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
Notice how fast his routine is compared with mine. See my times at 'Points, patterns, and total solutions' on my star definitions page.
Jonís email of November 8, 2003
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.
Are there any stats you would like me to collect when I start on the 12a stars?
Email of November 8, 2003
top point #stars
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.
An important email came from Jan on November 10, 2003.
generated all the lower order stars for 6, 7, 8, 9 points, 10a, 10b,11a,
11b, with numbers that match yours.
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.
Having written that, I'll just check your site... 27223+26305=53528, yep.
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!
An email from me to Jon on November 10, 2003 and his reply the same day.
your 2 emails. It is really encouraging to find others that are also
interested in the subject of magic stars.
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.
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!
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.
>> 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?
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 look forward to hearing more from you on this subject.
And this is Jonís reply (the same day)
Oh yes, here is the first I generate for the 13b/c/d/e stars...
(Points, clockwise, first, then valleys starting with the valley between point 1 and point 2)
And another email the same day (Nov. 10, 2003)
(Points first, clockwise, then valleys starting with the valley between point 1 and point 2)
15c: not quick
15f: not quick
16b: not quick
17d: not quick
17g: not quick
19d: not quick
19h: not quick
20c: not quick
20e: not quick
20h: not quick
I thought I'd better
get those in before you figure out my methods, Watson.
This is Jonís email of November 11, 2003.
Yes, of course you may
write a web page on this material. I'm flattered that you
I started 12b stars
but 10 hours only got me about one quarter through the
An email from Jon on November 16, 2003
points between lines on various sizes of magic stars, I
transform will lead to identical numbers of stars with
I will try to work out
the details of these star-type pair transforms - it
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.