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meursault, all of those are correct (21D being 12403)

Many thanks, Demeter.

For anyone still working on this, and in need of reference, I should add this for 6-digit squares :
http://benvitale-funwithnum3ers.blogspot.co.uk/2010/12/3172-100489-4882-238144-6582-432964.html

I couldn't find any list for polygon diagonals, for these and the type 's' numbers, spreadsheet seemed to be the easiest solution.

For type o, if 'a' is a value we have:

n(n-3)/2 = a so n^2 -3n - 2a = 0 and the quadratic formula gives:

n = (3 + sqrt (9 + 8a))/2

(note that 9+8a is always odd, so if it does have an integer sqrt, that too will be odd so adding 3 gives an even number so the entire fraction will be an integer.

So you can easily tell if a number is of type o; just multiply by 8 and add 9 and see if that is a perfect square.

E.g. 2: 2*8 + 9 = 25; 5*8 + 9 = 49.

Similarly for triangular numbers:
n(n+1)/2 = a => n^2 + n - 2a => n = (-1 + sqrt(1 + 8a))/2 so 1+8a must be a perfect square.

Hi all. Started this yesterday evening. Tried to go it alone as it were but am now really struggling. To further kick-start my attempt with this could someone let me know which property 2D has? Is it k or q? Any hints would be appreciated.

Goshawk, I have 2d as type o.

Thanks Dryden. Hmm was hoping it wasn’t that. That leaves a myriad of possibilities as I haven’t any entries in that portion of grid. Anyway thanks.

Goshawk, the NW corner involves a lot of exclusion. To get you started, you've probably worked out that 5D is a Fermat prime, since 3 has to be a triangular number (6 & 28 are perfect numbers, 1 & 36 are powers, leaving only 3, 10, 15, 21). There is only 1 solution for a 3-digit Fermat prime. Then for 1A, there are only 3 (quite surprised me) solutions for 6 digit squares with 5th digit 2. And 2 out of those 3 are invalidated due to having zeroes in positions which would start down clues.

You mean "there are only 3 (quite surprised me) solutions for 6 digit 4th powers with 5th digit 2." There are 151 6-digit squares with 5th digit 2.

Since there are just 14 6-digit 4th powers, it is not surprising that there are only 3 with 5th digit 2.

I’m quite new to the Listener (this is probably my fourth or fifth). If anything I found this the easiest so far, although these things are relative, but I’m afraid I did resort to a spreadsheet to generate possibilities for some of the big numbers, just so I could get on with the rest of my life.

What struck me as most obvious, though, was that this wasn’t so much a crossword as a crossnumber. I’m assuming from the lack of comment about it on here that this is quite normal for the Listener, and I suppose variety is a good thing, but I’m not sure about it really...

Buzzb, yes the grinding down of the NW corner took me to a point of tedium where I began to repress memory. Of course, 1A is a 4th power, the correct one being 923521. But it was good that you intervened to prevent my comment causing confusion. Thanks.

Mdmaylwin, the question of when is a crossword not a crossword is one close to my heart. Probably most Listener solvers have views on this, and the debate has been raised many times in the past. There are many different aspects.
The word 'word' is one of them.
The idea that a cryptic crossword, with definition and wordplay, and therefore self-checking, is another.
There is also the question of the people who want to be 100% on Listener puzzles. Some will just achieve this privately, but others must do so publicly and vie for a pewter goblet or some such object awarded at a snobbish event. At some point 4 numeric puzzles got thrown in each year by the Listener editors, and you have to assume that this was partly a means of thinning out the 100% entries. To which I might reply, "So, gather your 100%ers so far and then ask them to do 440 yards race and if there is a dead-heat they should each need to vault over a charging bull : this will surely establish who is the winner."

But when all of this has been said, there remains a minority of Listener solvers who love numerics. Who am I to argue against their enjoyment ? What I've said before is that there should be 2 separate puzzles.