Some Nifty Math Stuff

Here’s something nifty. Take a multiplication table and look at just the last digit on the right (i.e., 7 x 5 = 35, so you’d look at the 5, etc.). Let’s compare the various numbers side-by-side from 1 – 9:

1 2 3 4 5 6 7 8 9
2 4 6 8 0 2 4 6 8
3 6 9 2   8 1 4 7
4 8 2 6   4 8 2 6
5 0 5 0   0 5 0 5
6   8       2   4
7   1       9   3
8   4       6   2
9   7       3   1

Astute readers will already see some cool patterns in the above. The most obvious is the fact that the 9’s column is the inverse of the 1’s column. But when you look further and ignore the zeros (for the moment), you see that the 8’s column is the inverse of the 2’s column, the 7’s is the inverse of the 3’s, and the 6’s are the inverse of the 4’s.

And now comes the fun part. What happens if we fill in the blanks on the rest of the square we’ve got by padding it with the repetitions of the previous rows? We get this:

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

The 1’s column is still the reciprocal of the 9’s column, etc. But look at what else happens! The 1’s row is equal to the 1’s column; and that means that the 1’s row is also the reciprocal of the 9’s row! This carries through with the rest of the rows too. Finally, looks at some diagonals. For instance, take the diagonal at 1.

The diagonal at 1 is: 1 4 9 6 5 6 9 4 1, a sequence that is a palindrome (it reads the same forwards as it does backwards). This is true for all the diagonals.

This is also true if we consider other even bases. Let’s look at base-6 for instance. Here we get:

1 2 3 4 5
2 4 0 2 4
3 0 3 0 3
4 2 0 4 2
5 4 3 2 1

Note, too, how the mid-point oscillates between 0 and the midpoint, just as in base 10 (i.e., in base 10 it’s 5 0 5 0 5 0… in base 6 it’s 3 0 3 0 3 0…).

So what happens with and odd base, like base 7? Let’s find out!

1 2 3 4 5 6
2 4 6 1 3 5
3 6 2 5 1 4
4 1 5 2 6 3
5 3 1 6 4 2
6 5 4 3 2 1

Once again, the pattern holds, only this time there’s no central column that oscillates between 0 and the midpoint of the base.

I’ll let you do what you will with this :-)