A straight line of water lilies stretches across a broad river. The first lily is a distance of 2 inches from the bank, the next 10 inches , and so on, each 8 inches apart.

Freddie the frog is trying to hop to the shore by jumping from lily to lily. On his first hop, Freddie always jumps 8 inches, just enough to get to the next lily. On each subsequent hop, that champion jumper springs 8 inches further than he did on his previous jump; his second hop is 16 inches, his third 24 inches, and so on.

Such prodigious leaps would seem to ensure that Freddie will always land on another lily, or the bank itself, but not all lilies are equal. If Freddie lands on a lily whose distance in inches from the bank is a sum of two ODD squares, then the lily breaks and he falls into the river; if not, the lily is safe, and he may continue to his next hop.

First question: Given that Freddie can start on any lily except the first, can he ever reach the shore safely without falling in?

Second question: If the answer to the first question is 'No', then what is the greatest number of hops that Freddie can make before falling in?

(In reply to re: Some thoughts and questions by broll)

We had a difference of interpretation.  I was picturing the 2 inch lily as being near one shore with the goal being the far opposite shore.  After all every river has two banks.  The far shore, being the goal, was at an unknown distance.

From now on I'll stick with your interpretation as it makes more sense.

Is it still ok to jump in either direction?

Posted by Jer on 2011-11-16 09:07:08