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