wolf goat cabbage

This is a really interesting take on a very well known puzzle (courtesy of Ian Stewart’s Cabinet of Mathematical Curiosities).

The puzzle itself is pretty famous:

A farmer wants to cross a river and take with him a wolf, a goat, and a cabbage. There is a boat that can fit himself plus either the wolf, the goat, or the cabbage. If the wolf and the goat are alone on one shore, the wolf will eat the goat. If the goat and the cabbage are alone on the shore, the goat will eat the cabbage. How can the farmer bring the wolf, the goat, and the cabbage across the river?

And the standard way of solving it is trial and error with some logic thrown in.  However, as Ian Stewart points out, we can actually utilise 3 dimensional geometry to solve the puzzle.  We start with a 3D wolf-goat-cabbage (w,g,c) space (shown in the diagram).  All 3 start at (0,0,0).  0 represents this side of the bank, and 1 represents the far side of the bank.  The target is to get therefore to (1,1,1).  In (w,g,c) space , the x direction represents the wolf’s movements, the y direction the goat and z the cabbage.  Therefore the 8 possible triplet combinations are represented by the 8 vertices on a cube.

We can now cross out the 4 paths:

(0,0,0) to (1,00) as this leaves the goat with the cabbages

(0,0,0) to (0,0,1) as this leaves the wolf with the goat

(0,1,1) to (1,1,1) as the farmer would leave the goat and cabbage alone

(1,1,0) to (1,1,1) as the farmer would leave the wolf and goat alone.

which reduces the puzzle to a geometric problem – where we travel along the remaining edges – and the 2 solutions are immediately evident.

(eg.  (0,0,0) – (0,1,0) – (1,1,0) – (1,0,0) – (1,0,1)- (1,1,1)   )

What’s really nice about this solution is that it shows how problems seemingly unrelated to mathematics can be “translated” in mathematics – and also it shows how geometrical space can be used for problem solving.

For a more complicated problem (that can’t be solved without going into 8 dimensions geometrically – so don’t use this method!) try this puzzle. It has a nice animation which allows you to do the problem online.  The problem is as follows:
Find a way to move this group of people across the river. Only 2 persons on the raft at a time. The father cannot stay with any of the daughters without their mother’s presence. The mother cannot stay with any of the sons without their father’s presence.  The thief (striped shirt) cannot stay with any family member if the Policeman is not there.  Only the Father, Mother and the Policeman know how to operate the raft.