IBM Ponder This #090 · October 2005

Puzzle for October 2005.

This months puzzle concerns arrays of integers, a(i,j) with the following property.  For all pairs of distinct rows i1,i2 and distinct columns j1,j2 the diagonal sum a(i1,j1)+a(i2,j2) and the anti-diagonal sum a(i1,j2)+a(i2,j1) are unequal.  We are interested in finding such arrays with the entries chosen from as narrow a range of integers as possible.  For example the following is an example of such a 3x3 array with entries chosen from {0,1}

0  0  1
        0  1  0
        1  0  0

Find a 5x5 array with this property with entries chosen from {0,1,2}. If this is too easy try to find a 7x7 array with entries from {0,1,2,3,4}, a 11x11 array with entries from {0,1,...,6,7} and a 13x13 arrays with entries from {0,1,...,7,8}.  This will probably require computer assistance.

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The first 100 people who answer all four parts correctly will be listed. The answer will be posted a week after the 100th is received, or at the end of the month.

Solution

Best opened after a real attempt

To be added.