0%

Problem 113


Problem 113


Non-bouncy numbers

Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468.

Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420.

We shall call a positive integer that is neither increasing nor decreasing a “bouncy” number; for example, 155349.

As n increases, the proportion of bouncy numbers below n increases such that there are only 12951 numbers below one-million that are not bouncy and only 277032 non-bouncy numbers below 1010.

How many numbers below a googol (10100) are not bouncy?


非弹跳数

从左往右,如果每一位数字都大于等于其左边的数字,这样的数被称为上升数,比如134468。

同样地,如果每一位数字都大于等于其右边的数字,这样的数被称为下降数,比如66420。

如果一个正整数既不是上升数也不是下降数,我们就称之为“弹跳”数,比如155349。

随着n的增长,小于n的弹跳数的比例也随之增长;在小于一百万的数中,只有12951个非弹跳数,而小于1010的数中只有277032个非弹跳数。

在小于一古戈尔(10100)的数中有多少个非弹跳数?