
13 340

Processing time 5 hours 50 min. The output file has three characters per node and is 47,291 kbyte, zips to 7,488 kbyte. My own ordering of the nodes is clockwise around the outer points then clockwise around the valleys. (I could easily write a reordered file). Let me know if you want the file.
Jon

An important email came from Jan on November 10, 2003.
In it he confirmed my number of solutions for each order and pattern from 6 to 11.
See my ‘Points, Patterns and Total solutions in star definitions page to compare his search times with mine.

Yes I generated all the lower order stars for 6, 7, 8, 9 points, 10a, 10b,11a, 11b, with numbers that match yours.
For interest the speeds were:
9a: 40 sec
9b: 4 min
9c: 7 min
10a: 5 min
10b: 40 min
10c: not generated (=10a see below)
11a: 40 min
11b: 6 hours
11c,d: not generated (=11a,b respectively, see below)
12a: 5 hours 50min

Also, see attached pictures, I borrowed some diagrams from your web site to illustrate how to transform 10a stars into 10c, 11a stars into 11c and 11b stars into 11d. In each case I've used the labels from the first-named star type to show how those number fit into the second-named type.

For the 10a-10c transform and the 11b-11d transform the points stay points,so the top-point quantities should match. For the 11a-11c transform the points flip to valleys, so the 11a stars with top point = 1 should be the same quantity as the 11c stars with top point <> 1 (and vice versa). OR to put it another way, the total stars should be the sum of the number of 11a 1-stars and the 11c 1-stars.

Labels are from 10a

Labels are from 11a

Labels are from 11b

I shall check the geometry of the 12x stars and see what relationships we can expect there.

Also attached is the guts of the code - I've trimmed out some of the extra stuff which writes intermediate stuff to the spreadsheet. As you'll see if you try it, it doesn't produce stars in the order you're looking for. I shall consider how they might be reordered. To change from one type of star to another, change the "Points" constant at the top and "Skip" constant a few lines down. "Skip" is equal to 2 for type a stars, 3 for type b stars, 4 for type c, etc. Not too hard to change!

Thanks for your 2 emails. It is really encouraging to find others that are also interested in the subject of magic stars.
As you are probably aware, there has been very little published work on magic stars.
The work I have done (with results on my magic stars web pages) was completed about the end of 1998.

After receiving your messages, I dug out my old notes, because I was surprised that you found only 396930 basic solutions to the 12A star. I had estimated about 800,000 solutions.
However, I realize now that that number was based on the solutions for the 12B star, which is the only one I actually completed.
For the 12A star, I have only 257830 solutions. the last one starts off with numbers 2, 10, 20, 18 in the first line (my method of numbering).

My list confirms that there are 207,027 solutions that start with the number 1!!!! Unfortunately, my search did not reach the end of solutions starting with 2.

I am amazed at how fast you program runs!
If I read my notes correctly, it took over 3 months to find the first 257830 solutions.
I would appreciate more details on your program.

My programs produce the solutions in index order. When a program has found all the solutions for a particular order, I then run another program that reads the data file and finds the complement of each solution. (This is another solution already in the list.) One reason for this is to confirm I have all the solutions, and no duplicates.
This program also numbers the complement pair. and also totals the sum of the points.
The program then produces a new data file that includes the
-- Solution number (index order as explained on my web site)
-- Complement solution number
-- Solution pair number
-- Point total
-- Cell values shown in order as per my web page.

>> Are there any stats you would like me to collect when I start on the 12a stars?

It would be nice if you could convert your data files to the same format as mine are, so we can do a direct comparison. I have the impression that you like programming, so probably would appreciate the challenge!

First though. Have you searched for the complete set of basic solutions for the smaller orders. Number of solutions for orders 6, 7 and 8 were confirmed by many people in the 1960's. If you come up with the same totals, it is confirmation that your program has no bugs.

BTW How easy is it to convert from 1 order to another?
From 1 pattern to another in the same order?
My program is designed so it is relatively easy to do these conversions by copying the previous order, then simple editing (although it can take several hours for each conversion by the time I get to order 12).
So I end up with a lot of different programs, not just 1 general program.

Several years age I was contacted by Simon Whitechapel. He had started a search for stars of type A that were larger then my order 14 stars.
I posted a page on these at http://www.magic-squares.net/bigstars.htm By coincidence, I had an email from him on Nov. 1/03. He reported that he had made no progress with the other patterns but had found solutions for pattern A for orders 15 to 100. Also a few other isolated orders, for example order 166 which has the magic constant 666!
I have no idea how his program works.

(Points, clockwise, first, then valleys starting with the valley between point 1 and point 2)

Harvey
Getting carried away here... Here's the "Big stars" (15-20 points) requested, at least those that give the first solution quickly:

(Points first, clockwise, then valleys starting with the valley between point 1 and point 2)

15b:
1 7 2 3 12 4 20 9 23 11 17 13 5 8 6 28 25 30 18 26 21 15 14 16 22 10 24 27 19 29

15d:
1 2 6 16 4 3 12 13 19 17 8 5 15 14 9 28 23 22 20 30 25 21 7 11 26 24 27 18 10 29

15e:
1 8 19 6 2 7 3 9 14 5 11 10 23 17 4 28 21 13 25 22 30 24 16 26 27 15 12 20 18 29

16c:
1 11 7 2 3 14 8 4 17 9 12 13 5 10 16 6 30 21 27 32 20 24 28 26 19 18 23 25 29 15 22 31

16d:
1 11 19 8 2 3 18 7 10 4 14 5 9 12 15 6 30 13 23 27 32 24 17 21 28 25 26 29 16 22 20 31

