Magic Squares

This is a very old form of puzzle. In its simplest form, a magic square is a 2-dimensional square array of fields filled with numbers, where the sum over each row equals the sum over each column as well as the sum of the two diagonals.

  • The generic Magic Square
  • Magic Squares with initial constraints
  • The Prime Magic Square
  • The Knight's Tour
  • Magic Sequences
  • The Generic Magic Square Find for a given n a (n x n)-matrix such that
    • Every field of the matrix is an integer between 1 and n 2.
    • The fields of the matrix are pairwise distinct.
    • The sums of the rows, columns, and the two main diagonals are all equal.
    Magic Squares with initial constraints From the "Saarbrücker Zeitung"

    The objective is the same as in the last puzzle, but some values in these 4x4-squares are preset and/or positions are (by bold floor) constrained to hold just numbers between 1 and 8 (the lower half of the domain).

    The Prime Magic Square From Dudeney, Amusements in Mathematics, Number 410.

    Another type of initial constraint:

    Place the numbers 1 to 25 in the square so that every column, row, and the two diagonals add up 65, with only prime numbers on the shaded "T"

    (the available primes are: 2, 3, 5, 7, 11, 13, 17, 19, and 23).

    The Knight's Tour

    The goal here is to find an assignment of integers 1 .. 64 to the fields of a chessboard, such that a knight could make a tour over the whole board, using his normal moves (two forward, one sideways in any direction) and stepping on the field labeled n in the nth move.

    Magic Sequences

    A Magic Sequence if length n is a sequence of integers x0 . . xn-1 such that for all i=0 . . n-1:

    • xi is an integer between 0 and n-1
    • the number i occurs exactly xi times in the sequence.

    Markus Löckelt