I found this exceptionally vague question, but I can't come up with a mathematically robust solution to it.
Give it a go if you have the time!
(i) Find the least number of cuts required to cut 12 identical sausages so that they can be shared equally among 18 people
(ii) Find the least number of cuts, in terms of m and n, required to cut m identical sausages so that they can be shared equally among n people.
The question does not specify that:
- A cut applies to all sausages
- The resulting length of each sausage must be the same
So the resulting solution for part (i) would be 12 cuts:
- Cutting each sausage at the 2/3 mark
- Giving 12 people a 2/3 piece of sausage each, and the rest 2 x 1/3 piece
Assuming this obscure way of understanding the question, what would be the solution for (ii)?
[I'm thinking that there exist an M and N such that you will require M x N cuts to split equally (a proof by contradiction) but I can't find such a M and N, neither can I formulate a proof by induction for (ii)]
You can always do it with $N-1$ cuts. Imagine lining the sausages up, then cut off $\frac MN$ for each person. The number of cuts can be reduced if some of them come at the sausage boundaries. In your $M=12, N=18$ example five of the sausage boundaries are where you want to cut, so instead of $17$ cuts you only need $12$. You should be able to convince yourself that you can always reduce the number by $\gcd(M,N)-1$, which in your example is $5$. This proves you can get the cuts down to that, it doesn't prove you can't do even better.
In your $12,18$ problem I could claim that I can do it with one cut. Line up all the sausages side by side and make one cut across all of them at the $\frac 23$ point, then distribute the pieces as you suggested.