16e:
1 5 9 23 14 2 3 8 20 13 6 4 7 10 17 12 30 24 15 19 21 32 29 22 11 25 28 27 26 18 16 31

16f:
1 12 6 2 7 4 10 3 16 5 9 19 24 11 23 8 30 20 27 26 25 17 32 18 28 29 15 21 13 22 14 31

17b:
1 7 2 3 8 4 13 10 18 14 17 16 9 27 5 19 6 32 29 34 26 30 28 22 20 21 23 15 24 11 25 31 12 33

17c:
1 10 7 2 3 12 9 4 24 14 8 27 16 5 13 17 6 32 26 29 34 22 25 30 21 19 23 18 11 28 31 15 20 33

17e:
1 4 11 16 24 2 3 9 10 20 8 5 6 7 13 15 14 32 27 23 12 21 34 30 26 22 17 29 31 28 25 18 19 33

17f:
1 4 15 6 2 21 7 3 19 12 5 9 14 11 17 13 20 32 23 27 29 28 10 34 24 16 30 31 25 18 26 22 8 33

18b:
1 7 2 3 8 4 13 9 11 14 20 15 27 10 23 5 12 6
34 31 36 28 32 30 25 29 22 16 26 17 18 19 24 33 21 35

18c:
1 12 7 2 3 15 8 4 23 11 10 14 13 22 5 9 25 6
34 26 31 36 21 28 32 27 20 24 29 18 17 30 33 19 16 35

18d:
1 8 13 20 2 3 9 16 6 4 12 14 17 5 7 10 18 11
34 28 21 26 36 29 24 22 32 30 27 19 31 33 25 23 15 35

18e:
1 2 4 18 11 6 3 12 15 20 17 19 8 5 16 7 23 9
34 29 28 26 24 36 31 27 10 22 13 32 30 33 25 21 14 35

18f:
1 8 12 17 6 2 4 3 10 25 9 5 11 13 24 22 15 7
34 29 14 30 31 28 36 27 23 18 32 33 21 20 19 16 26 35

18g:
1 7 6 2 8 25 15 12 3 11 5 9 23 13 14 4 18 10
34 29 31 30 19 20 17 36 27 32 33 24 16 28 22 26 21 35
(a full set on the 18 stars!)

19b:
1 7 2 3 8 4 13 9 11 15 10 20 19 26 17 29 5 14 6
36 33 38 30 34 32 27 31 23 28 24 16 18 25 12 22 35 21 37

19c:
1 15 7 2 3 12 8 4 14 13 9 20 10 21 26 5 16 28 6
36 19 33 38 32 30 34 29 23 27 25 31 17 18 22 35 11 24 37

19e:
1 12 21 23 7 2 3 24 10 13 8 4 15 5 11 16 9 25 6
36 14 29 20 33 38 28 18 22 30 34 32 31 35 19 27 17 26 37

19f:
1 5 9 13 18 31 2 3 8 12 15 24 6 4 7 10 17 22 19
36 30 29 27 11 20 38 35 28 23 25 21 34 33 32 26 16 14 37

19g:
1 4 23 6 2 17 13 8 3 27 16 5 9 18 24 12 19 30 7
36 22 28 33 32 14 26 38 25 11 34 35 29 15 20 31 10 21 37

19h: not quick

20b:
1 7 2 3 8 4 13 9 11 10 12 17 18 25 19 16 22 5 14 6
38 35 40 32 36 34 29 33 30 28 24 26 21 20 27 15 31 37 23 39

20d:
1 12 13 7 2 3 18 19 8 4 14 10 11 9 16 5 15 25 22 6
38 21 26 35 40 31 24 32 36 34 30 20 29 28 33 37 17 23 27 39

20f:
1 4 11 15 26 30 2 3 9 10 16 17 8 5 6 7 13 14 22 21
38 33 29 23 12 20 40 36 32 28 27 24 35 37 34 31 25 19 18 39

20g:
1 4 17 25 6 2 8 15 3 16 18 10 5 9 22 14 23 12 11 7
38 30 19 27 35 34 21 40 32 28 20 36 37 31 13 24 26 33 29 39

Yes, of course you may write a web page on this material. I'm flattered that you
offered. I would prefer that the material on magic stars was in one place in any case.

I started 12b stars but 10 hours only got me about one quarter through the
1-stars. Much slower than the 12a stars. I'll have to write a restart routine.

Reviewing shared points between lines on various sizes of magic stars, I
believe the following star configuration pairs have a transform and so have
the same number of variations. Where I can I have indicated whether the
transform is point-preserving (P) or inverting (I) (where points become valleys):

(7a,7b) P
(8a,8b) I
(9b,9c) I
(10a,10c) P
(10b,10b) I (so every 10b star has a point-to-valley companion)
(11a,11c) I
(11b,11d) P (a little more difficult to tell without building the star)
12-point stars - can't find any transforms, although a&d are "related" as
are b&c in the pattern of point-sharing
(13a,13d) P
(13c,13e) I
(14a,14d) I
(14b,14e) P??
(14c,14c) I? - will this also have a huge number of solutions like 10b?
(15c,15f) P?
(15d,15e)) I

A point-preserving transform will lead to identical numbers of stars with
top point 1, 2, 3 etc. An inverting transform will lead to the two sets of
1-stars totaling to the complete number of stars. A self-inverting
transform (10b) will lead to the number of 1-stars being exactly half the
total number of stars.

I will try to work out the details of these star-type pair transforms - it
may be that some don't exist.

Refer to my Example stars-2 for 1 solution of each pattern of orders 12a to 14e.

Also my Bigstar page for pattern diagrams of the stars of orders 15 to 20.
(You may match a solution on this page to the corresponding pattern if you wish.)