# A 40 X 40 white sware is divided into 1 X 1 squares by lines parallel to its sides.﻿﻿﻿Some of these 1 X 1 squares are coloured red so that each of the 1 X 1 squares, regardless of whether it...

A 40 X 40 white sware is divided into 1 X 1 squares by lines parallel to its sides.

﻿﻿﻿Some of these 1 X 1 squares are coloured red so that each of the 1 X 1 squares, regardless of whether it is coloured red or not, shares a side with at most one red square (not counting itself). What is the largest possible number of red squares?

samhouston | Certified Educator

I began by drawing a 10x10 white square and divided it into 1x1 squares.  This was much more manageable than a 40x40 square.

Going from left to right:

a column of 10 white

a column of 2 red, 2 white, 2 red, 2 white, 2 red

a column of 10 white

a column of 2 white, 2 red, 2 white, 2 red, 2 white

a column of 10 white

a column of 2 red, 2 white, 2 red, 2 white, 2 red

a column of 10 white

a column of 2 white, 2 red, 2 white, 2 red, 2 white

a column of 10 white

a column of 2 red, 2 white, 2 red, 2 white, 2 red

This made 26 red small squares in the 10x10 square.  Multiply that by 4, and you get 104 red small squares in the 40x40 square.

The largest possible number of red squares is 104.

isamitto | Student

I got 28 on my 10 x 10 square. And I think it may be possible to fit even more, by staggering the red squares from row to row.

But I have yet to find a pattern.

On my 12 x 12 square, I fit in 40 red squares.

The answer is supposedly 420, so if anyone has any explanation on how they got this, I'd really love to hear any ideas. Thanks.