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Path: utzoo!watmath!clyde!cbosgd!ihnp4!bentley!kwh
From: kwh@bentley.UUCP (KW Heuer)
Newsgroups: net.math,net.puzzle
Subject: Bimagic squares
Message-ID: <666@bentley.UUCP>
Date: Wed, 26-Mar-86 13:47:50 EST
Article-I.D.: bentley.666
Posted: Wed Mar 26 13:47:50 1986
Date-Received: Fri, 28-Mar-86 07:14:57 EST
Organization: AT&T Bell Laboratories, Liberty Corner
Lines: 21
Xref: watmath net.math:3007 net.puzzle:1568

Some definitions.

An n x n array has 2n+2 _lines_: n rows, n columns, and two full
diagonals.  A square is _magic_ if the sum of the n elements in
a line is constant (i.e. independent of the line).  A square is
_bimagic_ if it is magic and the sum of the squares of each line
is also constant.  A square is _normal_ if its elements are the
first n^2 positive integers.  (Since magic and bimagic squares
preserved under affine transformations, you may substitute your
favorite arithmetic progression.  I often use [0,n) instead of
(0,n].)

What is the smallest normal bimagic square (after the trivial
case n = 1)?

More specifically, I've proved to my satisfaction that there are
none for 2 <= n <= 6, and I have an example for n = 8.  Is there
a bimagic square of order 7?  (The magic constant is 175 and the
bimagic constant is 5775.)

Karl W. Z. Heuer (ihnp4!bentley!kwh), The Walking Lint