Starting with x and repeatedly multiplying by x, we can compute x31 with thirty multiplications:
x2 = x �� x, x3 = x2 �� x, x4 = x3 �� x, ��, x31 = x30 �� x.
The operation of squaring can be appreciably shorten the sequence of multiplications. The following is a way to compute x31 with eight multiplications:
x2 = x �� x, x3 = x2 �� x, x6 = x3 �� x3, x7 = x6 �� x, x14 = x7 �� x7, x15 = x14 �� x, x30 = x15 �� x15, x31 = x30 �� x.
This is not the shortest sequence of multiplications to compute x31. There are many ways with only seven multiplications. The following is one of them:
x2 = x �� x, x4 = x2 �� x2, x8 = x4 �� x4, x8 = x4 �� x4, x10 = x8 �� x2, x20 = x10 �� x10, x30 = x20 �� x10, x31 = x30 �� x.
If division is also available, we can find a even shorter sequence of operations. It is possible to compute x31 with six operations (five multiplications and one division):
x2 = x �� x, x4 = x2 �� x2, x8 = x4 �� x4, x16 = x8 �� x8, x32 = x16 �� x16, x31 = x32 �� x.
This is one of the most efficient ways to compute x31 if a division is as fast as a multiplication.
Your mission is to write a program to find the least number of operations to compute xn by multiplication and division starting with x for the given positive integer n. Products and quotients appearing in the sequence should be x to a positive integer��s power. In others words, x−3, for example, should never appear.
The input is a sequence of one or more lines each containing a single integer n. n is positive and less than or equal to 1000. The end of the input is indicated by a zero.
Your program should print the least total number of multiplications and divisions required to compute xn starting with x for the integer n. The numbers should be written each in a separate line without any superfluous characters such as leading or trailing spaces.
- Power Calculus