Clock Right Angle
Logic
Level
3
Between 12pm and 6pm, how many times does the minute and hour hand of a clock form a angle?
10
12
13
none of these
11
9
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Here's an intuitive solution, where we think of the minute hand going around quickly while the hour hand doesn't move "much".
There will first be a 9 0 ∘ angle sometime shortly after the minute hand passes the hour hand, and then there will be another 9 0 ∘ angle when the minute hand is coming back around:
This would lead us to think there are 6 ⋅ 2 = 1 2 right angles. However , note that for 2pm, the right-angle #2 doesn't actually exist, since it is the right-angle #1 for 3pm. Thus, there are 1 2 − 1 = 1 1 right angles.