Snowball Problems Melted

The snow from the record setting snowfalls in Boston are pretty much gone, so here are solutions to the two snowball problems from the February.

Solution to Snowball Problem #1.

We claim that every point in space on the side of the plane with the rolling snowball and within a snowball diameter of the plane can be visited by the marked point.

Let C be the circumference of the snowball. Any positive distance less than 2C can serve as the base of an isosceles triangle with equal sides of length C. Draw this triangle in the plane and put the snowball at one endpoint of the base with its marked point touching the plane. Roll the snowball along the two sides of length C. It ends up at the other endpoint of the base with its marked point touching the plane. This shows that the marked point can visit any point within a circle of radius 2C from its starting point. By taking several such journeys, we see that every point in the plane can be visited. For any point P on the same side of the plane as the snowball and within a snowball’s diameter of the plane, position the snowball so that its marked point is at P. We can roll this snowball until the marked point is touching the plane and since all points in the plane are visitable, we conclude that P can also be visited.

Solution to Snowball Problem #2 (Noah Fechtor-Pradines).

We claim that the locus of visitable points is a cardioid of revolution.

Let’s mark the point on the sphere where the marked point on the snowball touches it at the start with a golden dot. Look at the situation from the point-of-view of the common tangent plane. The two balls roll as mirror images of each other. The marked points are reflections of each other in the common tangent plane.

In other words, no matter how the snowball rolls about the sphere, the marked point will be located at the reflection of the golden dot in the common tangent.

We conclude that the locus of visitable points is a surface with rotational symmetry (about the axis one of whose poles is the golden dot). A vertical cross section by a plane containing the pole of the stationary sphere is the curve traced out by a dot on a circle as it rolls about a circle of the same diameter, and this is known to be a cardioid.




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