Problem Description
It is possible to show that the square root of two can be expressed as an infinite continued fraction.
sqrt(2) = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...
By expanding this for the first four iterations, we get:
1 + 1/2 = 3/2 = 1.5
1 + 1/(2 + 1/2) = 7/5 = 1.4
1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...
1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...
The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.
In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?
C++
Get the pseudo-code of calculating the numerator and denominator of the fraciton first from the expression above. Then you could easily write the main code:
Main()
void Problem_57() { int count = 0; for(int i=1; i<= EXPANSION_TIME; i++) { Fraction fra(1, 2); int loop = 1; while(loop <i) { fra.Add(2); fra.Reciprocal(); loop++; } fra.Add(1); if(fra.IsOKForProblem()) { count++; // fra.Display(); } } printf("result = %d\n", count); }
You may need a Fraction class which manages two fields numerator and denominator.
Plealse note that the numbers in the calculation are far beyond the maxnium of a 4 bytes integer, you need a Big Integer type.
I would like to show my simple implementation below:
Fraction and BigInteger
class BigInteger { public: BigInteger(int a) { memset(m_digits, 0, 1000); int index = 0; while(a != 0) { m_digits[index] = a % 10; a = a/ 10; index++; } m_length = index; } void Add(const BigInteger& other) { int maxLength = max(m_length, other.m_length); for(int i=0; i<maxLength; i++) { int temp = m_digits[i] + other.m_digits[i]; if(temp >= 10) { m_digits[i+1]++; m_digits[i] = temp % 10; } else { m_digits[i] = temp; } } m_length = (m_digits[maxLength] == 0) ? maxLength : (maxLength + 1); } void Multiply(int num) { BigInteger other(*this); for(int i=0; i<num - 1; i++) { Add(other); } } int GetLength() { return m_length; } void Display() { char str[1000] = {0}; for(int i = m_length - 1; i>=0; i--) { str[m_length - i - 1] = (char)(m_digits[i] + '0'); } printf("%s", str); } private: BYTE m_digits[1000]; int m_length; }; class Fraction { public: Fraction(int numerator, int denominator) : m_bi1(numerator), m_bi2(denominator), m_isBI1Numerator(true) { } void Reciprocal() { m_isBI1Numerator = !m_isBI1Numerator; } void Add(int addend) { BigInteger temp(GetDenominator()); temp.Multiply(addend); GetNumerator().Add(temp); } void Display() { GetNumerator().Display(); printf("/"); GetDenominator().Display(); printf("\n"); } bool IsOKForProblem() { return GetNumerator().GetLength() > GetDenominator().GetLength(); } private: BigInteger& GetNumerator() { return m_isBI1Numerator ? m_bi1 : m_bi2; } BigInteger& GetDenominator() { return m_isBI1Numerator ? m_bi2 : m_bi1; } bool m_isBI1Numerator; BigInteger m_bi1; BigInteger m_bi2; };