Alphamagic square

Last updated

An alphamagic square is a magic square that remains magic when its numbers are replaced by the number of letters occurring in the name of each number. Hence 3 would be replaced by 5, the number of letters in "three". Since different languages will have a different number of letters for the spelling of the same number, alphamagic squares are language dependent. [1] Alphamagic squares were invented by Lee Sallows in 1986. [2] [3]

Magic square arrangement of numbers (usually integers) in a square grid

In recreational mathematics and combinatorial design, a magic square is a square grid filled with distinct positive integers in the range such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. The sum is called the magic constant or magic sum of the magic square. A square grid with n cells on each side is said to have order n.

Lee Sallows English recreational mathematician

Lee Cecil Fletcher Sallows is a British electronics engineer known for his contributions to recreational mathematics. He is particularly noted as the inventor of golygons, self-enumerating sentences, and geomagic squares.



The example below is alphamagic. To find out if a magic square is also an alphamagic square, convert it into the array of corresponding number words. For example,


converts to ...


Counting the letters in each number word generates the following square which turns out to also be magic:


If the generated array is also a magic square, the original square is alphamagic. In 2017 British computer scientist Chris Patuzzo discovered several doubly alphamagic squares in which the generated square is in turn an alphamagic square. [4]

The above example enjoys another special property: the nine numbers in the lower square are consecutive. This prompted Martin Gardner to describe it as "Surely the most fantastic magic square ever discovered." [5]

Martin Gardner recreational mathematician and philosopher

Martin Gardner was an American popular mathematics and popular science writer, with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literature—especially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton. He is recognized as a leading authority on Lewis Carroll. The Annotated Alice, which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies. He had a lifelong interest in magic and illusion and was regarded as one of the most important magicians of the twentieth century. He was considered the doyen of American puzzlers. He was a prolific and versatile author, publishing more than 100 books.

A geometric alphamagic square

Figure 1:   A geomagic square that is also alphamagic Sallows geometric alphamagic square.svg
Figure 1:   A geomagic square that is also alphamagic

Sallows has produced a still more magical versiona square which is both geomagic and alphamagic. In the square shown in Figure 1, any three shapes in a straight lineincluding the diagonalstile the cross; thus the square is geomagic. The number of letters in the number names printed on any three shapes in a straight line sum to forty five; thus the square is alphamagic.

Geometric magic square

A geometric magic square, often abbreviated to geomagic square, is a generalization of magic squares invented by Lee Sallows in 2001. A traditional magic square is a square array of numbers whose sum taken in any row, any column, or in either diagonal is the same target number. A geomagic square, on the other hand, is a square array of geometrical shapes in which those appearing in each row, column, or diagonal can be fitted together to create an identical shape called the target shape. As with numerical types, it is required that the entries in a geomagic square be distinct. Similarly, the eight trivial variants of any square resulting from its rotation and/or reflection, are all counted as the same square. By the dimension of a geomagic square is meant the dimension of the pieces it uses. Hitherto interest has focused mainly on 2D squares using planar pieces, but pieces of any dimension are permitted.

Other languages

The Universal Book of Mathematics provides the following information about Alphamagic Squares: [6] [7]

<i>The Universal Book of Mathematics</i> book by David J. Darling

The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes (2004) is a bestselling book by British author David Darling.

A surprisingly large number of 3 × 3 alphamagic squares exist—in English and in other languages. French allows just one 3 × 3 alphamagic square involving numbers up to 200, but a further 255 squares if the size of the entries is increased to 300. For entries less than 100, none occurs in Danish or in Latin, but there are 6 in Dutch, 13 in Finnish, and an incredible 221 in German. Yet to be determined is whether a 3 × 3 square exists from which a magic square can be derived that, in turn, yields a third magic square—a magic triplet. Also unknown is the number of 4 × 4 and 5 × 5 language-dependent alphamagic squares.

In 2018, the first 3 × 3 Russian alphamagic square was found by Jamal Senjaya. Following that, another 158 3 × 3 Russian alphamagic squares were found (by the same person) where the entries do not exceed 300.

Related Research Articles

Eight queens puzzle mathematical chess problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other

The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, column, or diagonal. The eight queens puzzle is an example of the more general n queens problem of placing n non-attacking queens on an n×n chessboard, for which solutions exist for all natural numbers n with the exception of n=2 and n=3.

<i>Conways Game of Life</i> 2D cellular automaton devised by J. H. Conway in 1970

The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970.

