Squares have to end with 1,4,5,6, or 9, therefore none of the digits in the two digit numbers can be 2,3,7, or 8.
This eliminates 25, 36, and 81 from the pool of 2 digit numbers, therefore the 2 digit numbers must be 16, 49, and 64.
My problem so far is that I get 6 different configurations of the 2 digit numbers in the three spots, and none of them seem to work to get the last two down 4 digit numbers. Taking into account that some of the places could be more than one number, I get 10 different grids, but none of them yield the two downward 4 digit squares. Either I have a logic issue, am missing something (probably ;-)), or am just blind.......
May be a day or so before I get much time to look again. Interesting problem. I have a feeling that I am going to do a face palm when I figure it out ;-)
I know that at least Dan is still working on this, so I'll offer the solution outline in steps, with each hint assuming that the previous has been understood and figured out, so don't look at subsequent hints if you still want to figure out the previous step.
Once you have 16, 49, and 64 as the only square that can be in fj and kl, Look at the even/odd status of the penultimate digits of the four digit squares that end in the second digit of each of the above. Obviously, that digit has to fit in space i. You could test them all, but looking at the odd-even status is a short=cut.
Any square ending in a 4 or a 9 has an even digit in the tens place, while any square ending in a 6 has an odd digit in the tens place. So 49 and 64 are fj and kl (but we don't know yet which is which.
This is where a little grunt work comes in. Excel is your friend!
At this, we fill in 49 and 64 (either place; doesn't matter for the next bit of work), and prepare lists of all the squares ending in 4, 9, 6, and 4, then start eliminating candidates based on the possibilities of matching with other squares on the interior four digits.
The previous step should have eliminated most of the possible squares, leaving only the first two digits to test. Test the square(s) that are candidates for cg the way you have it laid out, and if that doesn't work, switch fj and kl.
Here's my spreadsheet following hint #4. For each digit in 64 and 49, there is a column of numbers to be squared, the four-digit squares, and for those in red, a notation about the reason they have been excluded. The exclusions were don in the number order of the explanations.
I actually had this except for one thing that was a fundamental error. In my original part of the analysis, I talked about the only possible numbers for the ENDS. I forgot that the two digit on the left side was the STARTING number. All the rest I had and would have gotten it if I had not made that dumb error. ;-)
I had the tables of squares, and all the rest. Funny how you can get a faulty pattern in your thoughts and be unable to see beyond it. Oh, Well, that is what makes stuff like this fun. ;-)
