Bike polo mechanics & fixing

It's actually an interesting maths question.

You have a few different variables

1) The angle of the chain compared to the (virtual) chain stay will increase as the chainring gets bigger, the bigger the angle the less difference in length.

And the opposite as the cog gets bigger (assuming it's never bigger than a 1:1 ratio)

2) it will depend on the length of the chainstays too, as again that affects the angle of the chain.

It's say it's ultimately dependant on 4 factors:
l, length between the centres of the two circles
t1, the tooth count of the chainring (and therefore the diameter)
t2, the tooth count of the cog (and therefore the diameter)
p, the chain pitch (a constant in our case)

At some point when i'm super bored I might work it out.

#mathsnerd

It's not that interesting a maths question! It's just solving the long side of a right angle triangle (square root of the sum of the length from the center points of the cog and chainring and the difference between the two radius), doubled and added to half the number of teeth on the cog and chainring times the pitch.

But that's basically what you said, so I guess we just disagree on interesting maths questions! :)