Problem57

原题：

It is possible to show that the square root of two can be expressed as an infinite continued fraction.
2(开方) = 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?

 

分析：

根据题意可看出2的开平方的近似序列1为：
 3/2    7/5     17/12    41/29     99/70     239/169     577/408    1393/985 
 可以推导出：
  序列1的递推公式为f(N)=f(N-1)(a-分子,b-分母)=(a+2b)/(a+b)

 

代码如下：

public class Problem57 { //将每组数存进该map中，key-value 对应 分子-分母 static List<Scores> scoresList = new ArrayList<Scores>(); //将前一千组数放入该数组 static{ Scores beginNumber = new Scores(); beginNumber.setDenominator(BigInteger.valueOf(2)); beginNumber.setNumerator(BigInteger.valueOf(3)); scoresList.add(beginNumber); while(scoresList.size()<1000){ int size = scoresList.size(); Scores s = scoresList.get(size-1); scoresList.add(getSores(s)); } } //判断分子的位数是否大于分母的位数 static boolean exceed(Scores scores){ if(String.valueOf(scores.getNumerator()).length()> String.valueOf(scores.getDenominator()).length()) return true; return false; } //通过上个分数求下一个分数 static Scores getSores(Scores s){ Scores next = new Scores(); next.setNumerator(s.getDenominator().multiply(BigInteger.valueOf(2)).add(s.getNumerator())); next.setDenominator(s.getDenominator().add(s.getNumerator())); return next; } public static void main(String[] args) { //System.out.println(scoresList); int count = 0; for(Scores s :scoresList){ if(exceed(s)) count++; } System.out.println("count:"+count); } } class Scores{ private BigInteger numerator;//分子 private BigInteger denominator;//分母 public BigInteger getDenominator() { return denominator; } public void setDenominator(BigInteger denominator) { this.denominator = denominator; } public BigInteger getNumerator() { return numerator; } public void setNumerator(BigInteger numerator) { this.numerator = numerator; } @Override public String toString(){ return "该分数的分子为："+this.getNumerator()+" 分母为："+this.getDenominator(); } @Override public int hashCode() { final int PRIME = 31; int result = 1; result = PRIME * result + ((denominator == null) ? 0 : denominator.hashCode()); result = PRIME * result + ((numerator == null) ? 0 : numerator.hashCode()); return result; } @Override public boolean equals(Object obj) { if (this == obj) return true; if (obj == null) return false; if (getClass() != obj.getClass()) return false; final Scores other = (Scores) obj; if (denominator == null) { if (other.denominator != null) return false; } else if (!denominator.equals(other.denominator)) return false; if (numerator == null) { if (other.numerator != null) return false; } else if (!numerator.equals(other.numerator)) return false; return true; } }