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Perhaps if you give those 5 numbers and the original entries in the vacant lights we can offer further guidance ?

Reynard, I was in the same boat as you but noticed six of the numbers I hadn't yet filled in were at the back end of the list of numbers produced by the instruction. If you knock these six off you can complete it fairly quickly.

Thanks, Demeter, for your confirmation back on page 2, but I still can't produce a series that fits. I guess I could work backwards from the SE corner digit, which remains unchanged in the second fill, but I need to know the method of proceeding with this - is the second fill in conventional or "thematic" order? Anyone feeling merciful?

Durban, consider how the 2-digit number in the instruction relates to the number of grid entries - the endgame is basically nothing more than a jigsaw ;-)

Hi durban--yes the last bit is quite a head -scratcher. I will assume that you have decoded the grid correctly, and that you are left with word / 2-digit number / three words / 3-digit number / three words ?

A 38 item sequence starts with the 3-digit number mentioned above (which is in the form YZZ ), and you should find that the last number is a prime number under 20 . Then, remembering to leave the circled cells intact but erasing all the others, you should be able to fit everything in neatly.

Quite a breathtaking construction. Hope you get it...

Great puzzle! Or rather 3 puzzles in 1!

I thought I had cracked this but when I came to insert the last digit in 5dn I found I had several possibilities, which is presumably wrong ; but the hailstone sequence for the 3-digit number produces 50 numbers and even when you remove the 5 single digits that still leaves seven more than the preamble implies. I've fitted in all the 4-digit and 3-digit numbers but that leaves me with 4 possibilities for the 2nd digit of 5dn. Where have I gone wrong?

Hey barretter--you are nearly there---the whole sequence does contain more numbers than you need, so maybe try again using only the first 38 numbers in the sequence ( the three digit number down to a smallish prime) and you should be OK. It does work out perfectly.

Thanks very much Schorley, I was beginning to think along those lines but you have speeded my torpid thinking processes!

In the end I only had to change the digit in square no. 1.