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 more thoughts and more questions by Justin)

Justin,

This is just the sort of exploration I was hoping someone would attempt. For example, I found a 16 link chain starting at 201498426 (and starting on a 'safe' lilypad):

U:        2×3^2×71×157667
(U-8):   2×11×863×10613
(U-24):  2×3×7×4797581
(U-48):  2×3×33583063
(U-80):  2×7×67×214817
(U-120):  2×3×19×47×37607
(U-168):  2×3×13×43×60077
(U-224):   2×31×3249971
(U-288):  2×3^3×3731447
(U-360):  2×3^2×7×11×145381
(U-440):  2×23×4380391
(U-528):  2×3×7^2×685367
(U-624):  2×3×11×23×132739
(U-728):  2×19×41×283×457
(U-840):  2×3×33582931
(U-960):  2×3×1231×27281
(U-1088): 2×7×37×388991

But I can't believe that this is a minimum for those conditions. How close to the shore can Freddie start and still make, say, 16 jumps safely? Is there any way of ascertaining these minima other than by brute force/trial and error? (compare Factor numbers, an earlier problem). Are there good places to start from, e.g. a number that is a product of a lot of small primes, or just 6 (or twice some other power of 3) times a big prime?

The hopping back and forth variant is also fascinating - can Freddie reach the shore if he is allowed to hop in either direction?

I appreciate that there is a quick answer to both questions as first put, but sometimes the journey is no less interesting than the destination!

Edited on November 16, 2011, 11:41 pm

Posted by broll on 2011-11-16 23:06:37