What is the number of times the hour hand and the minute hand of a clock form
a right angle with each other between 0600 and 1200 on the same day?
Explain. (No second chances on this one)
Solution to the Problem:
The answer is eleven times.The times that the hour hand and the minute hand of a clock form
a right angle with each other between 0600 and 1200 are approximately
at 0617, 0649, 0722, 0754, 0828, 0900, 0933, 1005, 1038, 1111, 1149.
Note that this happens twice every hour, except between 0800 and 1000
when it happens only three times and not four times as expected.
This is because at 0900 exactly the hands form a right angle.
Thus between 0600 and 1200 it happens (6 x 2) - 1= 11 times.
James Alarie sent in this excellent analysis of the problem:
The hour hand circles the clock face (360 degrees) once every twelve
hours, so it moves 360/12 degrees (30 degrees) each hour.
The minute hand circles once every sixty minutes; it moves 360/60
degrees (6 degrees) each minute.
As the minute hand moves, the hour hand moves 1/12 as much. For them
to be at right angles, they must satisfy one of these equations:
30 * H + 6 * M / 12 - 90 = 6 * M
30 * H + 6 * M / 12 + 90 = 6 * M
Solving for M in each of these gives:
M = (60 * H - 180) / 11
M = (60 * H + 180) / 11
Putting hour 6 through 11 into these equations gives twelve results, but
the last one for 11 is past 12:00 and doesn't count. The exact values are:
6:16:21 9/11
6:49:05 5/11
7:21:49 1/11
7:54:32 8/11
8:27:16 4/11
9:00:00
9:32:43 7/11
10:05:27 3/11
10:38:10 10/11
11:10:54 6/11
11:43:38 2/11
Correctly solved by:
| 1. Martin Round | Studley, Warwickshire, United Kingdom |
| 2. James Alarie | Flint, Michigan |
| 3. Miko Zhang |
Delta High School, Delta, Colorado |
| 4. Keegan Genzer |
Mountain View High School, Mountain View, Wyoming |
| 5. Austin Smith |
Mountain View High School, Mountain View, Wyoming |