I don't think it's ever possible that ALL the animals will be able to cross that river
Proof
I'll name them BD, MD, SD / BC, MC, SC (big dog, medium dog, small dog / big cat, medium cat, small cat) and Sh1 and Sh2 (Shore 1 and 2)
Sh1 contains: BD, MD, SD / BC, MC, SC
Sh2 contains: no one
Out of the 15 possible combinations:
- BC,MC / BC,SC / MC,SC: 2 cats cannot go together since Sh1 will be left with more dogs than cats.
- BD,BC / BD,MD / BD,SC / MD,BC / MC,MC / MD,SC / SD,BC / SD,MC / SD,SC: only MD,MC is possible because at least one of the other animal kind will be left with a middle animal of the same kind
- BD,MD / MD,SD: Not possible since the middle dog will start the fight
- BD,SC: Seems possible at first, but when one of them return with the raft, they wil remain the middle dog and start a fight
So the only possible solution is MD,MC
Sh1 contains: BD, SD / BC, SC
Sh2 contains: MD, MC
Only MC can return the craft because if MD return it, Sh1 will contain more dog than cat
Sh1 contains: BD, SD / BC, MC, SC
Sh2 contains: MD
1 cat and 1 dog is not possible because Sh2 will have more dogs than cat
2 cats (BC,SC) also not possible because Sh1 will have more dogs than cat
Only solution here is BD,SD
Sh1 contains: BC, MC, SC
Sh2 contains: BD, MD, SD
Only MD can return the raft because if BD or SD return it, the other one will remain with the middle dog and start a fight
Sh1 contains: MD / BC, MC, SC
Sh2 contains: BD, SD
MD with any of the cat is not possible because Sh2 will have more dogs than cat
MC,BC / MC, SC also not possible because a fight will start with the middle cat
Only BC, SC is possible
Sh1 contains: MD / MC
Sh2 contains: BD, SD / BC, SC
Here it ends because no dog can return the shaft (Sh1 will have more dog than cat) and no cat can return it neither (a fight will start with the middle cat)