Pattern Recognition and Fun With Seven
I admit it. I’m a math and science geek. I am, undoubtedly, a software engineer as well so I better be darn good at working with numbers and recognizing patterns.
The other night, I was playing around with math. When I was younger I would play games with license plate tag numbers trying to make equations out of them. It was pretty easy. By assigning numeric values to the letters using a base-10 system with A = 1, you could take a tag number of AC214D and come up with the individual numbers 123144. You’d then have to figure out some way to make the equation work. The equivalency operator and parenthesis could go anywhere in the equation and you could combine numbers. This particular tag number supports:
(1+2-3+1)*4 = 4
1+2-3+1 = 4/4
12/3*1*4 = 4
You get the idea, but back to the point.
The other day I noticed something about the number 7. Particularly about 7 and 21. I found it odd that the sum of the digits of 21 (2+1) is also the multiplier of 7 required to equal 21. Or 7 * (2+1) = 21. I started playing randomly with other numbers to see if it would work and then I realized that it applies to the following:
7*(4+2) =42
7*(6+3) = 63
7*(8+4) = 84
7*(10+5) = 105
…
7*(18+9) = 189
(My math language is likely going to start to break down here so bear with me. Anyone have a whitboard?)
The pattern, in case you don’t see it yet, is that the two numbers in the sum, when concatenated as strings, equal the product of the sum and 7. Further, the first number of the sum is always a multiple of two and the second number of the sum is 1/2 the first number (9 is half of 18).
I was pretty excited (yes, excited) until I reached the following:
7*(20+10) != 2010
Rats. Unthwarted, I started to pursue this a little further and realized there was a more subtle pattern. Instead of concatenating the string representation of the numbers, what if the first number was moved over one digit (multiplied by 10) and then added to the second number.
This would mean:
7*(2+1) = 21 = (2*10)+1
7*(10+5) = 105 = (10*10) + 5
7*(20+10) = 210 = (20*10) + 10
Low and behold, it worked. And it works for much larger numbers as well.
7*(14296+7148) = 150108 = (14296*10)+7148
One thing I noticed that I thought was neat until I could easily explain it away is that the product of the numbers abiding by this pattern was always divisible by three. The reason is that your multiplying 7 by (2x+x) This is the same as multiplying 7 by 3x and so all numbers are divisible by 3. Incidentally, if the sum of the digits of a number are divisible by three, then the number itself is divisible by 3.
Turns out there IS a simple reason this occurs. If you multiply 7 by (2x+x) you essentially have (7 * 3x). Thusly, 7*3x = 21x = 20x+1x. So the reason the pattern occurs is pretty simple to understand but I still think it’s pretty neat.
Did I mention I love math?

