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It may help to review OEIS A143715. The sequence is explained there and will also help check for correct calculations.
The "sample output" in the problem description is in wrong order.
No, it is in the correct order.
The number of solutions is 1975 when M=99.
The number of solutions is 2060 when M=100.
It was fixed silently in the meantime ^^.
Solved this problem in F#. Insights:
If anybody is interested, I can write down my approach to solve this problem.
Curious to know how it works to compute million pythagorean triplets within the time limit. I've been using Euclid's formula, but that m += 1, n += 2 is supposedly killing it. It takes almost 5 seconds to generate triplets up to (with most of the time spent on finding the valid primitive triplets), let alone .
m += 1
n += 2
Perhaps it is because of the upper bound you mentioned? I don't know where that 1.1*sqrt(M) comes from, the bound I used was a while-loop with condition m**2 - (m-1)**2 <= M. When I look closely, this condition is unnecessarily loose, as it does not take into account the fact that either a >= b or b >= a should be true, while and .
m**2 - (m-1)**2 <= M
a >= b
b >= a
EDIT: I tried using this upper bound, and got WA for Test Cases #3 through #9.
EDIT 2: Tried with sqrt(3*M) and it works. Even sqrt(2*M) did not work, where it was missing some pairs like m=917, n=218. Maybe the difference is due to the way I handled the triples, as I append whenever a >= b_plus_c / 2 + b_plus_c % 2 or b_plus_c >= a / 2 + a % 2.
a >= b_plus_c / 2 + b_plus_c % 2
b_plus_c >= a / 2 + a % 2
I'm quite sure that this condition would fail again for much larger input. The naming here is quite confusing, as b_plus_c >= a / 2 + a % 2 in fact means b >= a_plus_c / 2 + a_plus_c % 2, but I just handle them all in one place.
b >= a_plus_c / 2 + a_plus_c % 2
EDIT 3: By the way, I find it interesting that no one mentioned another key insight involved in solving this problem, that is to visualise the unfolding of the 3D box. At the beginning, I was trying to solve the equation and find the derivative of it to look for a minimum, thinking about how this problem is supposed to be 35% difficulty.
Yes, the upper bound is crucial in finding all the Pythagorean triplets within the time limit.
Note that the M in my upper bound definition is not the same as the M in the problem statement. Sorry for the confusion about that. This might explain the difference in your findings about the upper bound.
The other insight you mention is crucial as well. I didn't want to spoil too much :-)
I hope this clears some things up.
. Would definitively advise to complete 75, and use roll distribution on the answer of 75.
I dont think there is a way to solve it without generating pythagorean triplet correctly.
in the diagram given in problem,how the shortest path is 10.
open the cubiod