At my annual physical exam, the nurse gave me a sheet of paper with a large circle, and she asked me to draw the time 11:10. I understand this is called the Clock Drawing Test, and it is a screening tool to help detect early signs of dementia.
I drew the two numbers 11 and 2 in the correct positions, and 12. I felt no need to draw all the other numbers. Then I drew a little hand (hour hand) approximately on 11, and a big hand (minute hand) on 2.
Of course, only I would then worry about the angle that the two hands make.
There are twelve evenly spaced integers on a conventional clock. At three o’clock, the hands are on the 12 and the 3, which are three numbers apart, and the hands make a 90 degree angle. For every difference of 1 on the clock, the hands are 360 times 1/12 = 30 degrees apart. My first thought was that at 11:10, the hands are also three numbers apart, so I assumed the hands also make a 90 degree angle. The 90 is the sum of 30 degrees between 11 and 12, plus 60 degrees between 12 and 2.
More precisely, at 11:10, the hour hand is somewhat past the 11. It is 10/60 = 1/6 of its way toward 12. So my initial assumption needs to be modified. Now I assume the angle that the two hands make should be 60 degrees (between 12 and 2), plus 30 reduced by a factor of 1/6 (between 11 1/6 and 12), or 60 plus 30 times 5/6, or 85 degrees. I do own a protractor (what do you mean – doesn’t everyone?), but I didn’t bring it with me for my physical exam. So I drew the two hands at something less than a 90 degree angle, although not as exact as I would have liked.
I did not share any of my thinking on this with the nurse, assuming she would not be interested. She may have wondered why it took me so long to draw the two hands.
It turns out there is a general formula. For a time of h hours and m minutes (with h∈{0,1,…,11}): "Angle"= ∣ 30h-5.5m ∣. At 11:10, ∣ 30*11 −5.5*10 ∣ = ∣ 330 − 55 ∣ = 275, and the smaller angle is 360 – 275 = 85.
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