Verbal arithmetic, also known as alphametics, cryptarithmetic, crypt-arithmetic, or cryptarithm, is a type of mathematical game consisting of a mathematical equation among unknown numbers, whose digits are represented by letters. The goal is to identify the value of each letter.

The equation is typically a basic operation of arithmetic, such as addition or multiplication. The classic example, published in the July 1924 issue of Strand Magazine by Henry Dudeney, is:

S E N D + M O R E = M O N E Y

The solution to this puzzle is O=0, M=1, Y=2, E=5, N=6, D=7, R=8, and S=9.

Traditionally, each letter should represent a different digit, and (as in ordinary arithmetic notation) the leading digit of a multi-digit number must not be zero. The puzzle should have only one solution.

History

---

Verbal arithmetic puzzles are quite old and their inventor is not known. An example in The American Agriculturalist of 1864 largely disproves the popular notion that it was invented by Sam Loyd. The name crypt-arithmetic was coined by puzzlist Minos (pseudonym of Maurice 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.

Solving cryptarithms

---

Solving a cryptarithm by hand usually involves a mix of clever deductions and exhaustive tests of possibilities. For instance, in Dudeney's example one can immediately conclude that the leading M of the result is 1, since it is the only carry-over possible in the sum of two numbers. It follows that S=8 or S=9, because those are the only values that can produce a carry when added to M=1 (and possibly a carry). And so on.

The use of modular arithmetic often helps. In particular, the familiar check of casting out nines can be applied to cryptarithms, too; in the example above, it says that S+E+N+D + M+O+R+E should be equal to M+O+N+E+Y modulo 9, that is, S+E+D+R-Y must be evenly divisible by 9.

In computer science, cryptarithms provide good examples for the backtracking paradigm of algorithm design. They also provide a pedagogical application for algorithms that generate all permutations (reorderings) of n given things.

See also

---

• diophantine equation
• mathematical puzzles
• permutation
• puzzles

References

---

• H. E. Dudeney, in Strand Magazine vol. 68 (July 1924), pp. 97 and 214.
• Martin Gardner, Mathematics, Magic, and Mystery. Dover (1956)
• Journal of Recreational Mathematics, has a regular alphametics column.
• J. A. H. Hunter, in the Toronto Globe and Mail (27 October 1955), p. 27.
• Louis Sachar, Sideways Arithmetic from Wayside School and More Sideways Arithmetic From Wayside School, children's books with cryptarithmetic puzzles.

External links

---

• Cryptarithms at cut-the-knot

Logic puzzles

Mathematisches Rätsel | Cryptarithme | Wiskundige puzzels | 覆面算 | Kriptaritem | 覆面算