January 2023 Puzzler

Question

It seems to have turned cold a bit early this year. Here’s a puzzle appropriate for the onslaught of winter.

A ball was floating in a pond when the pond froze. When the ball was removed it left a dent in the ice that is 6 inches deep and 20 inches across. What is the diameter of the ball?

Solution to the September Puzzler

The angle X is 54 degrees. We’ll use the diagram shown here, in which points H and F represent my head and feet, while M and A are the tips of my morning and afternoon shadows. Here are some facts we can derive from the problem statement.

• First, segments FH, FM, and HA are all equal to my height.
• That makes triangles FHM and HFA isosceles.
• So angle(FHM) and angle(FMH) are equal.
• Both angle(FHM) and angle(FHA) are complementary to angles of size X, so we can now say that angle(FHM), angle(FMH), and angle(FHA) (all shaded yellow in the diagram) are all 90 – X.
• The angles of triangle FHM must add to 180, so angle(HFM) must be 180 – 2(90 – X) = 2X.
• That means angle (HFA) is 180 – 2X, and since triangle HFA is isosceles, angle(HAF) is also 180 – 2X.
• The three angles of triangle HFA must add to 180, so we get an equation:

2(180 – 2X) + (90 – X) = 180

270 = 5X

X = 54

Correct Solvers:

First correct solution: Alex Blumenfeld, NMR Lab Manager, U of I Chemistry

Second correct Solution: Greg Stenback, B.S. Geological Engineering 1984, M.S. Statistics 1987