project euler 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？

package projecteuler;

import java.math.BigInteger;
import junit.framework.TestCase;

public class Prj113 extends TestCase {

/**[a1,a2,a3,....an],  1<=a1<=a2<=a3<=...an<=9
*   map to ---> [d + i -1]
* one to one map ===>rC(n+r-1)
*/
public void testNonBouncyNumbers() {

assertEquals(calNumOfNonbouncy(10, 6).toString(), "12951");
assertEquals(calNumOfNonbouncy(10, 10).toString(), "277032");
System.out.println(calNumOfNonbouncy(10, 100));
}

BigInteger calNumOfNonbouncy(int _n, int _r) {

BigInteger ret = BigInteger.ZERO;

int n = _n;
int r = _r;
BigInteger asc = calNr(n + r - 1, r);
// zero out
ret = ret.add(asc).subtract(BigInteger.ONE);

// [9,8,7,6,5,4,3,2,1]

n = n - 1;
//倒序补上0
for (int i = 1; i <= r; i++) {
BigInteger desc = calNr(n + i - 1, i).multiply(
new BigInteger(Integer.toString(r + 1 - i)));
ret = ret.add(desc);
}

ret = ret.subtract(new BigInteger(Integer.toString(_r * (_n - 1))));

return ret;
}

BigInteger calNr(int _n, int _r) {

BigInteger ret = BigInteger.ONE;
for (int i = 1; i <= _r; i++) {
ret = ret.multiply(new BigInteger(Integer.toString(_n + 1 - i)));
}
for (int i = 1; i <= _r; i++) {
ret = ret.divide(new BigInteger(Integer.toString(i)));
}

return ret;
}

}