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Hi Goshawk. Your advice should help me a lot but I find I am now even more confused. By leaving a gap in 1a, how do I then fit two 4-letter answers into the remaining 7 squares? I also thought I had the answer to 1ac but I can't get my answers to 1d, 2d and 3d to fit the letters. My answer for 1ac is "spay" and if that is wrong it is throwing me out. I have spent a lot of time on this fiendish puzzle and should probably give up and get on with my life but I am going to have one last go.

I do have the answer for 1d which fits but it is 2d and 3d that don't.

SE(space)T
TU E
AR P
LY I
EA D
RL
E

Sorry Catkin the computer moved my spacing

Tepid comes off the T of set (1a)

There is an empty cell between the E and T of set

And 2d is of course euryale. Spacing buggered up again by computer!

Phew. What a challenge! Have filled the grid but deeply puzzled by 20d which seems to be a complex abbreviation. No sign of it in TCD. So I probably have got it wrong! Can anybody offer a hint please? I should say that my 22a is a word also meaning water, maybe that's wrong too. Thank you!

It is all now falling into place. Thank you! Spay was such a good answer!!!

Have realised now where I went wrong. Doh!

Catkin is not alone in being unable to start populating the grid. Until I realised yesterday that an earlier post here about the space in 1a meant that the solution was a 3-letter word, not 4, I couldn't fit anything in either. Making steady progress now that I've ignored the maths element, but I'm confused as to which type of clue 29a is. My rapper without his title, plus a 'w', seems to work perfectly without any modification, although I wanted to use the 'd' from the message there.

Thanks to those who have responded to my question re filling in the grid. I have made an elementary error in mis-spelling 2d as Ureyale instead of Euryale. This has led me up the garden path wondering how to fit in answers! Really annoyed with myself. Anyway thanks again for the assistance. Will attack it with renewed vigour.