Project Euler - Problem 4

A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 X 99.

Find the largest palindrome made from the product of two 3-digit numbers.

通常解法：

def isPalindrome(n):    s = str(n)    return s == s[::-1]def findLargestPalindrome():    largestPalindrome = 0    for a in range(100, 1000):        for b in range(100, 1000):            if isPalindrome(a*b) and a*b > largestPalindrome:                largestPalindrome = a*b    return largestPalindrome

分析：
1、每个palindromic数可以表示为：
P=100000x+10000y+1000z+100z+10y+x
P=100001x+10010y+1100z
P=11(9091x+910y+100z)
且P=ab，由每个数可以分解为N个素数的乘积。可知，a和b中至少有一个可以被11整除。

2、a*b和b*a的结果可以只计算一次。

3、从大往小计算，可以更早的发现结果。

改进算法：

def findLargestPalindrome():    largestPalindrome = 0    a = 999    while a >= 100:        if not a % 11:            b = 999            db = 1        else:            b = 990 # 990是小于1000又能够被11整除的最大数            db = 11        while b >= a:            if a * b <= largestPalindrome:                break # a和b都是从大到小遍历，相乘的结果不断变小。            if isPalindrome(a*b):                largestPalindrome = a*b            b -= db        a -= 1    return largestPalindrome