# Problem 6

The sum of the squares of the first ten natural numbers is,

12 + 22 + ... + 102 = 385

The square of the sum of the first ten natural numbers is,

(1 + 2 + ... + 10)2 = 552 = 3025

Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 385 = 2640.

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

Both the required subroutines are straight forward to implement. The square of the sum grows much quicker than the sum of the squares.

# Solution

#!/usr/bin/python

def sumsquare(val): #Returns the sum of the squares.

ss = 1

for each in range(2,val+1): ss = ss + each**2

return ss

def squaresum(val): #Returns the squares of the sums.

ss = 1

for each in range(2,val+1): ss = ss + each

return ss**2

if __name__ == '__main__' :

import sys

val = int(sys.argv[1]) # Bad Code, assumes input is an integer

print (sumsquare(val))

print (squaresum(val))

print (squaresum(val)-sumsquare(val))

*The answer is 25502500-338350=25164150. *

# Approach

I first define two functions which compute the desired sums. The algorithms are clear and concise in python. I did notice that the for loop, which presumably gets allocated all at once, uses an enormous amount of memory.

# Benchmarks

Python3 really did poorly here and Jython crashed. Pypy was the most efficient as before. Java crashed with an out of memory error so I decided to look at memory usage. Python 3 was actually the most memory efficient followed by Pypy and Python 2. Jython memory usage was reported despite the crash. I have read that pypy is implemented on stackless python, and expected to see stack usage with the other versions. However, none of the python versions used any stack memory.

CPU Usage

Problem #6 Benchmarks

time python factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

real 0m37.971s

user 0m36.458s

sys 0m1.360s

time python -O factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

real 0m38.709s

user 0m37.198s

sys 0m1.408s

time python3 factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

real 1m16.192s

user 1m16.133s

sys 0m0.008s

time python3 -O factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

real 1m16.864s

user 1m16.805s

sys 0m0.008s

time pypy factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

real 0m10.756s

user 0m10.729s

sys 0m0.016s

time pypy -O factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

real 0m10.672s

user 0m10.649s

sys 0m0.012s

time jython factdiff.py 100000000

java.lang.OutOfMemoryError: java.lang.OutOfMemoryError: Java heap space

real 0m38.766s

user 1m48.291s

sys 0m1.680s

Memory Usage

/usr/bin/time -f "%M" python factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

"12750032"

/usr/bin/time -f "%M" python -O factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

"12754544"

/usr/bin/time -f "%M" python3 factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

"30688"

/usr/bin/time -f "%M" python3 -O factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

"39264"

/usr/bin/time -f "%M" pypy factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

"76832"

/usr/bin/time -f "%M" pypy -O factdiff.py 100000000

333333338333333350000000

25000000500000002500000000000000

25000000166666664166666650000000

"76864"

/usr/bin/time -f "%M" jython factdiff.py 100000000

Command exited with non-zero status 255

17467504

# Conclusion

The way I implemented this problem led to large memory usage. I have 16 GB of memory so I didn't run out, but I noticed a jump of several GB when I ran the benchmarks. Python 2 used almost 13 GB as reported by the time function. Python3 was the most memory efficient of the versions, but also the slowest to run.

The answer was *25164150.*