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We are interested in the pairing of a male and a female.

The information given allows us to derive

Diophantine Equationsfor the number of bachelors in the buttons department (m) in terms of the number of spinsters in the bows department (f).Or, conversely:

This Python program considers increasing values for

muntil a suitable value forfis found.It runs in 43ms.

Solution:There are 5 bachelors in buttons. (And 9 spinsters in bows).There are two further integer solutions to the Diophantine Equations, but these involve negative numbers.

They are:

m = –8, f = 3andm = 8, f = –3.Is it just chance that 3 × 5 + 8 = 23 and 3 × 8 + 5 = 29,

or could it be that your expressions linking f and m are unnecessarily complicated?

I can’t see how they’re derived (though I may be blind).

I did omit the comment from my code deriving the formula, but the formula comes from the given odds.

There are 3 females in the button department and

fin the bows department.There are 8 males in the bows department and

min the buttons department.If we pair a female with a male we get the following pairings:

The chances of a pairing in the same department is 29 to 23 in favour, so:

And this is the equation I started with.

Thank you, Jim.

I’ve just derived the same relationship, but you beat me to it.

My simplified expression works only because there happen to be three times as many lassies in bows as in buttons.