Thursday, December 29, 2016
Primitive Operations Leave the Sums Unchanged
We need to show that the primitive operations of transpose, vertical and horizontal reflection and the double shifts, acting on a magic square, leave the sums unchanged. To do this we perform the actions on a magic square and compare sums.
The sums for the rows, columns and the quarters after one of the primitive operations can all be found in the original magic square in rows, columns or quarters. For a transpose and a flip (multiplication by F) the sum for a diagonal is a sum for a diagonal in the original magic square and the same is true for the center and corners. For the double shifts (multiplication by G) sums for the diagonals, center and corners are evaluated above and remain unchanged. So all these primitive operations leave the sums for the magic square unchanged.
Supplemental (Dec 30): Had a little problem typing in the lower case i. It changed to what looked like an l. Retyping the i corrected the problem.
Turns out the lower case i was autocorrected to capital I. Decided to remove it from the autocorrection options.