Euler Problem

Problem 6: What is the di fference bet ween the sum of the squares and the square of the sums? First of all, one coul d use a brute force impl ementati on, becaus e 100 is not a rea lly high limit . Thi s could be done as follows: limit = 100 sum sq = 0 sum = 0 for i = 1 to limit do sum = sum + i sum sq = sum sq + i 2 end for print sum 2 − sum sq However, such an approach would definitely get in trouble when limit becomes very large. A closer loo k at the pr ogra m shows that the sum va riable , at the end, contains the sum of the integers from 1 to limit. As is widely kno wn, this sum can be dir ect ly calculated using the for mu la sum(n) = n(n + 1) /2. As you might have expected, such a formula also exists for the sum of squar es. Let us derive this formula. Thus, we are looking for a function f (n), that for any n gives the sum of 1 2 up to n 2 . Assume it is of the form f (n) = an 3 + bn 2 + cn + d, with a, b, c,d constants that we have to determine. This we can do because we can verify that f (0) = 0,f (1) = 1,f (2) = 5,f (3) = 14. This yie lds four equati ons in fou r variables, namely d = 0 a + b + c + d = 1 8a + 4b + 2c + d = 5 27a + 9b + 3c + d = 14 Solving this system of equations, we obtain a = 1 3 ,b = 1 2 , c = 1 6 , d = 0. This gives f (n) = 1 6 (2n 3 + 3n 2 + n) = n 6 (2n + 1)(n + 1). What remains is to show this f actually is what we want. This we prove by induction: Assuming f is the correct formula for n, we show it is also for n + 1. Then, because we know it is correct for n = 0, 1, 2, 3, we know that it’s correct for all n. Thu s, we hav e to show f (n + 1) = f (n) + (n + 1) 2 . By expa ndi ng both sides we get f (n + 1) = f (n) + (n + 1) 2 n 3 3 + 3 n 2 2 + 13n 6 + 1 = n 6 + n 2 2 + n 3 3 + n 2 + 2n + 1 Since both sides are equal, we have proven that f is the corre ct formula. This means that we can now write a very simple program to calculate the difference between the sum of the squares and the square of the sum: limit = 100 sum = limit(limit + 1)/2 sum sq = (2limit + 1)(limit + 1)limit/6 print sum 2 − sum sq This algorithm is limited only by the size of the integer types your programming language (and computer memory) support. Copyright Project Euler. Further distribution without the consent of the author is prohibited. Author: Lord Farin 1

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PE Problem solution pdf

Transcript of Euler Problem

Problem 6: What is the difference between the sum of thesquares and the square of the sums?

First of all, one could use a brute force implementation, because 100 is not a really high limit. Thiscould be done as follows:

limit = 100sum sq = 0sum = 0for i = 1 to limit dosum = sum + isum sq = sum sq + i2

end forprint sum2 sum sq

However, such an approach would definitely get in trouble when limit becomes very large.A closer look at the program shows that the sum variable, at the end, contains the sum of theintegers from 1 to limit. As is widely known, this sum can be directly calculated using the formulasum(n) = n(n + 1)/2. As you might have expected, such a formula also exists for the sum of squares.Let us derive this formula.

Thus, we are looking for a function f(n), that for any n gives the sum of 12 up to n2. Assume it is ofthe form f(n) = an3 + bn2 + cn+ d, with a, b, c, d constants that we have to determine. This we can dobecause we can verify that f(0) = 0, f(1) = 1, f(2) = 5, f(3) = 14. This yields four equations in fourvariables, namely

d = 0a + b + c + d = 1

8a + 4b + 2c + d = 527a + 9b + 3c + d = 14

Solving this system of equations, we obtain a = 13 , b =12 , c =

16 , d = 0. This gives f(n) =

16 (2n

3 + 3n2 +n) = n6 (2n + 1)(n + 1).What remains is to show this f actually is what we want. This we prove by induction: Assuming f is thecorrect formula for n, we show it is also for n+ 1. Then, because we know it is correct for n = 0, 1, 2, 3,we know that its correct for all n. Thus, we have to show f(n + 1) = f(n) + (n + 1)2. By expandingboth sides we get

f(n + 1) = f(n) + (n + 1)2

n3

3+

3n2

2+

13n6

+ 1 =n

6+n2

2+n3

3+ n2 + 2n + 1

Since both sides are equal, we have proven that f is the correct formula. This means that we can nowwrite a very simple program to calculate the difference between the sum of the squares and the squareof the sum:

limit = 100sum = limit(limit+ 1)/2sum sq = (2limit+ 1)(limit+ 1)limit/6print sum2 sum sq

This algorithm is limited only by the size of the integer types your programming language (and computermemory) support.

Copyright Project Euler. Further distribution without the consent of the author is prohibited.Author: Lord Farin

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