Three In A Line
This activity has a logic and reasoning focus
rotate pattern through quarter and half turns
be systematic to count possibilities
This problem is one of a series of 8 that builds up to some quite complicated maths based around the theme of no-three-in-a-line. That theme is not obvious here because we are putting three bottles in a line! However, this problem helps in the development of the theme as we move through the Levels. The other lessons in the series are Strawberry Milk, Strawberry and Chocolate Milk, Level 1; No Three-In-A-Line, Level 3; More No-Three-In-A-Line, Level 4; No-Three-In-A-Line Again, Level 5; No-More-In-A-Line Level 6; and No-Three-In-A-Line Game, Level 6.
We suggest that you have at least looked at the lessons from Level 1 before tackling this problem.
On the Statistics side, we are trying to count all possibilities. This is a precursor to determining probabilities, which is an important part of Statistics. On the Geometry side, we shall need to talk about symmetry in order to decide which arrangements of the bottles lead to different arrangements.
But above all, this problem is about students trying to go through the following steps:
- find some answers to a problem;
- think about whether there are any more answers or not;
- try to explain why there are no more answers.
We don’t necessarily expect students to find all of the answers by themselves. What we do expect though is that they will try to find more answers than they have got and in the end have some systematic idea as to why there are no more answers. This is because in the end these are three important skills that go throughout all mathematics (and maybe life itself); first being able to find some possibilities, then getting all possibilities and then justifying that there are no more.
Note that the basis for two arrangements being alike is discussed in Strawberry Milk.
There is a web site on the no-three-in-line problem. Its url is www.uni-bielefeld.de/~achim/no3in/readme.html. This is still an open problem in mathematics and has an interesting number of sub problems relating to symmetry.
Mary the milk lady had a square milk crate that could hold nine bottles. In how many ways can she put three strawberry milk bottles in the crate so that they form a line?
- Tell the class Mary’s problem.
How can you solve the problem?
What might you need to help you?
- After some discussion, let the class go into their groups or work alone.
- Help the students that need it.
- You may need to call them all together at some stage to see how many arrangements they have come up with. Get them to take turns in putting a picture of one of their arrangements on the board. Call each arrangement by the student's name (see Solution).
How do you know if there are any more?
How do you know if two answers are alike?
- Try to get them to see the systematic approach that we used in the Solution to get all possible answers.
- Students who are successful with this problem should go on to the Extension.
- Let a few groups/students report back to the whole class. Try to choose groups that have used different approaches to the problem.
Mary the milk lady had a square milk crate that could hold 16 bottles. In how many ways can she put four strawberry milk bottles in the crate so that they form a line?
The easiest way to be systematic here is to look at where the end bottle of the three in a line can be. Just take it systematically round the crate starting at, say, the top left-hand corner of the crate. This appears to give 3 answers but 1 and 3 are alike because a quarter turn of the crate will take 1 into 3. So we have two non-alike arrangements here. (See the Solution to the Strawberry Milk problem.)
we also now know that we don’t have to think of any other end bottle as being in a corner square. This is because any such possibility and 1 or 2 would be alike. Just try the quarter turn test test.
This now only leaves an end bottle of a line of three being in the middle square on a side. But this gives only one possibility. Anything else can be turned into this by a (series of) quarter turn(s).
So there are three different solutions, 1, 2, and 4.
By going through the methods of Strawberry Milk and the Solution above, we get the following three possibilities.
You might like to think about what happens with 5 bottles in a line in a 5 by 5 crate. We think that the answer is 4.
Do you get the same answer for 6 bottles in a line in a 6 by 6 crate? What is the general pattern?