# The Paintings Puzzle

An art gallery features a modern work of ‘moving art’. The artist stands by a stack of paintings, each featuring a different number. One of the paintings is displayed on the wall. At certain times the artist removes the painting from the wall and replaces it with a painting from the stack. At 11am, the artist hangs a painting of the number 30. At 4pm he hangs a painting of number 240. At 7.30pm he hangs a painting of number 315. What painting does the artist hang at 9.20pm?

The answer is 200. The explanation is that the painting number equates to the number of degrees between a clock’s hour-hand and minute-hand (measured in a clockwise direction). The first three examples are easy if you sketch the clock hands on a clock face and plot the hours around the clock face (bear in mind there are 360 degrees around a circle; the 12 on the clock-face equates to 360 (or zero) degrees, and each hour equates to 30 degrees, being one-twelfth of 360). The puzzle question (9.20pm) is more difficult to calculate than the first three time examples. Here are my two attempts to explain it:

method 1- Each hour on the clock face equates to 30 degrees (12 x 30 = 360). From 9 to 4 on the clockface moving clockwise is 210 degrees (7 hours x 30 degrees = 210 degrees). But the hour hand is not on the 9, it’s one-third of the way to 10 (the time being 9.20, not 9.00). This one-third (being 20 minutes of a 60 minute hour) equates to 10 degrees (10 is a third of 30 degrees). Therefore the angle in degrees between the hour hand and the minute hand at 9.20 is 200 degrees (210 – 10).

method 2 – At 9:20 the minute hand is at the number 4 which is 120 degrees from zero (4 is a third of 12, hence a third of 360 degrees is 120 degrees). The hour hand is at a position equating to 560/720 minutes (there being 720 minutes in 12 hours, and 9hrs 20mins being 560 minutes). 560/720 equates to 280/360 (360 is half of 720, and half of 560 is 280), so the hour hand is at 280 degrees from zero (remember zero is 12 on the clock face). Measured in a clockwise direction, the number of degrees (or angle) between the hour hand and the minute hand is 80 degrees around to to the 12 (at zero degrees), plus 120 degrees from the 12 to the 4 (which we previously established). Then simply add: 80 + 120 = 200.

### Your Answer

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