Egyptian Multiplication

Time Limit: 9000 ms Memory Limit: 32000 KiB

Problem Description

In 1858, A. Henry Rhind, a Scottish antiquary, came into possession of
   a document which is now called the Rhind Papyrus. Titled "Directions
   for Attaining Knowledge into All Obscure Secrets", the document
   provides important clues as to how the ancient Egyptians performed
   arithmetic.
   
   There is no zero in the number system. There are separate characters
   denoting ones, tens, hundreds, thousands, ten-thousands,
   hundred-thousands, millions and ten-millions. For the purposes of this
   problem, we use near ASCII equivalents for the symbols:
     * | for one (careful, it's a vertical line, not 1)
     * n for ten
     * 9 for hundred
     * 8 for thousand
     * r for ten-thousand
       
   (The actual Egyptian hieroglyphs were more picturesque but followed
   the general shape of these modern symbols. For the purpose of this
   problem, we will not consider numbers greater than 99,999.)
   
   Numbers were written as a group of ones preceded in turn by groups of
   tens, hundreds, thousands and ten-thousands. Thus our number 4,023
   would be rendered: ||| nn 8888. Notice that a zero digit is indicated
   by a group consisting of none of the corresponding symbol. The number
   40,230 would thus be rendered: nnn 99 rrrr. (In the Rhind Papyrus, the
   groups are drawn more picturesquely, often spread across more than one
   horizontal line; but for the purposes of this problem, you should
   write numbers all on a single line.)
   
   To multiply two numbers a and b, the Egyptians would work with two
   columns of numbers. They would begin by writing the number | in the
   left column beside the number a in the right column. They would
   proceed to form new rows by doubling the numbers in both columns.
   Notice that doubling can be effected by copying symbols and
   normalizing by a carrying process if any group of symbols is larger
   than 9 in size. Doubling would continue as long as the number in the
   left column does not exceed the other multiplicand b. The numbers in
   the first column that summed to the multiplicand b were marked with an
   asterisk. The numbers in the right column alongside the asterisks were
   then added to produce the result. Below, we show the steps
   corresponding to the multiplication of 483 by 27:
| *                             ||| nnnnnnnn 9999
|| *                            |||||| nnnnnn 999999999
||||                            || nnn 999999999 8
|||||||| *                      |||| nnnnnn 99999999 888
|||||| n *                      |||||||| nn 9999999 8888888
The solution is: | nnnn 888 r
 
   (The solution came from adding together:
||| nnnnnnnn 9999
|||||| nnnnnn 999999999
|||| nnnnnn 99999999 888
|||||||| nn 9999999 8888888.)
 
   You are to write a program to perform this Egyptian multiplication.
 

Input

Input will consist of several pairs of nonzero numbers written in the
   Egyptian system described above. There will be one number per line;
   each number will consist of groups of symbols, and each group is
   terminated by a single space (including the last group). Input will be
   terminated by a blank line.

Output

For each pair of numbers, your program should print the steps
   described above used in Egyptian multiplication. Numbers in the left
   column should be ush with the left margin. Each number in the left and
   right column will be represented by groups of symbols, and each group
   is terminated by a single space (including the last group). If there
   is an asterisk in the left column, it should be separated from the end
   of the left number by a single space. Up to the 40th character
   position should then be filled with spaces. Numbers in the right
   column should begin at the 41st character position on the line and end
   with a newline character. Test data will be chosen to ensure that no
   overlap can occur. After showing each of the doubling steps, your
   program should print the string: "The solution is: " followed by the
   product of the two numbers in Egyptian notation.

Sample Input

||
||
|||
||||
nnnnnn 9
||| n
n
9
|||
8

Sample Output

|                                 ||
|| *                              ||||
The solution is: ||||
|                                 |||
||                                ||||||
|||| *                            || n
The solution is: || n
| *                               nnnnnn 9
||                                nn 999
|||| *                            nnnn 999999
|||||||| *                        nnnnnnnn 99 8
The solution is: nnnnnnnn 88
|                                 n
||                                nn
|||| *                            nnnn
||||||||                          nnnnnnnn
|||||| n                          nnnnnn 9
|| nnn *                          nn 999
|||| nnnnnn *                     nnnn 999999
The solution is: 8
|                                 |||
||                                ||||||
||||                              || n
|||||||| *                        |||| nn
|||||| n                          |||||||| nnnn
|| nnn *                          |||||| nnnnnnnnn
|||| nnnnnn *                     || nnnnnnnnn 9
|||||||| nn 9 *                   |||| nnnnnnnn 999
|||||| nnnnn 99 *                 |||||||| nnnnnn 9999999
|| n 99999 *                      |||||| nnn 99999 8
The solution is: 888

Hint


Source