May 2023 Puzzler

Question

There is a time between 7:00 and 7:30 when the hour and minute hands of a clock are perpendicular to each other. To the nearest second, at what time does this occur?

Solution to the January Puzzler

The ball is 68/3 = 22.666... inches in diameter

Let's let r represent the radius of the ball. A good choice of coordinates really helps with this puzzle. If we think of the xy-plane as a cross section through the center of the ball (with the origin (0,0) at the ball's center) then the bottom of the ball is at the point (0,-r) and the top of the ice is 6 inches above this at the line y = 6 - r.

The hole left by the ball is 20 inches across, so the distance from the center of the hole out to its edge is 10 inches. That means that the line y = 6 - r will hit the edge of the ball at the point (10,6-r). (See the diagram.)

But now the profile of the ball in our diagram is a circle centered at the origin with radius r, so its equation is

x2 + y2 = r2

Plugging in the x and y values for our point (10,6-r) we get

100 + (6-r)2 = r2

100 + 36 - 12r + r2 = r2

136 = 12r

r = 136/12 = 34/3

So the radius of the ball is 34/3 inches, making its diameter equal to 68/3 inches

Correct Solvers:

First correct solution: Keith and Nancy Stutler, B.S. Mathematics, 1987

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

Others:

• Alex Blumenfeld, NMR Manager, U of I Chemistry
• Fred Burton, B.S. Mathematics 1968
• Mark Daily, B.S. Physics, 1981
• Carey Edwards, B.S. Forest Products 2002, GIS Certificate 2010
• Gary Green, B,S. Mathematics 1964
• Duke Hughes, B.S. Chemistry 1961
• Robert Johnson, Ph.D. Physics 1973
• Marianne Milander, Student (Animal and Veterinary Science, Microbiology)
• Roy Olson, M.S. Mathematics 1970
• John Stutz, M.S. Physics 1973