Project Euler — Problem 75 Solution

The prob­lem descrip­tion is here, and click here to see all my oth­er Euler solu­tions in F#.


I based my solu­tion on Euclid’s for­mu­la for gen­er­at­ing Pythagore­an triples.


And giv­en that max L is 1,500,000, the max­i­mum val­ue for m we need to con­sid­er is \sqrt{\frac{L}{2}} . Because L = a + b + c and a^2 + b^2 = c^2 , we can deduce that c < \frac{L}{2} ; and since c = m^2 + n^2 we also have m < \sqrt{c} and there­fore m < \sqrt{\frac{L}{2}} .


The above makes use of a recur­sive func­tion to cal­cu­late the GCD (based on Euclid­ean Algo­rithm):


For effi­cien­cy, we’ll cre­ate a cache to store the num­ber of ways L can be used to cre­ate inte­ger sided right-angle tri­an­gle. As we iter­ate through the m and n pairs we gen­er­at­ed above, we’ll take advan­tage of the fact that if a^2 + b^2 = c^2 then ka^2 + kb^2 = kc^2 must also be true and incre­ment mul­ti­ples of L by one.


Final­ly, to work out the answer:


This solu­tion runs for about 350ms on my machine.


The source code for this solu­tion is here.

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