Gridlines
How many triples of points are there in this 4x4 array that lie on a straight line?
Age
14 to 16
Challenge level
Problem
There are lots of ways of choosing three dots from this $4$ by $4$ array.
Image
How many triples of points are there where all three lie on a straight line (not necessarily equally spaced)?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Consider the top row of four dots. One can obtain a triple of dots by eliminating any one of the four - so there are four such triples. The same is true for each of the four rows, each of the four columns and the two main diagonals, giving $40$ triples.
In addition there are four diagonal lines consisting of exactly three dots, so there are $44$ triples in total.