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Should we expect the final letter answer to spell something legible? I thought I had finally arrived at the answer but it seems to read as nonsense...

Never mind, finally got it!

mooncow - fwiw i did it by hand (well calculator) - because i knew i was looking for the highest possible, i simply started from the max number possible and went down till i found one that worked (that had to be the highest). It sure would have been tedious to find all 5!

I did it by hand as well. I assume the other 4 are the ones in the grid so we knew there was a fairly high lower bound for the one we were looking for.

I enjoyed this. As usual for me, 95% of the time was spent faffing around and 5% on properly targeted effort. I could not have solved it without Excel.

Having recently won a Listener prize, I am rather chastened to be struggling with this one!

Nevertheless. for those who do not fully understand the algebra of surds (the roots of numbers that are not themselves perfect powers), the key equation is this (using Rt as the Square Root sign, & ^2 for squared):

(b + c x Rt66)(e + f x Rt66) = (b x e) + (((b x f ) + (c x e)) x Rt66) + ((c x f) x (Rt66)^2)

= (b x e) + (66 x c x f) + blah

(b x e) + (66 x c x f) = G -> crucial to the solve!!

blah = h x Rt66, and is irrelevant.

Hope this helps someone; in the meantime I am sure that I have nailed A, O , S & U to being consecutive larger numbers, but can't quite work out which is which...

BenWells: How can A, O, S & U be consecutive large numbers when they are digits?

I reread what you said...never mind

I do not believe you can deduce AOUS, unlike the other letters, by simply requiring all the answers to be 4-digit. There are at least 7 permutations of the choices for AOUS that result in all answers being 4-digits. However, there is only one that avoids zeros or duplicate digits in at least one answer. As far as I now you can only proceed by trial and error at the point you are at

For what it's worth, Excel users can rapidly get each letter's digital value by examining the 362,880 possibilities. The next stage is to identify the 608 possibilities for BCEFG .