Cover the Spot

Geometry Level 4

You go to a carnival and decide to play the Cover the Spot game.

The rules are simple: you must cover the largest circular spot possible using 5 identical circular disks. Which configuration can cover a larger spot?

A) All 5 disks pass through the center of the spot.
B) Exactly 3 disks pass through the center of the spot.

B Neither of these options A They are both optimal

5 solutions

Ayush Kumar
Apr 28, 2017

With configuration A, you have a lot of more overlap than option B. This means that the disks in configuration B can cover more space and therefore can have a lower radius for the disks.

This shows that B is the preferred version. It does not exclude the possibility that neither of these options is actually the optimal solution.

Marta Reece - 4 years, 1 month ago

Question is not about lowering the radius of disks.it only asks to cover larger circular spot with identical disks. I think, when we cross disks through the centre it defines the maximum radius for the circular spot.(that petal shaped area). So no matter where you put the two remaining disks(other 4 petals might increase the radius but max. R Is alredy decided, otherwise some area will be uncovered.). Then covered spot will be same as one in case A. Or is there something to the question i am missing....

pranav patil - 4 years ago

Maximum radius is not defined as the angle between the three disks that cross through the centre of the spot can be changed. In A, all disks have their centres on the points of a pentagon - i.e. they are 72 degrees apart. In B you can fix the top disk and reduce the angle between the two lower disks which increases the spot radius. I do not know a geometric proof for this, but by experiment I found that an angle of approximately 69.9 degrees between the lower two disks allows the remaining two disks to be placed so that a spot with a diameter approx. 1.31% larger can be covered.

Steven Linnell - 4 years ago

If you want to see the exact solution, check out this link

Geoff Pilling - 4 years ago

B IS prefered but both are optimal

Rachit Pulhani - 4 years ago

Actually, it turns out that B is the best you can do! Although I'm not yet sure how to prove that...

Geoff Pilling - 4 years ago

I don't think accounting for "a lot of overlap" is sufficient. E.g. the "4 disc through center" performs worse than "3 disc through center".

In part, because we want to cover the largest circle, hence there will also be wastage on the perimeter.

Calvin Lin Staff - 4 years ago

A and B are required to cover the same amount of space because remember we are COVERING (completely) a circle. The thing that gives it away is that the points of intersection between two discs, is always on the circumference of the circle. To answer the question intended, it should have been worded such as: Find the arrangement that has the most overlap in the circle, because the concepts are the same for the way the question was written.

Thomas Hill - 4 years ago

I agree with Pranav. If 3 of the 5 pink circles are required to pass through the centre of the blue spot, it doesn't matter that we can move the remaining to pink circles, if we increase the size of the blue spot it will result in a small area not being covered. In the above graphic, that are would be at the bottom of diagram B. Diagram B would allow you to cover a larger "area" but not a larger "spot". Therefore both result in the same size blue spot covered.

Tim Dowling - 4 years ago

I agree with Pranav and Tim, unless the 5 discs in B are larger than the 5 discs in A.

Richard Levine - 3 years ago

I added a visual explanation of why the optimal position of A can lead to a larger circle by relaxing the constraint to B.

Calvin Lin Staff - 3 years ago

Calvin Lin Staff
May 25, 2018

(Sorry for the really rough image)

Let's start with the optimal position of A. The large blue circle intersects the smaller circles at 5 points, which form a regular pentagon.

Because the side length of the pentagon is less than the diameter of the small circles (how do we know this?), if we move 2 non-adjacent circles so that their center lies on the side of the pentagon, then this gives extra space for the blue circle to expand outwards.

If you take the case of 100 circles with only 3 that have to pass through the same point, then this argument becomes much more obvious.

If you take the case of 4 circles, what happens then?

John Barron
May 17, 2017

We suspect that B covers a larger area. To show this intuitively, consider any two non-adjacent disks in configuration A. If we move these disks radially outward, the other three disks still cover the innermost area, and the outermost points at which these three circles intersect the two circles we moved changes. To enclose the largest possible area, we want these intersections as far from center as possible. In configuration A, the angle between adjacent circle centers is 360/5=72 degrees. A 72 degree angle subtended within a circle has terminal points less than the maximum distance apart that they can be. The maximum is achieved when a 90 degree angle is subtended within the circle. Hence we should move two non-adjacent circles radially outward until the outermost intersection points lie on diameters of these two circles. Configuration B is closer to this than configuration A, so configuration B covers a larger circle.

Nice write-up @John Barron !

Geoff Pilling - 4 years ago

Ishwar Karthik
May 15, 2017

$B$ has way less overlap than $A,$ so it's intuitive that a larger circular spot can be covered.

Yes, now can you prove your intuition?

Pi Han Goh - 4 years ago

M H
May 15, 2017

Bigger the spot, B covers more..

Not sure I understand the comment, you mean B covers more than a smaller spot, or B covers more than A?

Geoff Pilling - 4 years ago

Cover the Spot

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