Day 44: A Math Blog (NERD ALERT)

During my four years at Luther College I was convinced I was going to be a double-major in Math and Physics. Well as it turns out I'm not so good at the math aspect of this endeavor and I was content to settle with a Minor in Math instead. Anyway, so like I was saying during this time I got to take a few of the more abstract math classes (Group Theory, Discrete Structures) and from time to time would actually understand and appreciate a few nuggets that were taught. I'm going to try and relay one of the more memorable ones, though I didn't learn this in a class this was sort of an observation made and (still) never fully understood, but I'm not going to give it in mathspeak (mostly because I no longer possess the ability to Eigen anyone's Vectors).

Alright so I figured this out one day while I was tutoring physics (uber-nerd, yes) during the day after a test (meaning nobody would show up for tutoring, so I'd sit there for my 3 hours and do homework or whatever). These are a little bit abstract at first, but if you stick with it it'll make some sense (or at least you'll see the pattern) if you can make sense of it let me know.

So I'll try and explain this as best as I can.

Let's start with the number one:

If you take the numbers 1-10, and multiply each of them by the number 1, you'll get what you can see below, obviously. However, if you then add the resulting two digit product (if there is two digits), you'll get what you see after the arrows (-->) below. Clearly it doesn't change anything except for the 10x1=10-->1 case.

1x1=1-->1

1x2=2-->2

1x3=3-->3

1x4=4-->4

1x5=5-->5

1x6=6-->6

1x7=7-->7

1x8=8-->8

1x9=9-->9

1x10=10-->1

So in this case, the input constants (1,2,3,4,5,6,7,8,9,10) nearly matches the resulting output (1,2,3,4,5,6,7,8,9,1). If you were to continue past 10 to say 20, you'd see the same repetitive pattern through infinity.Let's look at another case, 2's.

2x1=2-->2

2x2=4-->4

2x3=6-->6

2x4=8-->8

2x5=1-->1

2x6=3-->3

2x7=5-->5

2x8=7-->7

2x9=9-->9

2x10=20-->2

In this case we get a much different pattern than we did with 1's as the input variable. In this case it sort of "filters out" the even numbers in numerical order and then places the odd numbers after it before beginning its pattern again. Rather than going through them all like this I've placed the individual cases in the table shown below.

In a few of the cases, such as say 8x6=48=12, this is continued since its still two digits; accordingly 8x6=48=12=3.

Now, I guess you're probably asking why is this interesting? Well, it may not be at all; however, if you look at some of the patterns that emerge then its at least a little more intriguing. I'll discuss a little bit here, but please weigh in on this because it something I've been wracked with for the past 10 or so years.

1: Already discussed these, it spits out the respective input constants, similar to an identity operator (as one would probably expect since it involves the number 1).

2: Again, discussed this already but its worth mentioning again how this one sorts the even/odd numbers in numerical order. (2/4/6/8/1/3/5/7/9/2/4...)

3: This particular one has a distinct short pattern of repeating 3, 6, 9 to infinity. Here you might expect something along those lines since those are all multiples of the number 3, so nothing too out of the ordinary.

4: Here's where it starts to get weird. Now we have no discernible pattern here, even though 4 is a multiple of 2 and 2 had a fairly obvious pattern, the 4's show no real pattern. There is an even,even,odd,odd pattern as you march through the numbers until you hit 9, so maybe that's something? (E/E/O/O/E/E/O/O/O/E/E)

5: Again, no real pattern emerges here. Here there is an odd/odd/even/even pattern that is somewhat continuous again, (O/O/E/E/O/O/E/E/O/O/O), however it ends with a string of 3 similar just as 4's did.

6: Another similar short repetitive pattern as was seen with 3's (6-3-9).

7: This is an interesting results especially considering which number it is (7). This pattern actually filters out the odd numbers first, then puts them in descending numerical order before doing the same with the even numbers. (7/5/3/1/8/6/4/2/9/7/5/3...)

8: Just as 7 has essentially the opposite effect as 2 does, 8 does the inverse of what 1 does. Here we have the numbers sorted by odd and even in sequential order, descending numerically.

9: The most strange of all the results. All 9's. No matter what happens here you'll always get a 9, it almost carries the same null properties that zero does (i.e. anything x zero = zero, in this "realm" anything x nine = nine; in addition to the zero property which I didn't put in any of the above figures simply because it's too boring).

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So why do you always get a 9? This is the part that baffles me, it doesn't make a lot of sense. Its not related to the fact that 9 is a multiple of 3, 3^2, or 3+3+3, or anything like that b/c then by the same logic you'd expect to see a similar result for 2's (as in 2^2 or 2+2+2) but you clearly don't.

One observation that I'm not sure how to explain, though it is slightly interesting is "mirror image" effect seen here. Assume for a moment that the results of the number 9 is the center, let's just say for the sake of argument that in this scenario its the number zero. From here if you descend in order (8,7,6,5) you'll notice that this is the mirror image of the effect that happens if you go up in order from 9, (1,2,3,4). Its a little difficult to explain but maybe this will help. Ive done this two different ways, the first one has the mirror images as the same colors by column. In the second figure I assigned a color spectrum value (according to ROYGBIV) to each of the values 1 through 9, and then filled it in. I thought the colors brought the pattern to light a little better since its easier to notice color patterns than numerical (at least for me). Also I trimmed off the #10 row of the ROYGBIV spectrum only because its essentially a repeat (10=1+0=1).

I've tried to highlight the symmetry lines in the ROYGBIV figure using the bold lines, it seems to help.

Anyway I don't really have a conclusion here, like I said I'm not sure I fully understand it at all, but its something I've observed. I was hoping that this could possibly spark some interesting debate about this, but maybe its a little too boring who knows.

*Also, turns out I don't know what the indigo color in ROYGBIV looks like so I assumed its that color seen on number 8.