“Antimagic square”的意思、由来-开放百科全书

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Antimagic square

释义

Examples
Order 4 antimagic squares Order 5 antimagic squares
Properties
Open problems
Generalizations
See also
References
External links

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.^[1]

Examples

Order 4 antimagic squares

2	15	5	13
16	3	7	12
9	8	14	1
6	4	11	10

1	13	3	12
15	9	4	10
7	2	16	8
14	6	11	5

Order 5 antimagic squares

5	8	20	9	22
19	23	13	10	2
21	6	3	15	25
11	18	7	24	1
12	14	17	4	16

21	18	6	17	4
7	3	13	16	24
5	20	23	11	1
15	8	19	2	25
14	12	9	22	10

Properties

In each of these two antimagic squares of order 4, the rows, columns and diagonals sum to ten different numbers in the range 29–38.^[2] In the antimagic square of order 5 on the left, the rows, columns and diagonals sum up to numbers between 60 and 71.^[2] In the antimagic square on the right, the rows, columns and diagonals add up to numbers between 59-70.^[1]

Antimagic squares form a subset of heterosquares which simply have each row, column and diagonal sum different. They contrast with magic squares where each sum is the same.^[2]

Open problems

How many antimagic squares of a given order exist?
Do antimagic squares exist for all orders greater than 3?
Is there a simple proof that no antimagic square of order 3 exists?

Generalizations

A sparse antimagic square (SAM) is a square matrix of size n by n of nonnegative integers whose nonzero entries are the consecutive integers for some , and whose row-sums and column-sums constitute a set of consecutive integers.^[3] If the diagonals are included in the set of consecutive integers, the array is known as a sparse totally anti-magic square (STAM). Note that a STAM is not necessarily a SAM, and vice versa.

References

1. ^¹{{Cite 201 web|url=http://mathworld.wolfram.com/AntimagicSquare.html|title=Antimagic Square|last=W.|first=Weisstein, Eric|website=mathworld.wolfram.com|language=en|access-date=2016-12-03}}
2. ^¹²{{Cite web|url=http://www.magic-squares.net/anti-ms.htm|title=Anti-magic Squares|website=www.magic-squares.net|access-date=2016-12-03}}
3. ^{{cite journal|last=Gray|first=I. D.|year=2006|title=Sparse anti-magic squares and vertex-magic labelings of bipartite graphs|journal=Discrete Mathematics|volume=306|pages=2878–2892|doi=10.1016/j.disc.2006.04.032|author2=MacDougall, J.A.}}

External links

{{MathWorld |urlname=AntimagicSquare |title=Antimagic Square}}

1 : Magic squares

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