Problem 112

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.

Clearly there cannot be any bouncy numbers below one-hundred, but just over half of the numbers below one-thousand (525) are bouncy. In fact, the least number for which the proportion of bouncy numbers first reaches 50% is 538.

Surprisingly, bouncy numbers become more and more common and by the time we reach 21780 the proportion of bouncy numbers is equal to 90%.

Find the least number for which the proportion of bouncy numbers is exactly 99%.

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。

显然不存在小于一百的弹跳数，而在小于一千的数中有略超过一半（525）的弹跳数。事实上，使得弹跳数的比例恰好达到50%的最小数是538。

令人惊奇的是，弹跳数将变得越来越普遍，到21780时，弹跳数的比例恰好等于90%。

找出使得弹跳数的比例恰好为99%的最小数。