Solving an alphametic
An alphametic consists of an arithmetic problem, where
the digits 0 through 9 have been replaced by letters.
Your job is to find the digit that each letter represents
and replace it, one by one, thus breaking the code used.
Letters always represent the same digit throughout a
puzzle, so if C represents 8 in one place, it
represents 8 everywhere. Of course, each puzzle uses a
different representation. Leading zeroes are not allowed,
and each puzzle has a unique solution.
The program "Crack a Puzzle
Online" facilitates enormously the replacement of
the letters with the corresponding digits, a tedious job
if made by hand. Wrong substitutions will not be
admitted, being signalled by the warning "Incorrect
solution!". Double attention must be paid when
solving the rare alphametics in which the letter
"O" stands for the digit "0". The
resemblance of both characters can get you confused!
At the right lower corner of the screen you see a little
narrow window that shows the current state of the key, a
table consisting of each of the letters used in the
alphametic, paired with its numerical equivalent. There
you have a record of which digits have already been
assigned and the ones still free. This tool helps testing
possible values for next assignment.
Hints will be given by double-clicking the letter you want
to see disclosed, though beware, this is a tool of last
resort to be scarcely used, otherwise it can steal the
exhilaration you derive from solving puzzles, that is, the
feeling of accomplishment after hard reasoning to
"crack the code".
The program detects the end of the solving process, when
the last substitution is being assigned, issuing a
"Congratulations!" message, and asking about
the userīs intentions - to solve another alphametic or
Letīs take this very easy problem : AD + DI = DID and
work it out in detail.
The initial configuration is:
+ D I
D I D
D + I = D is
the units column on the right side
A + D = I is the tens column in the
D (alone in the total) is the hundreds
column on the left side
You can guess that D in the hundreds
column equals 1 because it is the "carry 1"
from the tens column. In other words, A + D = I + 10
transfers a "carry 1" to the total in the hundreds
column. Can you see it?
Now you substitute all Dīs for 1, and the addition
+ 1 I
1 I 1
Take a look at the units colum, and you
will guess that I equals zero. In this column you see
that 1 + I = 1 so, if you try all digits not yet decoded,
you will find that the only one that matches this pattern
is zero. See it?
Now you replace all Iīs with 0, and the addition
+ 1 0
1 0 1
Then observe the tens column where we
have A + 1 = 10. So, A must be 9 to fit this pattern. And
substituting the A for 9 we have the whole problem
solved, that is:
+ 1 0
1 0 1