A total of points are equally spaced around a circle and are labelled with the integers 1 to , in order. Two points are called diametrically opposite if the line segment joining them is a diameter of the circle. If the points labelled 7 and 35 are diametrically opposite, then what is the value of ?
Problem 631
Official solution
The number of points on the circle equals the number of spaces between the points around the circle. Moving from the point labelled 7 to the point labelled 35 requires moving points and so 28 spaces around the circle. Since the points labelled 7 and 35 are diametrically opposite, then moving along the circle from 7 to 35 results in travelling halfway around the circle. Since 28 spaces makes half of the circle, then spaces make the whole circle. Thus, there are 56 points on the circle, and so .