Magic series
Encyclopedia
A magic series is a set of distinct positive numbers which add up to the magic constant
of a magic square
and a magic cube
, thus potentially making up lines in magic tesseract
s.
So, in an n × n magic square using the numbers from 1 to n2, a magic series is a set of n distinct numbers adding up to n(n2+1)/2. For n = 2, there are just two magic series, 1+4 and 2+3. The eight magic series when n = 3 all appear in the rows, columns and diagonals of a 3 × 3 magic square.
Maurice Kraitchik
gave the number of magic series up to n = 7 in Mathematical Recreations in 1942 . In 2002, Henry Bottomley extended this up to n = 36 and independently Walter Trump
up to n = 32. In 2005, Trump extended this to n = 54 (over 2×10111) while Bottomley gave an experimental approximation for the numbers of magic series:
In July 2006, Robert Gerbicz extended this sequence up to n = 150.
Magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15....
of a magic square
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
and a magic cube
Magic cube
In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic...
, thus potentially making up lines in magic tesseract
Magic tesseract
In mathematics, a magic tesseract is the 4-dimensional counterpart of a magic square and magic cube, that is, a number of integers arranged in an n × n × n × n pattern such that the sum of the numbers on each pillar as well as the main space diagonals is equal to a single number,...
s.
So, in an n × n magic square using the numbers from 1 to n2, a magic series is a set of n distinct numbers adding up to n(n2+1)/2. For n = 2, there are just two magic series, 1+4 and 2+3. The eight magic series when n = 3 all appear in the rows, columns and diagonals of a 3 × 3 magic square.
Maurice Kraitchik
Maurice Kraitchik
Maurice Kraitchik was a Belgian mathematician and populariser born in Minsk. His main interests were the theory of numbers and recreational mathematics.He is famous for having inspired the two envelopes problem in 1953, with the following puzzle in...
gave the number of magic series up to n = 7 in Mathematical Recreations in 1942 . In 2002, Henry Bottomley extended this up to n = 36 and independently Walter Trump
Walter Trump
Walter Trump is a German mathematician. He is known for discovering results on recreational mathematics.He has credits from square packing problem and magic tile problem...
up to n = 32. In 2005, Trump extended this to n = 54 (over 2×10111) while Bottomley gave an experimental approximation for the numbers of magic series:
In July 2006, Robert Gerbicz extended this sequence up to n = 150.