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?

Questions:  Must he start on a safe lily?
Can he jump backwards?


The numbers under 200 that are the sum of two odd squares:
2, 10, 18, 26, 34, 50, 58, 74, 82, 90, 98, 106, 122, 130, 146, 162, 170, 178, 194

Every one of these is of the form 8n+2 (which is easily proved) unfortunately the available lily pads are also of this form.   The first few that are left are 42, 66, 114, 138, 154, 186, 210.  They they are all spaced too far apart to be a starting pad.

The only significance I can see to these numbers is they are all even and square free.

The OEIS does not have either sequence (but maybe it should.)

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A simple answer to the first question: Yes, if he starts on the last lily he can jump directly to the shore.

What I thought would be another trick:  jump to the near shore and then jump around a bit on dry land.  Since the jumps keep getting bigger, eventually a single jump would be enough to reach the far shore.  Unfortunately I don't think he can reach the near shore.

Posted by Jer on 2011-11-15 16:07:16