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It's like a game of Top Trumps in here so... Despite being an Emeritus Professor in Graph Theory at MIT, I found the preamble virtually impossible to read. Hoping to find my spectacles soon.

Understanding the preamble is the hardest part. The second hardest part is finding the way in.

Once those feats have been accomplished, there is no need for a brute force approach at all, indeed it is much simpler to proceed without brute force because as has been said earlier, Oyler makes sure there is a solution path that does not require huge spreadsheets of possible answers.

My longest list was the list of triangles.

Thanks Oyler! I can return to the Magpie now.

candledave - thanks to your explanations, i made it!

excellent - glad to hear it.

As others have said deciphering the preamble and working out how to approach this puzzle was definitely the toughest part

stixmcvix - filling the grid doesnt really involve triangular numbers (which was my undoing) - the equation gives you key properties of the grid entries (given p, q, r and s are integers)

Regarding the difficulty that solvers are having with the preamble, in retrospect an example would have probably helped. Sorry. Although having said that my original preamble just mentioned 'with the help of some algebra'!

The original puzzle is a work of art as Adam managed to fit in all of the nine possible triples which were 2- and 3-digit integers with each term in a triple clued by a different solitary letter. The barring pattern was asymmetric and had a totally barred off cell as well which had to contain the result of a calculation based on some of the entries. The solution relied on solvers using number theory to come up with a set of formulae that generated the triples. No internet or personal computers to help back then. The best would have been a scientific calculator which would have cost you £50 which is £650 in today's money!

I think the pre-amble was really clear and it was a neat puzzle. Thanks!

Yesterday I gave up trying to understand the end game. This morning, with a coffee, I methodically re-read the preamble and completed the end game in about 3 minutes! A classic case of not seeing the wood for the trees. I agree with Dom that the preamble is really clear - once you understand it. What I like about this mathematical is that I found the route to solving it really very clear, starting with clue VII and ending with clue V, and each clue did not produce a myriad of possible solutions.

I have a full grid satisfying all the equations. However two of the clues lead to non-integer solutions for p, q and r, so I must have an error somewhere. The sum of all the individual digits in the bottom two rows is 62. I think that where my error might be, though it's surprising that there seems to be more than one solution to the grid.

62 seems right.

The numbers in the clues need to be odd numbers for p, q and r to be integers so if you have gone wrong you should be able to tell where but check first that you’re doing the maths right