No More in a Line
This activity has a logic and reasoning focus.
use symmetry and rotation to create geometri shapes that satisfy the no three in aline condition
explore symmetric arrangements
This problem is essentially the same as No-Three-In-A-Line Again, Level 5 but, of course, a little harder. It 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. These problems are Strawberry Milk, Strawberry and Chocolate Milk, Level 1; Three-In-A-Line, Level 2; No Three-In-A-Line, Level 3; More No-Three-In-A-Line, Level 4; No-Three-In-A-Line Again and Level 5 No-Three-In-A-Line Game<, Level 6. It might help if you have done the Level 3, 4, and 5 problems with your class before you start on this problem.
In this problem we revisit the three steps that were introduced in Strawberry Milk and Strawberry and Chocolate Milk. You may recall that these are
- 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.
Making a systematic check of all possibilities best produces the complete answer to this question. This approach has been used in No Three-In-A-Line, Level 3, More No-Three-In-A-Line, Level 4 and No-Three-In-A-Line Again, Level 5. In getting all the possibilities, we hope that the children will realise that there is a lot of symmetry in Mary’s square window. So their answers should take all the symmetries of the square window into account.
In the Extension to this problem we hope that, as a whole, your class will be able to come up with all possible numbers of red panes that fit Mary’s two conditions. At the same time, they should try to provide some systematic reason for why there are no more answers.
On the Statistics side, we are trying to count all possibilities. This is an important part of determining probabilities, which, in itself, is an important aspect of Statistics. On the Geometry side, we are concerned with the symmetry of a square.
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.
The Problem
Mary is now looking to build a larger square window made up of 16 red and white panes. What is the smallest number of red panes she can put into the window so that no three of them are in a line and so that she cannot put in another red pane without three being in a row?
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Teaching sequence
- Talk about square windows and their symmetry. Talk about similar problems and how they were tackled.
- Tell the class Mary’s problem and discuss any difficult ideas.
- Let the class work on the problem in their groups. It might be useful for them to write a computer program as this will save them the tedious checking processes. (Computers can be very good at doing boring systematic checking.)
- Help the children that need it.
- Call them all together from time to time to see to check on their progress.
- Try to get them to see how to use a systematic approach to get all possible answers. Let the more able students work on the Extension problem.
- Let a few groups report back to the whole class. Try to choose groups that have used different approaches to the problem.
- Discuss their conclusions.
- Give the children the opportunity to write up their solutions in a notebook.
Extension
(a) Mary now wants to put 25 panes in her red and white window. What is the smallest number of red panes she can put into the crate so that no three of them are in a line and so that she cannot put in any more red panes without three being in a row?
(b)Mary is now looking to build a larger square window made up of 16 red and white panes. What numbers of red panes can she put into the window so that no three of them are in a line and so that she cannot put in another red pane without three being in a row?
Solution
In the same way as before (see No-Three-In-A-Line Again, Level 5), we can show that three won’t work. However, 4 will and in two ways. This can be justified by working systematically as we have done before in that same Level 5 problem. The two possible answers are shown below.
| Attachment | Size |
|---|---|
| NoMoreinaLine.pdf | 48.34 KB |
| NoMoreinaLineMaori.pdf | 55.15 KB |
| NoMoreinaLineCM1.pdf | 46.42 KB |
| NoMoreinalineCM2.pdf | 47.39 KB |
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