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 more thoughts and more questions by Jer)
Running through all starting distances less than 1,000,000,000 inches, I can confirm there is no way to move in one direction every jump, and reach the shore. My best results are as follows:
Unsafe starting lily allowed:
Freddie can manage 61 jumps if he starts 923,316,066 inches off shore.
Only safe starting lilies allowed:
Freddie can manage 57 jumps if he starts 583,969,386 inches off shore.
Now, as for allowing both directions, it gets tricky. While I don't believe he will ever reach the shore, I can't rule it out. If Freddie begins just 13,938 inches off shore, my program crashes after jump #4221 due to limitations of the recursive approach I took, in the Python language. Another 2 rounds of 4220 additional jumps, and he has yet to fall in, leading me to believe from this starting distance and any distance contained in the subsequent jumps, he can perform an infinite number of jumps, getting further and further away from the shore.
Also, we can look at the number of integers that are the sum of 2 odd squares for different limits:
1000 = 81 numbers or 8.1%
1000000 = 53935 numbers or 5.39%
1000000000 = 43260782 numbers or 4.33%
With no further proof than the increasingly scarce distribution of unsafe lilies to land on, it would seem that for an indefinitely broad river, he will approach a 100% chance at landing on safe lilies, and all but guarantee an infinite sequence of jumps.
Edited on November 16, 2011, 8:48 pm
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Posted by Justin
on 2011-11-16 20:41:03 |
re: Some more thoughts and more questions
