Cutting A Cross

Probability Level 1

By using 2 straight lines, what is the most number of regions that we can separate the cross into?

Note: Using 1 straight line, it is possible to separate the cross into 3 regions.

4 5 6 8

5 solutions

Marta Reece
Apr 19, 2017

The two straight lines can intersect only once, and the most pieces are produced if this happens within the cross.

Do you know if there a more rigorous way to prove that the answer is 6?

Chung Kevin - 4 years, 1 month ago

Alan Parks
Apr 25, 2017

This is meant to be an arguement that you cannot exceed six , rather than showing you how to create six (see other solutions). Before adding any lines we have one distinct region, when we create a new region, I see this as shrinking an already existing region and placing in a new one. To create a new region we must add a line that joins two points of the perimeter, if you can leave the shape and re-enter it you can potentially make more than two intersection points and consequently more than one extra region.

At this point, you can check and convince yourself that firstly, the number of new distinct regions (ignoring the shrunken original(s)), is half of the number of intersections with the perimeter. And secondly, that for this shape it is not possible to place a line with more than four intersections with the perimeter. Hence for the first line you have a maximum of two new regions and three regions in total.

When placing a second line, aswell as maximising the number of perimeter intersections, we can also create more regions by maximising intersections with previously place lines. It is also worth convincing yourself that each intersection (if inside the shape) with an already placed line creates exactly one new region.

Putting these ideas together, you will find that for the second line you are restricted to four intersections with the perimeter and one intersection with already placed lines, giving you a maximum of three further new regions (4/2 + 1) and six regions in total . You can continue this arguement to show that there are 10 regions for three lines and 15 regions for four lines, however I haven't considered the limitations of intersection already placed lines for a large number of lines.

Counting the points of intersection is a good way to get started.

There is a bijection we can create, which is similar to your idea of "half of the number of intersections with the perimeter + number of intersections within the region". As with the more standard problem where we cut up the circle with lines, we can look at the "bottom most point of each region". The slightly tricky part is then determining what set this gives us a bijection with.

Chung Kevin - 4 years, 1 month ago

Venkatachalam J
Apr 24, 2017

Using one straight line,we can separate the cross into 3 regions. Two straight lines interest only one time. So, we will get max 6 regions in total.

Do you know if there a more rigorous way to prove that the answer is 6?

Chung Kevin - 4 years, 1 month ago

0 lines = 1 region, 1 line = 3 regions, 2 lines = 6 regions y = regions x = lines Y=x^2+x+1

Nathaniel Jett - 3 years, 5 months ago

number of regions=[x^2+x+2]/2 + x, where x is number of lines

Venkatachalam J - 3 years, 5 months ago

Alkis Piskas
Apr 28, 2017

I could create 6 pieces:

You need to show that creating more than 6 pieces is impossible. How would you show that?

Christopher Boo - 4 years, 1 month ago

Eddy Zavala
Apr 26, 2017

The pattern is 3n, where is n is the number of lines. In this case n=2 so 6.

That pattern does not hold for $n = 3$ . You can get 4 additional pieces with the 3rd cut.

Chung Kevin - 4 years, 1 month ago

Cutting A Cross

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