30 segments of lengths 1, √3, √5, √7, √9, ... ,√59 have been drawn on a blackboard. In each step, two of the segments are deleted and a new segment of length equal to the hypotenuse of the right triangle with legs equal to the two deleted segments is drawn. After 29 steps only one segment remains. Find the possible values of its length.

Let sqrt(m) and sqrt(n) be two arbitrary members of the set and apply the operation.  The hypotenuse is them sqrt[sqrt(m)^2+sqrt(n)^2] = sqrt[m+n]. This is very simple, the operation just adds the two radicands.

Posted by Brian Smith on 2025-04-29 13:01:10