Category: Captivating Circles

Captivating Circles BONUS

This bonus science challenge pays tribute to our old Captivating Circles series. It involves physics. You may ask, ‘will there be a Fascinating Frequencies bonus math challenge?’ Yes, 27 weeks after the last problem. Why 27? It’s about 6 months.

Illustration of Cylinders before they begin to roll.

In this bonus challenge, we have a cylinder. It has a smaller cylinder cut out of it. Inside this hole is an even smaller cylinder. The ratio of radii of the big cylinder to that of the hole to that of the small cylinder is 3:2:1, and the smallest one is 0.1 metres. Both the cylinders have equal density. The kinetic friction between the big cylinder and the ground is a kilogram per second. This unit seems weird because we haven’t multiplied by the velocity. That between the small and big cylinders is 5 kg/s. The ground is inclined by 10 degrees in the direction the cylinder is facing. It will begin to roll, and reach a stable speed eventually. The challenge is to figure out how fast.

solution

Captivating Circles 6

This is the final Captivating Circles challenge, this challenge requires the concepts of geometry and trigonometry.

Captivating Circle #6

We begin this challenge with 2 concentric circles. The smaller circle divides the larger 1 into 2 equal pieces. We then draw 2 parallel chords in the large circle, which divide it evenly into three. They do not divide the small circle into three equal pieces. The challenge is then to find the area of the biggest piece in the smallest circle if the large circle has radius 1.

Solution

Captivating Circles 5

You will need to use algebra to solve this challenge. Some programming would be extremely useful as well.

In the video-game “Battle of the Blobs”*, a blob is a circular creature that deals damage over time proportional to its current area. A blob’s area will shrink proportionally to damage dealt to it. The damage over time is not dealt at short intervals, rather is always being dealt, but with the same rate. Our challenge begins when 2 blobs, Jack and Jill, battle each other. The radius of Jack is 1, and the radius of Jill is 1.5. Clearly, Jill will kill Jack. The question is what will be Jill’s radius after fighting Jack, assuming the only change in a blob’s size is caused by damage from the other.

*”Battle of the Blobs” is not a real video-game.

Solution

Captivating Circles 4

You will need all your skills in graph theory, and pathmaking, algebra and combinatorics to solve today’s challenge. Good luck!

Captivating Circle #4

This captivating circle has been divided in to many sections. By combining these sections, you can make many different composite shapes. The challenge is to see how many shapes you can make. This includes the entire circle.

Solution

Captivating Circles 3

In this captivating circles challenge, you will need to use some algebra, some combinatorics, and some serious brainpower.

Captivating Circle #3

For our newest captivating circle, we are dealing with splitting oranges. The orange starts with N slices in a circle. It is then divided into 2 rings with A slices and B slices. In the diagram, N=9, A=3, and B=6. We then count one 3 and one 6. One of these rings is splitted to make 3 rings in all. We then count these rings. This continues until we can no longer split any rings as they all contain 1 slice. It happens that at the end, the number 1 is counted at least 3 more times than 2, which is counted at least 3 more times than 3, etcetera until the largest number counted. The challenge is this: what is the maximum value for N?

Solution

Captivating Circles 2

Today, we have a new captivating circle, and it is the first challenge to require the use of combinatorics. It also requires some trig and pathmaking.

Captivating Circle #2

Our second captivating circle is a circle with 7 points on its circumference. The points are equally spaced. Each point connects to 2 others through the circle and 2 others through a straight line. No points can connect to another through both a line and arc. The circle has circumference π. How many possible different sets of connections can be made? And of these, what is the highest possible distance that must be traversed to get between 2 points?

Solution

Captivating Circles 1

Today we begin a special series, captivating circles. This challenge requires trigonometry and geometry.

Captivating Circle #1

The first captivating circles challenge is to find the area of the shaded region in the circle in the diagram. All vertices of the white triangle are on the circumference of the big circle, and the small circle is tangent to all 3 sides of the triangle. You may find that this problem requires theorems and formula’s not normally required. Do not be afraid to research the tools you need to solve this problem or other captivating circles, it’s knowing which tools to use that’s the challenge.

Solution