Cryptarithm: Addition and Digits

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Cryptarithmetic, also known as alphametics, , crypt-arithmetic, Verbal arithmetic , cryptarithm or word addition, is a type of mathematical puzzle in which most or all of the digits in a mathematical expression, such as a sum, are substituted by letters or other symbols.  In a typical puzzle, there is a one-to-one correspondence between the numbers and the letters or symbols replacing them.  That is, the same digit is always represented by the same letter or symbol.  The objective of the puzzle is to break the code used.  That is, to replace each letter of the cryptarithm by a numeral so that the resulting mathematical expression is true.  Cryptarithms may have several solutions.  However, there are elegant ones with only one unique solution. The equation is typically a basic operation of arithmetic, such as addition, multiplication, or division. For example:

9567 + 1085 = 10652
can be represented like this:
abcd + efgb = efcbh
One of the best known cryptarithm is 
| S| E| N| D|
+| M| O| R| E|
M| O| N| E| Y|
first published in the July 1924 issue of Strand Magazine.  This puzzle was made by the famous puzzlist H. E. Dudeney.  This puzzle is also called an alphametic because the the letters in the puzzle form a sensible phrase SEND MORE MONEY.  This puzzle has the solution of | 9| 5| 6| 7|

+| 1| 0| 8| 5|
1| 0| 6| 5| 2|

This alphametic or cryptarithm is popular not only because the letters in the puzzle form a very popular phrase, but it has also a unique solution. History
The invention of Cryptarithmetic has been ascribed to ancient China. This art was originally known as letter arithmetic or verbal arithmetic. In India, during the Middle Ages, were developed the arithmetical restorations or "skeletons" a type of cryptarithms in which most or all of the digits have been replaced by asterisks. In 1864 the first cryptarithm appeared in the USA, in American Agriculturist. The word cryptarithmetic ("cryptarithmie" in French) was introduced by M. Vatriquant, writing under the pseudonym Minos, in the May 1931 issue of Sphinx, a Belgian magazine of recreational mathematics published in French from 1931 to 1939. A type of alphametic addition puzzle termed "doubly-true" was introduced in 1945 by Alan Wayne. It is made up of "number words" that, when read, also form a valid sum. In 1955, J. A. H. Hunter coined the word alphametic to designate a cryptarithm whose letters form sensible words or phrases. The world's best known alphametic puzzle is undoubtedly SEND + MORE = MONEY. It was created by H. E. Dudeney and first published in the July 1924 issue of Strand Magazine associated with the story of a kidnapper's ransom demand. Cryptarithm Rules

* Each letter represents a unique digit.
* Numbers must not start with a zero.
* The solution is unique (unless otherwise stated).

1. Preparation
Rewrite the problem, expanding the interlinear space to make room for trial numbers that will be written under the letters. For example, the puzzle SEND + MORE = MONEY, after solving, will appear like this:

9 5 6 7
+ M O R E
1 0 8 5
1 0 6 5 2
2. Remember cryptarithmetic conventions
* Each letter or symbol represents only one digit throughout the problem; * When letters are replaced by their digits, the resultant arithmetical operation must be correct; * The numerical base, unless specifically stated, is 10;

* Numbers must not begin with a zero;
* There must be only one solution to the problem.

3. See subtractions as "upside-down" additions
Ease the analysis of subtractions by reading them as upside-down additions. Remember that you can check a subtraction...
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