Cutting Circles!

Geometry Level 4

Consider a circle with no lines passing through it, which only has 1 region.

Now, draw a line through that circle. Then this line can cut the circle into 2 regions at most.

Draw another line, and these two lines together can cut the circle into at most 4 regions. (Note: the regions can’t overlap but don’t have to be congruent.)

Keep continuing this process such that you maximize the number of regions for each cut.

What is the number of regions after 1000 cuts?

Bonus: How many regions can we make with $n$ lines?

The answer is 500501.

1 solution

Zico Quintina
May 21, 2018

In order to maximize the number of regions, we must ensure that every new cut crosses every existing cut without going through any intersection points of earlier cuts.

Suppose you are making the $k$ th cut. As soon as you start the cut at the circumference, you are cutting one existing region into two, thereby increasing the number of regions by one; every time you cross an earlier cut, you are again cutting an existing region into two, so again the number of regions goes up by one. Since there were $(k-1)$ cuts before you started, the $k$ th cut increases the number of regions by $1 + (k - 1) = k$ regions.

Then the total number of regions after $n$ cuts is $1 + 1 + 2 + 3 + 4 + \dots + n$ (the first $1$ being the original circular region), which can be written as $1+ \dfrac{(n)(n + 1)}{2}$ .

Thus after $1000$ cuts we could make $1 +\dfrac{(1000)(1001)}{2} = \boxed{500501}$ regions.

Cutting Circles!

The answer is 500501.

1 solution

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