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Thanks - the operation I carried out after conducting the cryptic instruction was clearly wrong. But then faced with strings of numbers, it's hard to see what else you could do with them! I tried applying the cryptic rule to the actual squared digits instead, but that led nowhere. I'm clearly missing something that's staring me in the face. I'm pretty sure the numbers are correct, otherwise I wouldn't have been able to solve the cryptic rule!

Thanks Mack - I'm sure that the second operation I carried out after applying the cryptic rule was incorrect. Maybe it's simply done by inspection but nothing is obvious!

Check you have the instruction correct: the numerical value of the first letters of the 3 words totals 39.
Apply the instruction to each of the numbers in set 2.
Combine pairs of the resulting numbers using a very elementary process (the sort taught in primary schools). That's 6 calculations at most.
The results for two pairs should be highly significant.

I agree with 39 - I solved the anagram correctly and applied the rule to each number in each set, Then I did a simple thing that you would do with a string of digits and certainly found matching pairs and non-matching pairs, but they were not consistent numbers from set to set. And with the 3 solutions for 7 down there were two lots of three matching numbers. This did not allow me to know which number to enter in the grid at 7 down.

rhsl (#67), nothing I wrote was intended to suggest that the answer was obvious, only that the process involved is a simple one. I'm sorry if that didn't help.

Mack - thanks for your help. It forced me to check what I had done the first time round which had been correct. However I spotted an error in my initial approach when I carried out the obvious second step in the pairing process. Can you believe it! - this error of one digit was in Set 7, and now that it's corrected it has resolved which of the three solutions go in 7 Down! Phew!!

Hi,
It might be worth checking the second operation that you did in Set 7 after following the cryptic instruction. I found an error of one digit which led to the anomalous result for 7 Down. When corrected it became clear which of the three solutions should be entered in the grid.

As so often happens, posting (and a night's sleep, being in Sydney) did the trick.

And, of course, once you've seen the relationship it is simple and unambiguous.

I apologise, Elap, for my somewhat grumpy post. It is, indeed, a very clever puzzle. As others have said, finding the clued entries remained a challenge almost until the end. Along the way, the constraint of having to get a sensible message did help.

ByronK, it looks as though you've found the relationship despite my "help".

Done! Thanks in no small part to contributors on here, especially for the last step, simple as it seems with the benefit of hindsight. I have absolutely no idea how that can be constructed so I join others in doffing my cap to Elap. Great job!

Coming here late (it took me a while!) to congratulate Elap on a stellar puzzle.

In my mind this is what every numerical should be - an unrelenting level of challenge, each break leading to a new confusion, right to the last cell.

And a puzzle which doesn't require hundreds of primes / triangles with a brutal period of 'process of elimination' (i'll confess we normally use excel). Just a few pieces of A4 and a print out of a few squares. Brilliant!