As often happens when I think of math, I keep thinking of math. In this case, I’ve come up with a few more observations about the square that I described earlier, namely this one:
1 2 3 4 5 6 7 8 9 2 4 6 8 0 2 4 6 8 3 6 9 2 5 8 1 4 7 4 8 2 6 0 4 8 2 6 5 0 5 0 5 0 5 0 5 6 2 8 4 0 6 2 8 4 7 4 1 8 5 2 9 6 3 8 6 4 2 0 8 6 4 2 9 8 7 6 5 4 3 2 1
First, there’s an obvious reason as to why 1/9, 2/8, 3/7, etc. are reciprocal columns/rows. Each are the same distance away from the 0 position. One is above, one is below, but both columns are the same actual distance. So the patterns should be similar, and in fact the only difference is the inversion to describe the one that is above and the one that is below.
After this, I thought it might be fun to add up the various values in the square. I’ll just do the rows here:
1 2 3 4 5 6 7 8 9 T = 45 2 4 6 8 0 2 4 6 8 T = 40 3 6 9 2 5 8 1 4 7 T = 45 4 8 2 6 0 4 8 2 6 T = 40 5 0 5 0 5 0 5 0 5 T = 25 6 2 8 4 0 6 2 8 4 T = 40 7 4 1 8 5 2 9 6 3 T = 45 8 6 4 2 0 8 6 4 2 T = 40 9 8 7 6 5 4 3 2 1 T = 45
Furthermore, the long diagonals both add up to 45 as well.
Now the odd values that don’t add up to 45 seem somewhat out of place. And combining what I discovered earlier with the above chart, you can easily see why. For when we look at columns 1/9, 2/8, 3/7 etc., we see that the column pairs add up to 10 in each case. But when we get to the fifth column, it pairs to nothing.
So what happens if we treat the 0 in the above as if it held a value identical to 5? We’d get:
1 2 3 4 5 6 7 8 9 T = 45 2 4 6 8 0 2 4 6 8 T = 40 + 1 Zero character = 45 3 6 9 2 5 8 1 4 7 T = 45 4 8 2 6 0 4 8 2 6 T = 40 + 1 Zero character = 45 5 0 5 0 5 0 5 0 5 T = 25 + 4 Zero characters = 25 + 20 = 45 6 2 8 4 0 6 2 8 4 T = 40 + 1 Zero character = 45 7 4 1 8 5 2 9 6 3 T = 45 8 6 4 2 0 8 6 4 2 T = 40 + 1 Zero character = 45 9 8 7 6 5 4 3 2 1 T = 45
That’s right! If we treat the zero as if it were a five, the columns add up to 45 in each case. That these two numbers go hand in hand becomes more obvious when we look at the pattern of their placement in the square. If we replace the every character that is NOT a 5 or a 0 with an *, we get the following:
* * * * 5 * * * * * * * * 0 * * * * * * * * 5 * * * * * * * * 0 * * * * 5 0 5 0 5 0 5 0 5 * * * * 0 * * * * * * * * 5 * * * * * * * * 0 * * * * * * * * 5 * * * *
Obviously, only a 5 or a 0 appears on the cross line that creates this square. Let’s look at two more patterns:
1 * * * * * * * 9 * * 3 * * * 7 * * * * * * * * * * * * * * * * * * * * * * 9 * * * 1 * * 3 * * * * * * * 7 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 * * * 9 * * 7 * * * * * * * 3 * * * * * * * * * * * * * * * * * * 9 * * * * * * * 1 * * 7 * * * 3 * *
As you can see, they’re sprinkled symmetrically throughout (which is to be expected).
In any case, as I said before, you can make of this what you will! :-)
