No Three on a Line

In a recent post, I mentioned K-12 Unsolved, the project I’m involved in that aims to publicize 13 unsolved math problems, in the hope that an appropriate version of each problem will find its way into K-12 classrooms. One problem we looked at was posed by Henry Dudeney in 1917. Here is the problem:

Consider an n by n lattice. Is it always possible to choose 2n points in it so that no three points are in a line? 

Here is a non-solution in a 10 by 10 grid:


It fails because there are lines with more than two selected points, and in any case, only 19 points are selected.

Here is a solution with 8 points in a 4 by 4 grid:


Solutions for n=4 are readily found by first graders. I found n=5 easy, but I have not yet solved n=6.

What is great about this problem in elementary school is that it helps establish a foundation for the concept of slope. When discussing whether three points are in a line, students make arguments like: “over 2, down 1; over 2, down 1”.

In middle school or even high school, this would be a fun contest or project that would not need to take much classroom time — just a few minutes to introduce it, and a challenge to find the largest possible array. And it would have the same benefit of reinforcing the concept of slope.

Acknowledgments: I got the graphics from Gord Hamilton. He is the initiator and leader of the K-12 Unsolved project. He introduces the problem to 1st graders by telling them they are building skyscrapers. See the video. (Another possible story might be about eight teachers who want to be able to keep an eye on each other, so no one should be hidden behind someone else.)

Find another lattice problem here.


PS: I have many times done a not-unrelated activity about slope, which also turns out to be about decimals and fractions, with 9th graders. (It could be done in 8th or 7th grade.) On an 11 by 11 geoboard, what are all the possible slopes between 1 and 2? I ask for the answers in both decimals and fractions. Students will tend to do a lot of calculations, but of course all they need to do is set up two rubber bands like this…



…and look at the pegs between them. However this strategy will yield duplicates: points which lie on the same line through the origin will give us the same slope. (Thus the connection with the Dudeney problem.) A good follow-up is to ask for all the possible geoboard slopes between 0.5 and 1. Find this activity, and others like it, in Geometry Labs, a free download on my site.

Of course the geoboard is not required: dot paper will also work.

— Henri

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