Lots of regions

Probability Level 2

In the figure, 25 pink lattice points compose a $4 \times 4$ square grid. The pink lines connect all of the outer points. Then the 4 parallel blue segments and another 4 parallel green segments divide the whole figure into 25 regions.

If the grid is $2018 \times 2018$ and lines are drawn in the same fashion, how many regions will there be?

Clarification: The starting and ending points of each segment have a difference of 1 in their $x$ - or $y$ -coordinates.

The answer is 4076361.

2 solutions

X X
May 2, 2018

The formula is $4n+(n-1)^{2}$ . $(n-1)^{2}$ is the inner square.The other parts can be devided into four same triangles,every triangle contains $n$ regions,so it's $4n$ .

Could you maybe explain how you derived this formula (although I think I know how you did)? I found a different way of deriving the formula which also gives the same when rearranged

Stephen Mellor - 3 years, 1 month ago

$(n-1)^{2}$ is the inner square.The other parts can be devided into four same triangles,every triangle contains $n$ regions,so it's $4n$ .

X X - 3 years, 1 month ago

This was how I thought you did it. Maybe you could put this sentence into your solution and then other people wouldn't need to look in the comments

Stephen Mellor - 3 years, 1 month ago

@Stephen Mellor – I edited now.

X X - 3 years, 1 month ago

Nahush Kanitkar
May 9, 2018

This problem also has a simple solution. No of sides in example is 25 which is equal to number of dots of sides square (5^2). Therefore it could be reimagined as a 5*5 square . Therefore number if number of sides is n then number of regions would be (n+1)^2 . Or you could just solve it as xx by the formula 4n+(n-1)^2, this formulas derivation is given in the below solution by xx.

Lots of regions

The answer is 4076361.

2 solutions

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