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this is a useful website:
https://www.calculator.net/factoring-calculator.html

and so is this:
http://nonagon.org/ExLibris/fermat-sum-two-squares-calculator

Those websites do not deal with the constraint the the two numbers are distinct. For example, 162 = 9^2 + 9^2 (and that is the only way to get 162) but the numbers aren't distinct. So the algorithm can rule some numbers out, but will not detect those cases like 162.

Since we are not given the values for a^2 + b^2, but rather either a+b or a*b, these sites won't help that much for the grid fill. They are useful for the final step though!

Very relieved that Krauton divulged the useful websites. My calculator is very old and can only deal with 8-digit numbers. Completed the final step now and really filling in the grid was very tedious.

I'd be grateful for confirmation or otherwise of a couple of entries. Likely I started the puzzle too late in the evening and have made an error(s) somewhere...

For 20 I'm left with ?61 and have matched up on a spreadsheet the possible sums of squares for 1-39. Alas, no solutions ?61 that I can see.

For 17 I have 51973 and for 21 I have 6073.

You haven’t gone wrong anywhere. You need another look at your spreadsheet

Hi Keepatit, many thanks this very welcome news. So it's not an 'evening before' error, but a 'morning after' one...

Once again when the number puzzle appears I'm in the middle of nowhere with no chance of buying the Times. If anyone could email me the puzzle at robbies.blakwhole at gmail I'd be very grateful.

Blakwhole - on its way!

Got it ok, many thanks smartie