Can New Theorems Predict Incomplete 5x5 Magic Square Solutions?

[mfb]

It's getting difficult to disentangle the different cuts. What do we have so far:
* The upper left corner is the largest element on the diagonals (excluding the center, but if the center is 13 it's trivially true for that as well). This covers rotation and the corner-connector symmetry
* The upper right corner is larger than the lower left corner. This covers mirrored cases.
* The upper left connector is larger than the lower right connector. This covers the 2/4 exchange symmetry.

If we apply the 1<->25 swap to a solution then we have to rearrange everything to fit that pattern again. We would need an inequality that's guaranteed to swap.

Here is an approach that separates the symmetries better:
* The sum of the corners needs to be larger than the sum of the connectors (can they be equal? That would need to be treated separately). This covers only the corner-connector symmetry.
* The upper left corner is the largest corner
* The upper right corner is larger than the lower left corner.
* The upper left connector is larger than the lower right connector.

If we apply the 1-25 swap then the first condition gets reversed, so clearly we need to do the corner/connector swap. In general we might have to rotate and mirror the magic square, and it's possible that we need to use the 2/4 exchange.

I didn't find a nice way to throw away 1/2 of the magic squares, but I found a classification: Consider the center row/column. If the sum of the outer numbers is larger than the sum of the inner numbers, then this is still true after the 1-25 swap and applying the corner/connector symmetry. This splits our sample of magic squares (with 13 in the center) into two groups. Maybe one of the groups is empty, or has some nice patterns?

 

[Vanadium 50]

Here is what I do:

A B C D E
F G H I J
K L M N O
P Q R S T
U V W X Y

First place M.

Second, place main diagonal, requiring:

• A is smallest of {A,G.S,Y}
• S > G

Third, place opposite diagonal, requiring:

• A is smaller than {E,I,U,Q}
• U > E

Fourth, place center vertical column.
Fifth, place center horizontal row requiring

• 3M+2(A+E+U+Y)+(C+W)+(K+O)=195 (Parity + NineMinusFour rolled into one)
• A range check on the 8 remaining numbers, which gives me a 1% speedup. For example, if 65-A-C-D = 4 and the smallest of the eight remaining numbers is 4, I know there's no solution.

Finally, with the 8 remaining numbers, pick 2 for B and F and complete. Test for validity and uniqueness. Print base square and the 7 (3 if M=13) that derive from it.

You are right that I could divide my M=13 set into two based on C,K,O and W (or H, L, N and R). The problem is that this information arrives late in the process. By the time I get to the 6th step, I have only 56 tests remaining.

I'm doing some detailed profiling now, but most time is spent in Steps 4 and 5.

I'd also need to do a bit of work to place a cut in Step 4. I started off using a stl::list of rows, filtering at each step. I discovered half the time was spent in the row constructor. So I replaced the Step 5 stl::list with an stl::vector. This helped (although I am spending a frightful amount of time clearing vector elements) but for some reason cuts at Step 4 cause a double free somewhere. So a test would be slow - I'd need to clean up that mess first.

The way I load-balance is to have thread X look at rows in Step 1 with number mod N (number of threads) equal to X. As you see, it works pretty well, but not perfectly. Some rows take longer to go through than others, and I am counting on them averaging out. Putting more intelligence in would help: e.g. if I know that for M=1 thread 6 takes the least time and 9 the most, and for M=2, 11 takes the least and 4 the most, I would reassign the work so that the thread that handled 6 (9) for M=1 gets 4 (11) for M=2.

 

[Vanadium 50]

Some timing:

Steps 1-3 take < 1% of the time.
Step 4 - 14%
Step 5 - 64%
Step 6 - 22%

I thought Step 6 is smaller than it was because the cost of determining you don't have to do a calculation is comparable to the cost of doing the calculation.

With -Ofast, gprof tells me 92% of the time is spent finding squares and 3% checking that candidate magic squares are in fact magic. Another 5% is spent checking rows to see if they can be placed (steps 2,3 and 5).

Without-Ofast, 28% of the time is spent handling C++ overloading. I can get cells with with x,y coordinates or a single 5*x+y coordinate. 6.5% is spent seeing if numbers are used, and everything else is a few percent her and a few percent there.

 

[Vanadium 50]

So I was motivated by the work of @.Scott . I took all the solutions mod 5 and plotted the patterns looking only at the 3's on the diagonals. There are 511 such patterns. An astonishing 179 have no solutions.

 

[.Scott]

So I was motivated by the work of @.Scott . I took all the solutions mod 5 and plotted the patterns looking only at the 3's on the diagonals. There are 511 such patterns. An astonishing 179 have no solutions.

That's an interesting result - and perhaps it has direct application to the original post.
I've kind of dropped the ball on this. "Real" work has gotten in the way.

 

[Vanadium 50]

It has some application, although in the clear light of day it's less exciting.

The 512th pattern is all 3's. There are nine of them, and there are only eight numbers congruent to 3, so that never makes it into the sample.

Many of the 179 have a diagonal with exactly five 3's. This is helpful, but there are not many rows that are all 3's: there is 3, 8,13, 18 and 23, plus their permutations. Also, many of the 179 have a diagonal with exactly four 3's. This is even less useful, because there are no such rows: if you have four 3's and a sum to 65, what must the fifth one be?

Where it gets more interesting is the remaining set. It appears (checking now) that there needs to be 5 or fewer 3's on the two diagonals. That will take a bigger bite out of the number of possibilities. I would want to prove it before using it. Otherwise it feels like cheating.

For kicks, I am running a similar check on the placement of zeros mod 5. (Which is related to the placement of 1's)

 

[Vanadium 50]

Well, that was a bust.

The X-patterns that lead to no solutions have 6 or more elements with the same remainder mod 5. There are a very very small number (like 2) of patterns with 4 or 5 common remainders, but the bulk is 6+. How many does this eliminate that aren't eliminated some other way? Zero.

 

[mfb]

It appears (checking now) that there needs to be 5 or fewer 3's on the two diagonals.

Kind of obvious with just 5 elements that have the same remainder mod 5?

 

[Vanadium 50]

Kind of obvious with just 5 elements that have the same remainder mod 5?

In retrospect, yes.