# Mysteries of Odd Mathematics

## Math is the Language of All Sciences

Anything that has a pattern has a mathematical explanation to it. Some things I have yet to figure out, other things I've figured out a while back.

When I was in elementary school, someone told me that if you add all the digits of a number together (for the number 12, add 1 and 2 to get 3) if the number was divisible by 3 the sum would be 3, 6, or 9. They went on to state the reverse that if the sum is either 3, 6, or 9 that number would be divisible by 3. It doesn't matter if there are ten digits to add together or just two, the rule has never proven wrong for me.

And I was curious to see if there were any more patterns. I wanted to solve for them as opposed to just look them up. Most that I did eventually solve I found had been solved for years earlier in other places and by less pretty people than myself. For example, I discovered that if you add the digits of any number together and they sum up to 9, that number is divisible by 9. That was my first one and I since have seen it in many number theory sources. Oh well.

But there are others. Many, many others. For example... my first "big" solve. I was playing with exponents. I listed off all the numbers that were squares. 1, 4, 9, 16, 25, 36, 49... so on and so on. The differences between the numbers gets bigger. Between 1 and 4 the difference is only 3 where between 36 and 49 it is 13. I discovered that the differences between the squares was sequential, but I didn't have the mathematical background to describe it well. It hides in plane sight that the differences grow two at a time (from 3 to 5 to 7 to 9 to 11 to 13). Pretty sweet, eh? So if you ever want to predict what the next square in the sequence is, simply find the difference between the last two known squares, add two to that difference, then add that to the last number in the sequence. See for yourself in the following example: I want to know what 8 squared is going to be. I find 7 squared and subtract 6 squared (49-36 =13) I add 2 to 13, which is 15. I then add 15 to 49 to get 64, which is 8 squared.

And that was the beginning of a bigger adventure. This shit never gets old to me.

## The First Leg of the Grand Adventure

Studying squares quickly led to a dead-end, except for if you looked at cubes as well. For those of you who don't know, a square in math terms is not a shape, it's when you multiply a number by itself once, say 8*8=64. A cube is when you multiply a number by itself twice, say 8*8*8=512. Beyond that, they say eight raised to the fourth power for 8*8*8*8 = 4,096 or raised to the fifth power. Every once in a while you'll hear someone refer to it as eight raised to the seventh exponent. Same thing.

Anyway, I quickly got bored listing off as many numbers squared as I could to see if the rule held true. It is, just take my word for it now. But cubes? At first test, I found that the differences were junk. At second glance, I realized not so much. You had to take the differences between the differences to solve the pattern. There is no way I can explain it in words, so let me explain the image below. The first line is a row of first few numbers cubed. They are highlighted red. The differences between the numbers in the first row are listed below and in between their respective numbers and are highlighted in green. The difference in between the differences are below those and are highlighted in blue.

## Not the Obvious

So you can see that the green numbers don't look like they have a pattern, but they do. The blue numbers are all 6 apart from each other. If I were to want to predict the next blue, it's easy: 36+6=42. If I want to predict the next green, it's easy: 127+42=169. If I want to predict the next red, it's still pretty easy: 169+343=512. Sure enough, 8 cubed is 512.

There was something I hadn't discovered until I got at least this far. The larger exponent you're working with, the more differences between differences you'll need. When working with an exponent of 2, you only need 2 layers: the red and the green. When working with an exponent of 3, you need 3 layers: red, green, and blue. I went on to study exponents of 4, 5, and 6 before I figured out that yes, for whatever exponent you are using, you need that many layers. To predict the next exponent in the sequence, you need the same amount of numbers already solved. For example, if you are using exponents of two, you'll need to take the difference between two numbers. If you're using exponents of three, you'll need to take the differences between three numbers. If you are using exponents of four, you'll need to take differences between four numbers. So on and so on.

And that ain't the end of it, sister. Notice how the differences between the blues is sequential, but has 6 in between each one? When working with squares, the differences were also sequential but were only 2 apart? The number you will need when you reach your last "layer" is what mathematicians call the factorial of your exponent. This means that you multiply a number by all the numbers lower than it and is designated by the exclamation point. 4 factorial would be represented as 4! = 4*3*2*1 = 24. So when you're trying to solve for the next number raised to the fourth power, you need to go to the fourth layer and add 4!. Score!

