Enigma 1017: Paint the line
29 March 2019
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From New Scientist #2173, 13th February 1999 [link]
Pussicato, the great artist, is starting his new commission. The canvas is a horizontal line, 6 metres long, and he has to paint parts of it red according to a rule he has been given. He selects a point P on the line and measures its distance, x metres from the left hand end.
He then works out the number:
1/(x – 1) + 2/(x – 2) + 3/(x – 3) + 4/(x – 4)
If the number is 5 or more then he paints the point P red, otherwise he leaves it unpainted.
For example when x = 2.1 he gets the number 15.47… , which is more than 5, and so he paints P red. And when x = 1.7 he gets –9.28…, which is less than 5, and so he leaves P unpainted.
Pussicato repeats this for every point of the line, except those with x = 1, 2, 3 or 4, which he has been told to leave unpainted.
When he has finished he finds that four parts of the line are painted red and their total length is a while number of metres. (Pussicato could have worked all that out without doing the painting).
What is the total length of the red parts?