In tic tac toe, you draw 4 lines, 2 horizontal and vertical to create 9 boxes. What’s the maths for figuring out how many boxes or sections you’ll create by drawing a certain number of intersecting lines

Question

I hope that question makes sense. I’m an adult and maths may as well be a foreign language for me. This is my secret shame.

Anyway, let’s say I have a sheet of paper and I want to draw 4 vertical and 4 horizontal lines, how do I figure out that I will actually get 25 squares without counting?

In: Mathematics

Answer 1

Let n be the number of vertical lines you draw and m be the number of horizontal lines you draw. Any space* will be cut into (n+1)*(m+1) areas. Multiplying this out give you n*m+n+m+1 spaces. As to why you get n+1 and m+1 spaces, just look at the case where you draw 1 vertical line and no horizontal lines. The one vertical line gives you 2 areas; this can also be extrapolated from the fact that a line cuts a plane into two half-planes.

In the case of 4 vertical lines and 4 horizontal lives you get 5*5=25 or 4*4+4+4+1=16+4+4+1=25. This works for any number of vertical or horizontal lines.

* Just a note here, when I say any space I really mean any 2D space. 3D or higher spaces tend to get really picky about lines actually cutting/separating things into unique spaces.