Another very useless way to write an equation to solve for an exponent is say 8^4 = (#^4)^(Log(base#)8). The reason it's useless is because you'll spend five times as much effort to solve it with this equation as you would just crunching out the numbers.

## Comments

I FIGURED THIS ALL OUT IN 3RD GRADE! ALL OF THAT! AND MORE!

hey do you know if there is a more specific name to this number theory because i came a cros this and wanted to do it for my extended essay since im an IB student and need to do some research into this

Also, there is more to the trick where you add the digits of a number and if it is 9, the number will be divisible by 9. This is a trick of the counting base you are using, such that there will be a special number this will work for in each base(such as base 16(hexadecimal), base 13, base 44, or whatever base you want. It will always work for the last symbol in that base, in other words, the base number -1 (for example, it works for 12 in base 13). If the digits of a base 13 number add up to 12, it means it is divisible by 12. Also, factors of this "magic number" in each base will also be "magic numbers". In other words, if you use base 13, if the digits add to 6, it will be divisible by 6, and all the following : 4,3,2 are "magic numbers" too.

The way you add digits in higher bases can be considered equivalent to adding base 10 numbers, EXCEPT when you are dealing with the digits B and above(B is 11 in base 10). This is kind of tricky to explain, but when a high digit such as B is next to another, such as B5, the sum of these digits should be considered 13 ("13" is base13 for B+5), then add 1+3 like normal, giving you 4. This tells you that "B5" is divisible by 4.

It's no problem: 50^2 equals 2500, 49^2 equals 2401. The method here is to start with 2500, 50^2, subtract the difference between the number you are squaring and 50 multiplied by 100 from 2500: 50-49=1*100=100 ; 2500-100=2400, then add the difference^2 between 50 and your chosen number. 50-49=1 1^2=1 2400+1=2401. 49^2=2401, this can be adapted by choosing different starting points other than 50, such as 25. 17^2=?. 25^2=625; 25-17=8; 8*50: (when the starting point is something other than fifty the difference is 100*the ratio of the new base to 50. 25:50 equals .5. .5*100=50. ) 8*50=400, 625 - 400=225, add the difference squared 8^2=64: 225+64=289. If the number is higher than your base add the difference multiplied instead of subtracted.

After a short amount of practice, a day or so, I was able to calculate any number between one and one hundred in under thirty seconds, and any number between 100 and 1000 in a little over one and a half minutes.

53: 2500+3(100)+3^2=2809

72: 5625-3(150)+3^2=5184

it's helpful to know the bases squared, 25^2=625, 75^2=5625

it gets tricky with numbers such as 372: you have to quickly square either 350 or 375 to get a base, the work with a tricky subtraction or addition: 140,625- 3(750)+3^2=138,384. It is admittedly hard to keep track of the numbers mentally sometimes.

Perhaps I should add that this method is used primarily to square numbers in your head, and can be adapted to allow for numbers, at the very least from 1 to 1000.

Are you familiar with the method of squaring numbers based on increments of 50?

I believe that is only if they add up to a multiple of three, then they are divisible by three. I think this is a special case.

Sorry about my last response, I was wrong in interpreting what you were referring to. I'm not sure about the conditions in that particular pattern.

Yes, you misunderstood entirely. That isn't even the topic we are discussing. We are talking about common differences between squares and cubes...etc. You are implying that a number should be divisible by the sum of its digits.

23 goes to give 2+3 =5

23 is not devisable by 5 ?

same goes for 32, 41, 56 etc,

did i just misunderstand your orignal equation!?

I took a math course over the summer and then watched a documentary on mathematics. One day during my physics class I was bored and I stumbled upon this. The math background is where I became interested.

You said that the final difference is the factorial of the exponent. This isn't true however. It is the factorial of the exponent multiplied by the interval between numbers raised to the exponent. Say that you decide to do #^2 for every odd number. Your first four results would be 1,9,25,49. The first three differences are 8,16,and 24 respectively. The difference between these is 8. 8 = 2! * 2^2. When you do it for consecutive numbers the final difference is just x! * 1^x, which equals x!.

Same here unfortunately. Did you notice though that if you skip numbers, such as doing every third number. The common difference will be the factorial of that power times the number of integers you skip to that power. for example with n^2 for every other number the common difference is 2!*2^2. If you do every third number it is 2!*3^2.

I came across this, while looking for the very thing. I was trying to find out what this is called if anything. Do you know if there is a term for it?

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