perplexus.info :: Just Math : Recursive Pythagoras

Recursive Pythagoras (Posted on 2025-04-29)

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.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

Comment 2 of 2 |

Suppose √a and √b are any two of the segments.

They will be replaced by √(a+b).

If the next is √c, then √(a+b) and √c will be replaced by √(a+b+c).

The final segment will be √(1+3+5+...+59)

= √900 = 30 regardless of the order of which pair of segments is replaced

Posted by Larry on 2025-04-29 13:01:42

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