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 Some thoughts and questions by Jer)

Jer,

As to part 2 of your thoughts and questions, Freddie can start on any lily except the first (see problem) i.e. the lily two inches from the shore; so he can't hop straight onto terra firma. And the river is indefinitely broad, so no jumping around on the other bank!

As to part 1, A017089 is all the lily pads, A097269 is the lily pads other than {42, 66, 114, 138, 154, 186, 210...}. Sequences of hops are certainly possible: for example if Freddie starts 426 inches from the bank, then he can safely jump to 418, then 402, then 378; but then he would fall in at 346 inches from the bank.

 Can he do better than that, though...?

Posted by broll on 2011-11-15 22:56:23