Crossword type of puzzle in which the player uses clues to put words in a grid

A crossword is a word puzzle that usually takes the form of a square or a rectangular grid of white-and black-shaded squares. The game's goal is to fill the white squares with letters, forming words or phrases, by solving clues, which lead to the answers. In languages that are written left-to-right, the answer words and phrases are placed in the grid from left to right and from top to bottom. The shaded squares are used to separate the words or phrases.

Graeco-Latin square

In combinatorics, a Graeco-Latin square or Euler square or orthogonal Latin squares of order n over two sets S and T, each consisting of n symbols, is an n×n arrangement of cells, each cell containing an ordered pair (s,t), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells contain the same ordered pair.

A pandiagonal magic square or panmagic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009. Sloane is president of the OEIS Foundation.

Quantity is a property that can exist as a multitude or magnitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature, while others are functioning as states of things such as heavy and light, long and short, broad and narrow, small and great, or much and little.

An antimagic square of order n is an arrangement of the numbers 1 to n2 in a square, such that the sums of the n rows, the n columns and the two diagonals form a sequence of 2n + 2 consecutive integers. The smallest antimagic squares have order 4.

D. R. Kaprekar Indian mathematician

Dattathreya Ramchandra Kaprekar (1905–1986) was an Indian recreational mathematician who described several classes of natural numbers including the Kaprekar, Harshad and Self numbers and discovered the Kaprekar constant, named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles.

Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids.

The programming language APL is distinctive in being symbolic rather than lexical: its primitives are denoted by symbols, not words. These symbols were originally devised as a mathematical notation to describe algorithms. APL programmers often assign informal names when discussing functions and operators but the core functions and operators provided by the language are denoted by non-textual symbols.

In elementary mathematics, a variable is a symbol, commonly a single letter, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown. Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation. A typical example is the quadratic formula, which allows one to solve every quadratic equation by simply substituting the numeric values of the coefficients of the given equation to the variables that represent them.

Mystic square

The square array of the integers 1 through n2 that is generated when a method for constructing a 4 × 4 magic square is generalized was called a mystic square by Joel B. Wolowelsky and David Shakow in their article describing a method for constructing a magic square whose order is a multiple of 4. A 4 × 4 magic square can be constructed by writing out the numbers from 1 to 16 consecutively in a 4 × 4 matrix and then interchanging those numbers on the diagonals that are equidistant from the center.. The sum of each row, column and diagonal is 34, the “magic number” for a 4 × 4 magic square. In general, the “magic number” for an n × n magic square is n(n^2 + 1)/2.

Rep-tile a tiling of the plane in which a prototile is recursively subdivided into copies of itself

In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by the American mathematician Solomon W. Golomb, who used it to describe self-replicating tilings. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.

Self-tiling tile set

A self-tiling tile set, or setiset, of order n is a set of n shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of n shapes. That is, the n shapes can be assembled in n different ways so as to create larger copies of themselves, where the increase in scale is the same in each case. Figure 1 shows an example for n = 4 using distinctly shaped decominoes. The concept can be extended to include pieces of higher dimension. The name setisets was coined by Lee Sallows in 2012, but the problem of finding such sets for n = 4 was asked decades previously by C. Dudley Langford, and examples for polyaboloes and polyominoes were previously published by Gardner.

In logic, quantification specifies the quantity of specimens in the domain of discourse that satisfy an open formula. The two most common quantifiers mean "for all" and "there exists". For example, in arithmetic, quantifiers allow one to say that the natural numbers go on forever, by writing that for all n, there is another number which is one bigger than n.

Matt Parker Australian comedian and math communicator

Matthew Parker is an Australian recreational mathematics author, YouTube personality and communicator. Parker is the Public Engagement in Mathematics Fellow at Queen Mary University of London. He is a former maths teacher, and has helped popularise maths via his tours and videos.


  1. Wolfram MathWorld: Alphamagic Squares
  2. Mathematical Recreations: Alphamagic Square by Ian Stewart, Scientific American: , January 1997, pp. 106-110
  3. ACM Digital Library, Volume 4 Issue 1, Fall 1986
  4. Double Alphamagic Squares Futility Closet, November 16, 2015
  5. Gardner, Martin (1968), A Gardner's Workout: Training the Mind and Entertaining the Spirit, p. 161, A K Peters/CRC Press, Natick, Mass., July 2001, ISBN   1568811209
  6. The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes, by David Darling, p. 12, Hoboken, NJ: Wiley, 2004, ISBN   0471270474
  7. Encyclopedia of Science, Games & Puzzles: Alphamagic Squares