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This one looks terribly difficult. Is the best place to start the top and bottom rows? Any help much appreciated.

You could start by considering clues 1a, 3d and 16d. Together they tell you the nature of the two 3-digit grid entries at 1a and 20a. I suggest you draw up a list of possible pairs for these two grid entries. Then consider more specifically what the entry at 20a must be given that its last digit bears a special relationship to the last digit of 1a ( you need to consider the significance of clue 18d during this).

Good luck !

I thought that the clue for 18 dn was a misprint. Still don't understand it!

It simply means each digit is the encoded version of the other. e.g. if 1 encodes to 2 then the encoded version of 12 is 21

But is this right? 1 may encode to 2 but 2 may not necessarily encode to 1.

This is what's been holding me up. The reverse coding applies to 18dn, but what about elsewhere? If x becomes y, does y always become x?

I'm feeling really dim, as I can't find a logical path at all. I have just one coded entry I'm confident about - the one mentioned by Cruciverbalist, as it happens, so I must be starting off in the right way. Should I be trying to crack a clue that intersects with it, or should its counterpart be obvious to me? I have three out of four possibles to play with (the fourth one creates a duplication that is disallowed by the preamble), but trial and error with each of them has got me nowhere.

I don't think it always applies, although the calculations I have made so far may well be wrong. I have also got reverse coding for the number in 18 down which is reassuring. I was able to work out 17d, 3d and 1a after I got 20a but just don't ask me how! Now stuck again.

A good way to start is to fill down a column in Excel from 10 to 31. In the next column, have the squares of those numbers (=A1^2) and pull down. That will show you that 19 and 361 satisfy the condition that the first digit and the last digit of the root and the square are the same. You will later need to do the mapping. Frankly, I find this puzzle boring, and will go no further with it.

I did something similar, but then got flummoxed by the requirement that no entry or answer must begin with zero. In my mind this forces the zeros to locations where square roots can't be found. I've missed something - hint appreciated

"...no clue answer or grid entry starts with a zero."
has to be taken literally - don't read too much into it.