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.

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.