More No three in a line
This activity has a logic and reasoning focus.
create geometric shapes that satisfy the no three in a line condition on 4 by 4 grids.
devsie a systematic approach to find possible outcomes
This problem is one of a sequence of 8 that starts reasonably simply and builds up to some quite complicated maths based around the theme of no-three-in-a-line. The full set of problems is Strawberry Milk, Strawberry and Chocolate Milk, Level 1; Three-In-A-Line, Level 2; No Three-In-A-Line, Level 3; No-Three-In-A-Line Again, Level 5; No-More-In-A-Line Level 6; and No-Three-In-A-Line Game, Level 6.
Before you tackle this one we suggest that you look at some of the earlier ones in the sequence, especially Strawberry Milk and No Three-In-A-Line. These two problems cover the essential ideas that are needed here. In particular they deal with systematic searches to justify that all cases have been considered as well as the idea of when such square arrangements are alike.
In the main part of this problem we simply expect students to come up with a few possibilities. We don’t necessarily expect them 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 then they will try to see whether or not these answers are the only ones. In trying to get 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 this into account. Because one answer will often become another by simply rotating it through a quarter turn either clockwise or anticlockwise, the students should see those two answers as being essentially the same. This concept was discussed in the two Level 1 problems.
However, here for the first time, symmetry through a line in the plane of the square becomes important. 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 are concerned with the symmetry of a square.
In the Extension to this problem we hope that as a group, your class is able to come up with all possible arrangements of the five red panes that fit Mary’s condition of there being no three in a row. At the same time, they should try to provide some systematic reason for 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. There is a web site that discusses the no-three-in-line problem. Its url is www.uni-bielefeld.de/~achim/no3in/readme.html.This no-three-in-line problem is still an open problem in mathematics and has an interesting number of sub-problems relating to symmetry.
Mary wants to make a square window made up of 16 smaller red or white square panes. What is the biggest number of red panes that Mary can put in the window so that no three of them are in a line? And in how many ways can this be done?
- Remind the class of other problems in this sequence.
- Tell the class Mary’s problem and make sure that they understand it.
- After some discussion, let the class work on the problem in their groups.
- Help the students that need it.
- Call them all together from time to time to see how many arrangements they have come up with. Get them to take turns in putting a picture of their arrangement on the board. Call each arrangement by the student's name.
Are all of these arrangements different?
How might we think of some of them as being the same?
- Let groups who finish early and have a good grasp of the situation go on to 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. Get the students to write up what they did in their notebooks.
- This problem can be solved by using a computer. It might be worthwhile for some of the class to try to do it this way.
- You might also let the students construct their own windows using transparent coloured paper.
Mary is now working on a square window that has 25 smaller red and white square panes. What is the biggest number of red panes that she can put in the window so that no three of them are in a line? In how many ways can this be done?
The method of solution here is the same as for No Three-In-A-Line. As a result we give an outline to the solution.
The first thing to note is that we cannot have more than two red panes in each row of the window. This means that we can’t fit in more than 8 red panes without forcing three to be in a line. So we systematically try to insert two red panes in each row to see if that is the largest number of red panes that Mary can use. And it seems that it is. Here is the full set of answers.
Note that 1 and 2 are the right- and left-hand forms of the same situation. If we were talking about crates of milk, these two arrangements would be different as we couldn’t rotate or reflect 1 into 2 without all of the bottles falling out of the crate (seeNo Three-In-A-Line, Level 3). However, because we are dealing with a window, we can flip 1 over about an axis of symmetry in the plane of the window, so that it becomes 2. Hence for windows these two arrangements are the same. So there are 4 different solutions to Mary’s problem: 1, 3, 4, and 5.
Extension: The solution method may be becoming routine now, so we just give the answers. With 25 possible squares we can put at most two red panes in a row so that no three are in a line. So let’s try to put 10 red panes in Mary’s window. Amd we cam do this in 5 different ways. Any other way that you can find can be rotated or reflected into one of these 5.