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There are 3 solutions to Barry's puzzle: 
64, 36 or 16 
Cuckoo's speeds: 8, 6 or 4 ; wind's: 1 
 
8/1 = 8: 8^2 = 64 
6/1 = 6: 6^2 = 36 
4/1 = 4: 4^2 = 16 
 
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@Denis "It turned out that the cuckoo's natural speed divided by the wind speed was a square number less than 100" - 8 and 6 don't fit. Not to mention other conditions...

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Thanks Peter. I was (erroneously) using the square root as equal to natural/wind. My bad.

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Sorry, forgot to post corrected version. 
 
Solution is length = 49 
 
 
 
Easy enough:  
c=cuckoo speed, w=wind speed 
a=distance against wind, b=with wind 
 
a/(c-w) = b/(c+w) : c = w(a+b)/(b-a) [1] 
 
c/w=a+b : c = w(a+b) [2] 
 
[1][2]: b-a = 1  
From there, it's a breeze! 
 
Of course,without the prime restriction, 
there are 3 other solutions: 
a=4,b=5 : 9 
a=12,b=13 : 25 
a=40,b=41 : 81

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