Verbal arithmetic

Verbal arithmetic, also known as alphametics, cryptarithmetic, cryptarithm or word addition, is a type of <a href="/facts/Mathematical_game/96L20FzX">mathematical game</a> consisting of a mathematical <a href="/facts/Equation/pxFzLAVR">equation</a> among unknown <a href="/facts/Number/auQSAE2m">numbers</a>, whose <a href="/facts/Numerical_digit/JJdLBoGT">digits</a> are represented by <a href="/facts/Letter_(alphabet)/uRnrbGIJ">letters</a> of the alphabet. The goal is to identify the value of each letter. The name can be extended to puzzles that use non-alphabetic symbols instead of letters.
The equation is typically a basic operation of <a href="/facts/Arithmetic/hvHwkbMV">arithmetic</a>, such as <a href="/facts/Addition/apFWJBFB">addition</a>, <a href="/facts/Multiplication/VtcUjpNv">multiplication</a>, or <a href="/facts/Division_(mathematics)/FqLKNkmq">division</a>. The classic example, published in the July 1924 issue of <a href="/facts/The_Strand_Magazine/z5LnlGrH">The Strand Magazine</a> by <a href="/facts/Henry_Dudeney/YSSBQgoK">Henry Dudeney</a>, is:

 
 
 
 
 
 
 
 
 
 
 S
 
 
 
 
 E
 
 
 
 
 N
 
 
 
 
 D
 
 
 
 
 
 +
 
 
 
 
 M
 
 
 
 
 O
 
 
 
 
 R
 
 
 
 
 E
 
 
 
 
 
 =
 
 
 
 M
 
 
 
 
 O
 
 
 
 
 N
 
 
 
 
 E
 
 
 
 
 Y
 
 
 
 
 
 
 
 {\displaystyle {\begin{matrix}&&{\text{S}}&{\text{E}}&{\text{N}}&{\text{D}}\\+&&{\text{M}}&{\text{O}}&{\text{R}}&{\text{E}}\\\hline =&{\text{M}}&{\text{O}}&{\text{N}}&{\text{E}}&{\text{Y}}\\\end{matrix}}}

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 an ordinary arithmetic notation) the leading digit of a multi-digit number must not be zero. A good puzzle should have one unique solution, and the letters should make up a phrase (as in the example above).
Verbal arithmetic can be useful as a motivation and source of exercises in the <a href="/facts/Education/WIwsQKF9">teaching</a> of <a href="/facts/Elementary_algebra/973nufT7">elementary algebra</a>.

Verbal arithmetic open-in-new

Verbal arithmetic