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        riddles >> hard >> Bimagic Square Reconstruction
        
(Message started by: Barukh on Aug 9^th, 2014, 7:22am)

Title: Bimagic Square Reconstruction
Post by Barukh on Aug 9^th, 2014, 7:22am

A bimagic square is a magic square that retains its properties after every element is squared.

That is: A bimagic square is a square matrix n x n with the following properties:

1. Every number 1, ..., n² appears in the matrix.
2. The sum of numbers at every row, every column and two main diagonals is the same.
3. The sum of squares of numbers at every row, every column and two main diagonals is the same.

That bimagic squares exist is not immediately evident (at least for me). However, it was proved that they exist for every n > 7.

Now - the problem. You are given a partially populated bimagic square of order 8. Put in the missing numbers.

It is very probable that solution may be obtained - and quite quickly - by programming. The question is: how far can we go without any programming at all?

Note that this bimagic square was built 124 years ago!

Title: Re: Bimagic Square Reconstruction
Post by rmsgrey on Aug 10^th, 2014, 7:43am

nitpick: two of the square's properties (1 and 3 in your list) are not retained when every element is squared...

Title: Re: Bimagic Square Reconstruction
Post by dudiobugtron on Aug 10^th, 2014, 5:14pm

on 08/10/14 at 07:43:46, rmsgrey wrote:

nitpick: two of the square's properties (1 and 3 in your list) are not retained when every element is squared...

I took 'that is' to be an indicator that the properties listed were a more rigourous definition; a rewording of the earlier definition rather than a continuation.