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Property F is a bitch. For numbers that are not squares or 2 x a square, the hint is useful. But for a^2 and 2a^2 the hint is useless. 16 and 36 fail but 25 succeeds as does 100. 8 and 18 fail but 50 succeeds. So for all squares and double squares you have to do an exhaustive search - unless some other theorem applies. We need a master list of Pythagorean triples where C < 1000

Cockie, I'm exactly where you are, I think. I believe I've checked quite carefully, and it seems like an extremely remote possibility that I could have a full grid but for the unchecked digit.

I'm thinking that we have to eliminate all but one of three options, but, as has been pointed out elsewhere, the "exactly one" property criterion would seem to be inconsistently applied (i.e., if fourth powers are also squares, then certain answers must meet more than one criterion; accordingly, on what theory do we eliminate the various options at 37?).

riversk - I've just checked the whole thing and found, to my annoyance, that I did not read condition i carefully: it's not just prime. Two of the three other primes that fit have a repeated digit.

The instructions do NOT say or imply that each entry complies with exactly one property.

For each entry you have to choose a single one of the properties among the (possibly many) properties that both the entry and grid number conforms with in such a way that when you have made all the choices the enumerations given for each property hold.

cockie, don't forget that 16301 and 66301 are inadmissible due to repeated digits :-)

Can anyone give me any help with 10 across. I presumed it was either type g or type h, but with the 3dn and the 5dn (the latter surely the only possibility), i can’t get anything to fit.

Thanks. Missed the repeating digit. Thank God I don’t have to retrace my steps.

Sorry cockie, didn't see your latest post.

Regarding the wording of the preamble, why include the word 'exactly' at all, or why not replace it with 'at least' ?

I agree with buzzb. There are six squares, but 1 and 16 are also the two fourth powers. Associating property c with 1 and 16 means that property a must be associated with 4, 9, 25 and 36.

I found it useful to prepare a grid of all 38 clue numbers and the 20 properties, showing which properties were satisfied by each number (for example, 1 satisfies as many as eight of the properties). This grid showed that some numbers satisfied only one property while other properties (for example, property c, the fourth power, and property n, the perfect number) were satisfied by exactly the number of answers shown in my grid. Using the limitations imposed by the number of cells in each answer and avoiding "leading zeroes", this allowed me to fill in 24, 17, 30, 6, 28, 27, 32, 8, 1 and 5 fairly quickly.

However, without the various lists of numbers on the internet (particularly lists of prime numbers), this puzzle would have been much harder. However, it's all come out nicely after around four hours of work.

10a has property (h) with the middle term 54