I have got a puzzle as a Christmas present from a friend of mine and I have managed to solve it while skipping this part, but I would still be interested in how this represents the information that I have figured out by myself as well.
The only instruction for it in the accompanying letter was this: "You will require some algebra."
What I've also got was a crossword and clues for the missing words. After getting all the words I arranged them into the crossword through trial and error, and after solving their order I came back to this piece of paper.
What I know right now is that the correct order was
3, 23, 10, 13, 20, 24, 16, 8, 12, 9, 14, 11, 21, 15, 2, 5, 22, 6, 19, 4, 7, 18, 17 which, as I have noticed, corresponds to the sequence that can be read by removing the
15 from every pair and then going left-to-right, top-to-bottom in a row major consecutive manner starting from the
(3, 15) entry, inserting
15 at the end and then wrapping around.
I have also noticed that the pairs are ordered such that the lower number is always on the left and as I have already mentioned, the number
15 is present in every pair. (Also, there are only 22 pairs for 23 words.)
The only thing I have found to use this parenthesis pair notation is for denoting the greatest common divisor of two numbers, but that doesn't really seem to be of much help here.