More No three in a line

The Problem

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?

Teaching sequence

1. Remind the class of other problems in this sequence.
2. Tell the class Mary’s problem and make sure that they understand it.
3. After some discussion, let the class work on the problem in their groups.
4. Help the students that need it.
5. 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.
6. Ask
   Are all of these arrangements different?
   How might we think of some of them as being the same?
7. Let groups who finish early and have a good grasp of the situation go on to the Extension problem.
8. 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.
9. 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.
10. You might also let the students construct their own windows using transparent coloured paper.

Extension

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?

Solution

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.