Siamese and Uniform Step Squares - Analogy to Fractals (Part III Con't)

25^th and 49^th Order Squares

This section is a continuation of Part III which covered the 5^th, 7^th and 9^th squares. This section will be used to predict which of the 25th and 49th order Staircase and Uniform Step squares are magic and which are not. However, none of the squares predicted as magic will be posted. The tables posted will be presented in Soduko fashion with four symbols represented by:

in which one of these symbols is placed into the same cell of a square where the initial number 1 is to be placed. A diabolic square, labeled here as an O with horns (el Diablo), is a magic square in which every row, column, two major diagonals and its negative diagonals (all the diagonals going from the left to the right) sum to the magic number. These large order squares, not divisible by 3, can be built up from the lower order 5^th and 7^th which are shown below. For example, a 25^th order square requires 5×5 of these subsquares and the 49^th order square 7×7 subsquares and is a way of predicting the magic state of the square within a particular cell.

Just by looking at the squaretype table one can tell immediately if the square is magic without any previous knowledge of abstract mathematics. The squaretype tables are nothing more than large structures built up from smaller substructures similar to the fractals where the component parts resemble the whole.

As mentioned in Part III an axis of symmetry runs along the right diagonals of tables Staircase 5 and 7 but not in the Uniform Step 5 and 7 subtables. This means that the cells above and below the axis of symmetry in the Staircase squaretype tables (those constructed from the lower subtables) are mirror images of each other. This, however, is not possible for the Uniform Step squaretype tables which have no similar axis of symmetry.

For example, the staircase 25 squaretype table below shows what a block of 5×5 subtables looks like where the main right diagonal serves as a boundary for the two mirror images. The same cannot be said for the Uniform Step 25 squaretype table which lacks an axis of symmetry.:

This method also applies to the 49th order squaretype tables which are similarly constructed.

This completes this section (Part III Con't). To go back to Part III. To go to Part IV. To return to homepage.