Leonard Euler, a Swiss mathematician, established a new mathematics discipline. He built a square using Greek letters with one of each letter in every row and every column. He did the same with Latin letters in a second square. When he merged these two squares, matching every cell in one square with every cell in the other, if every Greek letter matched up with every Latin letter he called it an orthogonal pair of squares, orthogonal in this context meaning perfect. Today they are called orthogonal pairs of Latin squares. Euler and Franklin were contemporaries, Euler 1707-83 and Franklin 1706-90. I believe Franklin built two squares with two different number sets and merged them to produce a consecutive set of numbers in his magic square which is similar to Euler’s constructions. Franklin’s original 8x8
Franklin most likely built his square by adding the numbers in squares A and B. Since the patterns are easier to see I reduced those numbers by dividing A by eight and subtracting one from B to produce C and D. I will use the number sets zero to seven and zero to fifteen when discussing all his squares. These numbers sets are probably not the numbers he used. Complimentary pairs are important in his constructions. A complimentary pair has the same total as the sum of the smallest and the largest numbers in the number set. For the set zero to seven that total is seven. Complimentary pairs are zero and seven, one and six, two and five, and three and four. Four complimentary pairs make up a row and row total is 28. Any feature that is in both component squares will be in the combined magic square. A feature in only one square will not be in the magic square. If a magic square is broken down every feature in the magic square will be in both component squares. There can be hidden features in the magic square that are in only one of the component squares. In square A he started a consecutive sequence in row one column three. That gave him half rows with half row totals. He placed the compliments of the numbers in row one in row two. I call that process Franklin’s complimentary construction. He repeated that process to fill the square. Rows one, three, five, and seven are the same. Rows two, four, six, and eight are also the same. Every half column starting at an outside edge has two complimentary pairs and will have a half row total. Every 2x2 is made up of two vertical pairs. Since all vertical pairs are complimentary pairs all 2x2’s will have half row total as well. In square B he placed complimentary pairs in column one in rows one and three, two and four, five and seven, and six and eight. He completed the square with complimentary construction. All half rows, half columns, and 2x2s have two complimentary pairs and have half row totals. By putting a complimentary pair in square B in column one rows one and three the three repeats in odd columns in row one and in even columns in row three. Since rows one and three in square A are the same the three matches up with all the numbers in square A when the squares are merged. The four repeats in even columns in row one and in odd columns in row three. It also matches up with all the numbers in square A. Since all complimentary pairs are both in odd rows or both in even rows in column one of square B all numbers in B match up with all numbers in A. Either square can be multiplied by eight and added to the other square plus one and a complete consecutive sequence from one to 64 will be produced in the magic square. The half diagonal from the upper left corner to the middle of the square in square B is three, two, four, and five. The three and four are in column one and are a complimentary pair. The two and five are compliments of the five and two in column one, a complimentary pair. Since the compliments of a complimentary pair are also a complimentary pair the half diagonal is made up of two complimentary pairs and has a complimentary total. All half diagonals in square B starting in row one or row five, down to the left or down to the right, will have half row totals. Therefore all bent diagonals and all broken diagonals will have row total. The problem with the diagonal totals is in square A. The half diagonal from the upper left to the middle of the square is six, zero, zero, and six. The compliment of six, one, is under it in row two. The zero is one cell to the right and because of the reverse sequence in row two is one less than complimentary. The full diagonal is made up of four pairs of numbers that are all one less than complimentary and totals four less than a complimentary total. The upper right to lower left diagonal totals four greater than complimentary in a similar manner. Some materials on this site are Copyright © Donald Morris 2005 all rights reserved |