20180809, 17:45  #1 
6809 > 6502
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Aug 2003
101×103 Posts
23611_{8} Posts 
Mythic squares/rectangles
While we have heard of magic squares. I started to wonder about a different kind of square (or rectangle). Most of us have worked a word search in our day. I thought about a number search.
The question is what is the minimum size and proper configuration such that you can find within the square/rectangle all numbers from 1>n? This is using the conventional word search rules: straight lines only, continuous, either direction. The following is the minimum for 1>9. The configuration is irrelevant. Code:
1 2 3 4 5 6 7 8 9 Code:
5 9 3 0 6 1 2 1 7 8 4 1 I have put only a very little effort into this. And I have to be AFT for a few hours. Figured that it would be an interesting challenge, either by hand, programmatically, or in theory. 
20180810, 03:35  #2 
Mar 2006
2×3^{5} Posts 
I misunderstood your "straight lines only" rule and thought you meant only horizontal and vertical lines. Once i realized you also include diagonal lines, I see how your 4x3 could get from 1>21. I found a small improvement to get from 1>24 with:
Code:
5932 6124 7810 I really like this puzzle. I think the flat x,y grid will be able to get up to around n = 2*x*y and the torus x,y grid might be able to get up to around n = 4*x*y These are just wild guesses. Hopefully someone can come up with better bounds. I'm going to try running a python program to see if I can come up with some other/better solutions. 
20180810, 04:49  #3  
Jun 2003
2×23×113 Posts 
Quote:
Code:
9532 8124 7610 

20180810, 22:30  #4  
6809 > 6502
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Aug 2003
101×103 Posts
29×349 Posts 
Quote:
Quote:
Someone had posted on FB, "like and share if you can find XXX". My thought, I wonder how big of rectangle one would need to get all of the numbers. So to get to 30, would a 5 x 3 work? Or would a 4 x 4 be required? Once we get to 5 x 5 the number of potential connections goes way up. And passing through the 50's will see a gain in XY, YX pairing (ie. 45 and 54 are the same, just in a different direction, same for 91 and 19). 

20180811, 01:54  #5 
Jun 2003
2·23·113 Posts 
Code:
1.... 41221 330.. Code:
3043 .12. .21. 3..3 I am assuming both of these can be improved  these were found quickly by manual search. 
20180811, 04:27  #6 
6809 > 6502
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Aug 2003
101×103 Posts
23611_{8} Posts 
I like what you did there in that last one. It pulls ahead of WraithX's prediction of n = 2*x*y by having 43 > 2*4*4. So maybe 6 x 6 might be the break through that gets all the way to 100.
Last fiddled with by Uncwilly on 20180811 at 04:28 
20180811, 12:03  #7 
"Tilman Neumann"
Jan 2016
Germany
13×37 Posts 
Here is a 5x3 rectangle with score 36:
Code:
9 0 3 6 8 1 2 3 1 2 7 4 1 5 2 Genetic algorithms might perform pretty good on this kind of problem. Last fiddled with by Till on 20180811 at 12:04 Reason: data formatting 
20180811, 14:28  #8 
"Tilman Neumann"
Jan 2016
Germany
1E1_{16} Posts 
A 5x5 square derived from axn's 4x4 with score 76:
Code:
. 0 4 3 6 5 1 2 4 6 6 2 1 9 5 0 7 8 3 5 3 5 4 6 3 And a 6x6 with score 99: Code:
9 9 2 8 5 1 6 7 3 4 0 6 2 6 6 3 4 7 2 8 5 9 1 7 1 2 8 0 5 5 1 0 7 5 4 1 Last fiddled with by Till on 20180811 at 15:08 Reason: put placeholder in upper left corner 
20180811, 15:06  #9 
"Tilman Neumann"
Jan 2016
Germany
13·37 Posts 
I found a 6x6 with score 100, derived from the 5x5 stated before:
Code:
. 7 0 6 . . . 3 0 4 3 6 9 5 1 2 4 6 9 6 2 1 9 5 0 7 7 8 3 5 8 8 5 4 6 3 The dots can be filled with any number, hence there is still a lot of freedom. On the other hand will it be difficult to get much better because then we need many "10x" strings. 
20180811, 17:00  #10 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2789_{16} Posts 
Great work!
Did you do that all by hand? I noticed that the inner square of 1's and 2's carried over. At some point rather that keeping them together, it might be that we don't care about them being so close as we have 3 or 4 of them. The initial thread title was a play on magic squares. But, there has to be a better title for these. Sentences that have all the letters are pangrams or holoalphabetic sentences. Holonumeric squares might be a good sterile name. Compact number squares. Convonumbero squares. Getting all Latin Omnibus numeris quadratum. 
20180811, 17:27  #11 
"Tilman Neumann"
Jan 2016
Germany
13×37 Posts 
Thanks.
I wrote a little program that randomtests rectangles that contain each of "0123456789" at least once. That gave me the 5x3 solution with score 36 and the 6x6 with score 99. The better 5x5 and 6x6 results were derived by hand. axn's 4x4 is quite ingenious I'ld say. My random test achieved no more than score 36 after >10million 4x4squares. 
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