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I have a full grid and the two symmetrically placed 9 digit numbers, but I cannot find the two pairs of positive integers. I'm at a loss why is specifies positive integers, and maybe this is what is befuddling me.

There’s no need to be sidetracked by the word positive. Do you know what the integers are, but cannot locate them? Or do you not know them?

In theory, you could use the sum of a positve and a negative integer to produce a clue value, so it is just ruling that out. It has nothing to do with your problem - keep looking, they are there in a straightforward arrangement.

Yes, I have found them now. I missed the solution for the lower number, so I tried (more carefully) again for the higher number and found it. It then became straightforward.

As a CMD I found this a tedious slog which I wouldn't have had a hope in hell of breaking without MS Excel. The thing that angers me is that I don't even have the satisfaction of having learned anything from the process. I'm still no closer to understanding how the puzzle could have been broken using logic, as opposed to (fairly) mindless brute force.

*CMD - Complete Maths Dunce

s_pugh: some of the clues have very few possibilities because:

If half of the clue squared (e.g. 1acr: 75^2 = 5625) is too large than you know immediately that the entry must be a sum of two factors squared and there are always a lot few pair of factors than addends - there are 74 ways to break 150 into a+b but only 6 where a*b=150. Moreover the larger factor must be less than sqrt(1000) which narrows 1a to 3 possibilities. Since you know the last digit of 1ac from 2dn, presumably, you will see that there is only 1 of the 3 that works,

This type of reasoning applies to most of the clues (i.e. half the clue is to big to allow addends). In two cases it only allows 1 possible entry.

If you know the last digit of an entry that narrows the possibilities for a & b where n=a+b or a*b. E.g. if the last digit is 0 then the last digits of a & b in some order must be (0,0) (1,3), (1,7), (2, 4), (2,6), (3,9), (4,8), (5,5). This helps you prune your possibilities also.

It's still a slog though with no satisfactory aha moment for me.

Thanks for that Buzzb - much appreciated (I've taken a copy for future reference!)

I can’t see a way to solve this except for brute force. It’s not a difficult task to derive a complete list of all the possible entries for any clue, just a tedious one if you can’t find some ways to shorten the process. Identifying the clues where the number must be the product of the two integers rather than the sum is helpful. Even more helpful is knowing how to use a spreadsheet or a simple program to run through all the possibilities for a clue quickly.

I thought that as a puzzle this was not all that interesting, but the final highlighting reveals the grid to be a very neat piece of construction, so for me there was that aha! moment. And another aha! moment when I realized what the mathematical theme behind the puzzle is (which is connected both with the common feature of the entries and with the title). You don’t need to find that theme to solve the puzzle, and in fact finding it won’t help solve the puzzle at all, but I enjoyed learning about it all the same.

For those who care the title is an anagram of a famous mathematician who had a 'little theorem' related to what's going on (but I doubt it is particularly useful as a solving aid).

I’ve only been at this a short while, and normally award myself a week off for the number puzzles, but decided to give this a go because it looked penetrable. Haven’t finished yet but pretty sure it will just be a matter of time.

Being a mathematical simpleton, I don’t know whether it is a fluke or a delightful inclusion but: the two options for 13a struck me as quite neat.

No doubt the numericals are every bit as sophisticated and clever as word puzzles, I can understand how this is but, unfortunately, my own limited capabilities can’t ascertain why.