Cryptarithms


Cryptarithms, alphametics, verbalarithmetic are some of the names of a type of puzzle, where two, three or more words are given, and each letter must be replaced by a single digit. The most well known of this is:
1) Dudeneys classic**

 SEND
 MORE
-----+
MONEY

Replace each letter with exactly 1 digit and make it a correct addition. The example above is from Henry Dudeney.

(Solution: 10)

Verbal arithmetic puzzles are quite old and their inventor is not known. An example in The American Agriculturist[2] of 1864 makes the popular notion that it was invented by Sam Loyd unlikely. The name crypt-arithmetic was coined by puzzlist Minos (pseudonym of Simon Vatriquant) in the May 1931 issue of Sphinx, a Belgian magazine of recreational mathematics. In the 1955, J. A. H. Hunter introduced the word “alphametic” to designate cryptarithms, such as Dudeney’s, whose letters form meaningful words or phrases.

There are several types of cryptarithms. One of them is them is the double true. This type of cryptarithm shows an addition that is true in words, but can also be deciphered as a cryparithm. This puzzle comes from N. Tamura, who wrote a program to search for puzzles with a unique solution.
2) DOUBLE TRUE:**

 THREE
  FIVE
  FIVE
 SEVEN
   TEN
------+
THIRTY

(Solution: 30)

3) Math formulas* are another subcategory.
This one was first published in Sphinx magazine, and republished by Jorge A C B Soares. He mentions M. Van Esbroeck as the author, and january 1933 as the date of first publication.

 A B C = C4
 B C A = D4

(Solution: 40)

Cryptarithms are of course language dependent. However, they are not limited to the English language. The booklet CIJFERWERK (digit work), written by J. van der Horst, published by Born periodieken, date unknown, isbn 8 710838 100910, has some 200 dutch cryptarithms. In Germany they are called “Kryptogramm”, in French “Cryptarithme”, or d’alphamétique. I see articles on them in the Japanese wikipedia, where they give the following example:
4) Japanese*

 大宮
×大宮
大井町
横浜 
-----+
浜松町

Please forgive me that my Japanese is insufficient to discover the author.

(Solution: 70)

5) List**
Lists are another subcategory of this puzzle type. Lists are composed of a number of items in category, followed by the name of the category, which equals to the sum of the items. In a good list items are unique, taht is, they appear only once.

Here is a non unique list, that is a list with items that appear more than once:

APPLE
 PEAR
 DATE
APPLE
 PEAR
 DATE
APPLE
 PEAR
 DATE
APPLE
 PEAR
 DATE
-----+
FRUIT

(Solution: 50)

Links

  1. online puzzle solver
  2. Mike Keiths site
  3. Sphinx collection
  4. http://bach.istc.kobe-u.ac.jp/puzzle/crypt/out/eg-num.out
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