Math Detours · Grade 5 · Around the corner · 19

The wolf, the goat and the cabbage

A boat that holds two, and things that must not be left together. Draw every position as a dot and the puzzle becomes a map.

Name ____________________Date ____________

Tonight's idea

A farmer must take a wolf, a goat and a cabbage across a river. The boat holds the farmer and one other thing. If the farmer is away, the wolf eats the goat, and the goat eats the cabbage.

Do not guess. List every position: who is on each bank. Write F, W, G, C for farmer, wolf, goat, cabbage. The start is FWGC | –.

Four things on two banks make 16 positions. Six are unsafe, like WG | FC, where the wolf eats the goat. That leaves ten.

FWGC | – WC | FG FWC | G W | FGC C | FWG FWG | C FGC | W G | FWC FG | WC – | FWGC start finish
All ten safe positions. Left of the bar is the near bank, right of the bar is the far bank. Each line is one boat trip.

Draw each safe position as a dot. Join two dots when one boat trip changes one into the other. Now the puzzle is a map. Solving it means walking from start to finish along the lines.

The shortest walk uses 7 lines, so the puzzle takes 7 crossings. There are two such walks.

Watch one: the upper route

Follow the top path on the map. Each step is one crossing.

1. Take the goat over.   WC | FG
2. Come back alone.   FWC | G
3. Take the cabbage over.   W | FGC
4. Bring the goat back.   FWG | C
5. Take the wolf over.   G | FWC
6. Come back alone.   FG | WC
7. Take the goat over.   – | FWGC

Step 4 is the trick. The goat has to travel backwards once. Guessing rarely finds that step. The map shows it straight away.

Your turn

  1. The first crossing has to be the goat. Say what goes wrong for each other choice: the farmer takes the wolf, takes the cabbage, or goes alone.
  2. Write out the lower route on the map as seven steps, like the example.
  3. (a) The position FC | WG is not on the map. Why not? (b) Is FG | WC safe? Is G | FWC safe? (c) Find all six unsafe positions.
  4. (a) The farmer starts on the near bank and ends on the far bank. Why must the number of crossings be odd? (b) Use the map to explain why nobody can solve the puzzle in 5 crossings.

Stretch: a bigger boat

The farmer finds a bigger boat. It holds the farmer and two other things. What is the smallest number of crossings now? Write out the steps, and say why it cannot be done in one.

Answers

  1. Takes the wolf: the goat is left with the cabbage and eats it. Takes the cabbage: the wolf is left with the goat and eats it. Goes alone: all three are left, so both things go wrong. Only the goat can go, because the wolf does not eat cabbage.
  2. 1. Take the goat over (WC | FG). 2. Come back alone (FWC | G). 3. Take the wolf over (C | FWG). 4. Bring the goat back (FGC | W). 5. Take the cabbage over (G | FWC). 6. Come back alone (FG | WC). 7. Take the goat over (– | FWGC).
  3. (a) The wolf and goat are on the far bank without the farmer. (b) Both are safe. In FG | WC the wolf and cabbage are alone together, which is fine. In G | FWC the goat is alone. (c) WG | FC, GC | FW, WGC | F, and the same three with the banks swapped: FC | WG, FW | GC, F | WGC.
  4. (a) Each crossing moves the farmer to the other bank. After 1, 3, 5, 7 crossings he is on the far bank. After 2, 4, 6 he is back on the near bank. (b) The map shows every safe move. Every walk must use the first two lines and the last two lines. It must also use one side of the diamond, which is three lines. That is \(2 + 3 + 2 = 7\) lines. The fewest is 7, so 5 is impossible.

Stretch. Three crossings. Take the wolf and the cabbage over, leaving the goat alone. Come back alone. Take the goat over. One crossing is not enough because the boat carries only two of the three things.

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