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Problem 145


Problem 145


How many reversible numbers are there below one-billion?

Some positive integers n have the property that the sum [ n + reverse(n) ] consists entirely of odd (decimal) digits. For instance, 36 + 63 = 99 and 409 + 904 = 1313. We will call such numbers reversible; so 36, 63, 409, and 904 are reversible. Leading zeroes are not allowed in either n or reverse(n).

There are 120 reversible numbers below one-thousand.

How many reversible numbers are there below one-billion (109)?


有多少小于十亿的可逆数?

有些正整数n满足这样一种性质,将它的数字逆序排列后和本身相加所得到的和[ n + reverse(n) ]的十进制表示只包含有奇数数字。例如,36 + 63 = 99 以及409 + 904 = 1313。我们称这样的数是可逆的;因此36,63,409和904都是可逆的。无论是n还是reverse(n)都不允许出现前导0。

在小于一千的数中,一共有120个可逆数。

在小于十亿(109)的数中,一共有多少个可逆数?