# unusual forcing matrix properties

This 4 x 4 matrix has an unusual property. $$\begin{pmatrix} 1 & 2 & 3 & 4\\ 5 & 7 & 9 & 11\\ 12 & 17 & 21 & 26\\ 27 & 40 & 46 & 59\\ \end{pmatrix}$$

If we take all 24 combinations where there is only one value from each row and one from each column then the sum of each combination gives a unique total.

I.e.

Total = Elements
57 =27 + 17 + 9 + 4
58 =27 + 17 + 3 + 11
59 =27 + 7 + 21 + 4
61 =27 + 2 + 21 + 11
63 =27 + 7 + 3 + 26
64 =27 + 2 + 9 + 26
65 =12 + 40 + 9 + 4
66 =12 + 40 + 3 + 11
etc...

There are some gaps in the totals (e.g. no number 60) which is OK but I am looking for a generalized way to produce tables with this property for any NxN matrix. (I came across the above example by trial and error but cannot see any logical patterns within the numbers that would suggest it may have this unusual property.)

• Removed wrong tag. – Martín-Blas Pérez Pinilla Jun 21 '17 at 7:46
• I am a newbie on this site. sorry about the presentation of the matrix and table. I could not get a table to appear correctly. – Gordon Jun 21 '17 at 7:48
• – Martín-Blas Pérez Pinilla Jun 21 '17 at 7:50

Here is one boring way to generate such a matrix: fill the matrix with different powers of a prime number $p$. Then in base-$p$ expansion, the sum of entries in each traversal gives a unique sequence of zeros and ones.