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Phew - thanks for the clarification Dryden - that makes me much happier! After giving up yesterday having reached a total impasse after god knows how many hours I started again today and have just crawled over the line. Here's the (not) funny bit.

I gave up yesterday as I couldn't resolve 33ac, as in my addled state I couldn't make it tally to twice the sum of 30 AND yet be a prime number as well ......

...... so you can probably imagine my language when I realised after restarting from scratch that it was never meant to be a prime and that my original near complete but discarded solution had nothing wrong with it all along! Oh how I laughed ..... ;¬(

I'm glad you got there (as I'm glad I did in the end after several impasses). We all make silly mistakes, and the consequences in a numerical puzzle are far more serious and far more difficult to trace than in ordinary crosswords. My daft error was to compile a list of possibilities for 16 based on squares of numbers ending in 6, completing forgetting that I should have included squares of numbers ending in 4.

Now to double-check everything.

For those of you interested the original L3277 that appeared in 1994 can be found in the first Times book of Listener Crosswords. I have a jpeg of it and can email you a copy if anyone would like a go. I would add that setting a puzzle like this is not easy. I have set a few like this and know from personal experience!

Hi Unclued, yes 11213 is correct (it was my last entry also).

Wintonian, I have a note of using the digit sum (16) > digit sum (10). As far as I can see it was useful in reducing the possible answers for 16.

In terms of whether this was a good puzzle or not, I readily concede that it would have taken some time to set, but as stated previously, much of it was a tedious number slog for the solver. And due to limited self-checking, one omission in looking up candidate squares cost me a lot of time.

Overall, this probably belongs in the Hermann Hesse basket. In one of his books, he rails against people who spend all their free time solving puzzles rather than confronting the more meaningful issues of their existence. I suspect that Hesse might have had stronger words yet about this puzzle.

Hi, Meursault,

Yes, I can see that knowing that the digit sum of 16 is greater than the digit sum of 10 will help, even if you don't know the first digit of 10 yet, as I didn't when I worked out 16 (the first digit of 10 has to be 5, 6, 7, 8 or 9 because 20, which is 2 x 10, has four digits, but it can't be 5 because then 10 would be prime).

I'm afraid that I found 16 by brute force calculation rather than logic, so I didn't use this condition. With numerical puzzles, it's often quicker to set up a spreadsheet to calculate all possible numbers or combinations than to try to reduce the possibilities using logical reasoning.

I would love a copy of the original Properties of Numbers puzzle if anyone has a forwardable copy! osirun at gmail dot com if anyone would be willing...

Can I give a plug for Oyler's "crossnumbers quarterly" magazine. If you have enjoyed this puzzle, then contact him for more details. (I'm not his agent!) Hopefully he'll have a feature on Picadilly puzzles in the future as they seem to go back a long way into the past according to the Listener website information. Maybe he'll even persuade P to produce one for the magazine.

I totally agree Wintonian - most of my work was done in either Word or Excel. Back in the day when I had a job I used to 'borrow' the company mainframe to crack numeric puzzles, sadly progress (a.k.a. redundancy) lost me that privilege. How anyone could crack a puzzle like this with pen and paper is frankly beyond me. Just glad to put it to bed now!

This is only the second numerical I have tackled and I was really gripped. Very enjoyable. I felt as though I was taken on a journey - tackling the problem in the sequence intended by Piccadilly. Goodness knows how you go about setting a crossword of this type.
Still on 100% completion for this my first full year of doing the Listener though admittedly with help from this forum on occasions. All my own work today though.

I have managed to complete the crossword except for 21. I have 3 possible numbers - 11549, 12548 or 13547 which all seem to fulfil the requirement for "20+21