# 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?

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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.**

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.