# Lake Crossing II

*Keywords:*

This activity has a logic and reasoning focus.

follow a chain of reasoning

recognise logical arguments

This is a pure logic problem that doesn’t seem to relate to any other part of the mathematics curriculum. Nevertheless it is an important part of the curriculum. We have included two more problems along the same line. They can be found under Lake Crossing I and Space Crossing. Your students will likely rebel at the suggestion but they should write down all the steps they make in this problem. There is no other way to be sure that they have got things right otherwise. So an answer can’t be accepted unless it is written down. Writing clear solutions is a vital piece of the mathematical puzzle and is necessary for every problem on the web site. Until a solution is there in black and white it is possible that there is a mistake lurking around. Complete written solutions should be encouraged at all stages. The reason that we mention it here is that students seem to like to do logic problems like this ‘in their heads’ or with equipment. Along the way they tend not to write anything down. So at the end, there is no way to check what they have done. Many of them will claim to have solved a problem and it will only be when you go through with them step by step that you will see an error. Now we don’t expect that the answer will be written down in everyday language in all it’s gory detail. Some sort of diagrammatic answer will do. (See the Solution below.) We suggest also that acting it out might be a good strategy, at least for one group of students to explain their answer to the rest of the class.

#### The Problem

Three couples, the Smiths, the Jones and the Browns wanted to go to the Brown’s yacht that was moored a little way out in the lake. There was a dinghy to take them to the yacht but it only held two people at a time. What’s more, each of the wives was jealous of their husbands and wouldn’t let him be with any of the other wives unless she was there.

Can they all get successfully to the yacht?

### Teaching sequence

- Ask for six volunteers to play act each of the characters. (A name tag hung around their necks helps students keep track of who is who.)
- Pose the problem and ask for initial thoughts on the problem.
- Get the students to make suggestions for the first couple of "moves".

*How could we keep track of the moves taken?*(diagrams, symbols and words) - Let the students continue working on the problem. When the students have solved the problem ask them if this is the quickest way to get everyone across.
- Can you get everyone across in fewer moves?
- Remind the students that they need to justify their answer. This means that they will need to record the steps taken.
- Share answers. You may like to get one of the groups to "move" the volunteers while the rest of the class counts the trips taken.

*Did anyone else use the same number of trips? Did you use the same method?*

Did anyone get a different number of trips?

Are you convinced that you have found the fewest number of trips? How do you know? - Look at the various written records. Get students to see if they can follow the reasoning used by other pairs.

#### Extension

Can this problem be done if there are four couples instead of three? (Suppose the new couple is Mr. And Mrs, White.)

Can you make up some more problems like this? Can you solve them?

#### Solution

In the shorthand solution below, Sm, Sf = Mr & Mrs Smith, respectively, Jm, Jf = Mr & Mrs Jones, respectively, Bm, Bf = Mr & Mrs Brown, respectively.

On the land |
On the water |
On the island |

Sm, Sf, Jm, Jf, Bm, Bf |
||

Sm, Sf, Jm, Jf, |
1. Bm, Bf (to yacht) |
(Bm, Bf temporarily) |

(Sm, Sf, Jm, Jf, Bf temporarily) |
2. Bf ( to land) |
Bm |

Sf, Jf, Bf |
3. Sm, Jm (to yacht) |
(Sm, Jm, Bm temporarily) |

(Sm, Sf, Jf, Bf temporarily) |
4. Sm (to land) |
Jm, Bm |

Sm, Sf |
5. Bf, Jf (to yacht) |
(Jm, Jf, Bm, Bf temporarily) |

(Sm, Sf, Jm, Jf temporarily) |
6. Jm, Jf ( to land) |
Bm, Bf |

Sm, Jm |
7. Sf, Jf (to yacht) |
(Sf, Jf, Bm, Bf temporarily) |

(Sm, Jm, Bm temporarily) |
8. Bm, (to land) |
Sf, Jf, Bf |

Jm |
9. Sm, Bm (to yacht) |
(Sm, Sf, Jf, Bm, Bf temporarily) |

(Sm, Jm temporarily) |
10. Sm (to land) |
Sf, Jf, Bm, Bf |

11. Sm, Jm (to yacht) |
||

Sm, Sf, Jm, Jf, Bm, Bf |

Can you do this in fewer than 11 crossings?

#### Extension:

With four couples we proceed by induction. In other words we can use what we have already done with three couples. Just before crossing 7, there are two couples on the land. With four couples go through exactly the same procedure but all the while keep the Whites (Wm, Wf) together on the land. Just before crossing 7 there will be three couples on the land and then we can repeat steps 1 to 11with a different three couples. This gives 17 crossings.

You should then see the pattern for 5, 6, …, n couples.

Attachment | Size |
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LakeCrossing2.pdf | 65.08 KB |

LakeCrossing2Maori.pdf | 72.17 KB |