*Originally posted by kes29*

**Thanks for the puzzle, very original! It would be interesting to know the solution you have, maybe there is some quick "clever" way to solve it? The way I did it was purely mechanical, a lot of algebra, few combinatorial identities and a bit of number theory in the end.
**

BTW, perhaps someone knows how to "hide" the answer? Couldn't find anywhere in the Help section. Thank you.

hope this is legible, it is pasted in from my notes when constructing the question. It's messy because you can't hide multiple lines with one command:

Reveal Hidden Contenty = x.slope + K

Reveal Hidden Contentaverage y = (n.slope + K + slope + K)/2 = (n+1)slope/2 + K

Reveal Hidden Contenty_variance = sum{x = 1..n}(x.slope + K - (n+1)slope/2 - K)^2/n

Reveal Hidden Contenty_variance = sum{x=1..n}(x.slope - (n+1)slope/2)^2/n

Reveal Hidden Contenty_variance = sum{x=1..n}(x^2.slope^2 -x*(n+1)*slope^2 + (n+1)^2.slope^2/4)/n

Reveal Hidden Contenty_variance = slope^2/n.(sum{x=1..n}(x^2) -(n+1)sum{x=1..n}(x) +n(n+1)^2/4)

Reveal Hidden Contentexpand those series. One for x^2 is not obvious but can be easily googled:

Reveal Hidden Contenty_variance = slope^2/n.(n.(2n+1).(n+1)/6 - (n+1).n.(n+1)/2 + n(n+1)^2/4)

Reveal Hidden Contenty_variance = slope^2(n+1)/12n(2.n.(2n+1) - 6.n.(n+1) + 3.n.(n+1))

Reveal Hidden Contenty_variance = slope^2(n+1)/12n(4n^2 + 2n - 3n^2 - 3n)

Reveal Hidden Contenty_variance = slope^2(n+1)/12n(n^2 - n)

Reveal Hidden Contenty_variance = slope^2(n+1)/12n(n.(n-1)

Reveal Hidden Contenty_variance = slope^2(n+1)(n-1)/12

Reveal Hidden Contenty_stdev = slope.sqrt((n+1)(n-1)/12)

Reveal Hidden Contentif y_stdev = mult.slope then

Reveal Hidden Contentmult = sqrt((n+1)(n-1)/12)

Reveal Hidden Contentmult^2 = (n+1)(n-1)/12

Reveal Hidden Content12.mult^2 = n^2 - 1

I did an Excel sheet search here, which found mult can be 0,2,28,390,5432...With corresponding values for n. I'd be interested to see your number theory.