Multiplying numbers using square numbers.

To do this, you have to know a lot of square numbers. You can look up a list online if you want. All that is needed is to take the square of the average of the two numbers, then subtract by the square of the amount of numbers away from the average the two are. For example, given 51x45, the average is 48, and they are 3 numbers away from it. The square of 48 is 2,304 and the square of 3 is 9. Subtract the two numbers to get your answer, 2,295. I have to manually count -1 for this number, +1 for that over and over to get the average number and the distance from it, maybe you can do it quicker. For that reason, it's the quickest if the 2 numbers are close to each other.

So far this only works if the numbers are either both even or both odd so that the average number is an integer, a number without decimals. It probably works if this isn't true, but it involves fractions and is more complicated.

Also, remember my previous post about multiplying one number by something and dividing the other by the same amount, if your two numbers are far apart and you want to make them closer. This is a link to the first thousand squares, which leads to one million. Also, I admit, I cheated a little on this one. I noticed that the pattern was 1 different for the first number, 3 more different for the second, 5 more for the third, 7... and I didn't realize that this was a square as well, and so I has some help on Quora. Also, the +1 formula I posted in 2012 is similar to this, it just involved one, though.

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March Seventeenth, 2019

I found (also on Quora) a way to multiply two-digit squares. A summary: Take the last digit, square it. Then multiply the first and last digits, then double that number. Then square the first number. Now you have three numbers. Place them in order, from right to left. Then combine them into one by adding, as the first number is in the ones place, the second is in the tens and the third is in the hundreds. If you want examples, they're in the comments.The two digits could be anywhere, including a number like 1.6, since you can just multiply numbers, ignoring periods, calculating them at the end of the process. Since the square method has a 50% chance of ending in .5 if you started with whole numbers (if the two numbers had differing parities), I'll try to find a method for multiplying 3-digit numbers.

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The Next Day

Today, I calculated the results when squaring numbers ending in .5. 1.5, 2.5, 3.5, etc. When I subtracted by the non-decimal portion squared (As in, 7.5²-7²), I got the number (7 in the example) +.25 as the result. So now, you can multiply any two digit numbers in your head!

Square Root Omniscience

14 = 3.5 x 4, which are the closest numbers I know of that equal it. If you want to take the square root of 14, the answer is between those 2. The closer the numbers to each other, the more exact the square root. The actual number is 3.741657387, which is slightly less than the average, the halfway point. If you take another number that has 2 numbers half a number away, like 33=5.5x6, the square root is almost the same, 3.744562647. The numbers grow father away from the halfway point the farther from each other they are, as follows:

√48 (6x8) = 6.92820323
√30 (6x5) = 5.477225575
√180 (15x12) = 13.41640
786

Subtracting with Exponents

If you take a lot of numbers and then square them and them subtract the result of each number from the one above it you will notice a pattern.

1²=1     4-1=3
2²=4     9-4=5
3²=9     16-9=7
4²=16     25-16=9
5²=25     36-25=11
6²=36     49-36=13
7²=49     64-49=15
8²=64     81-64=17
9²=81     100-81=19
10²=100

Every time the number you are going up by is increasing by 2. That phrase sounds a lot like my last post. So, let's see if there's something similar for the cubes.
1³=1     8-1=7
2³=8     27-8=19
3³=27     64-27=37
4³=64     125-64=61
5³=125     216-125=91
6³=216     343-216=127
7³=343     512-343=169
8³=512     729-512=217
9³=729     1,000-729=271
10³=1,000
This looks a lot like the exponents from before, except different numbers. So maybe we need to do it again?

19-7=12     127-91=36
37-19=18     169-127=42
61-37=24     217-169=48
91-61=30     271-217=54
It worked! We have the numbers which are increasing by a number again. The difference is that this time it's 6. So it's probably either going up by 4 or multiplying by 3. I'll do one more to see, though I'll probably have to subtract 3 times this time.
1⁴=1     16-1=15     65-15=40     110-40=70     84-70=14
2⁴=16     81-16=65     175-65=110     194-110=84     108-84=20
3⁴=81     256-81=175     369-175=194     302-194=108     132-108=24
4⁴=256     625-256=369     671-369=302     434-302=132     156-132=24
5⁴=625     1,296-625=671     1,105-671=434     590-434=156     180-156=24
6⁴=1,296     2,401-1,296=1,105     1,695-1,105=590     770-590=180     204-180=24
7⁴=2,401     4,096-2,401=1,695     2,465-1,695=770     974-770=204
8⁴=4,096     6,561-4,096=2,465     3,439-2,465=974
9⁴=6,561     10,000-6,561=3,439
10⁴=10,000

Well, I had to do 4 subtractions instead of the 3 I anticipated and something unexpected happened. After 2 numbers it got stuck in a rut of 24. I'll see what happens if I add the 11th number.
11⁴=14,641. 14,641-10,000=4,641. 4,641-3,439=1,202. 1,202-974=228. 228-204=24!
Well, I think it's broken. If you want to do the next stage, do it, but since it might be 8 subtractions, it will be a lot of work. I'm not going to do it, but if you do, please post your findings in a comment for this post.

A pattern that has two ways you can describe it as!

Take this sequence: 2, 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288...

Can you figure out the equations?

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Hint: The first way deals with exponents and is a geometric sequence.
The other way is by adding and involves an arithmetic sequence.

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Answer:

#1: Take every even number (or any number multiplied by 2) and square it, then divide by 2. You can see it as (2x)². This also equals 4x², which simplifies to 2x². This means you
                                 ------                           -----    could alternatively take every number
                                    2                                2       and multiply by 2. They both get to the
                                                                               same number every time!

#2 Notice how the numbers go up. 2 + _6_ = 8                         8 + _10_ = 18
       18 + _14_ = 32                       32 + _18_ = 50                    50 + _22_ = 72
       72 + _26_ = 98                       98 + _30_ = 128                 128 + _34_ = 162
     162 + _38_ = 200                   200 + _42_ = 242                  242 + _46_ = 288
The first number it goes up by is 6, then that number goes up by 4 every time. This number goes by the arithmetic equation of 4x+2. The only way I can think of to find a number in the sequence without going through the whole thing is to make a new equation using the one I just mentioned. For the 4x, it is (4x1)+(4x2)+(4x3).... which = 4(1+2+3...) To add up all those numbers quickly if you want to do a big number, add the first and the last, the second and the second to last, the third and the third to last... You will notice that all those numbers are the same. Now just take that number and multiply by the number of pairs you have. Now you have to add the twos. Add 2 multiplied by whatever number in the sequence you want, then plus 2 because the starting number is 2, not 0.
So, to get the number in the sequence, all you have to do is multiply 4 by the total of the 1 and 2 and 3 and all that and then add 2x+2 or 2(x+1).

Now, lets try with 15. The number above only go up to 11, so lets do some more. 288+50=338 +54=392 +58=450 +62=512 So now we know that the answer should be 512. So first do 4 multiplied by 1 to 15. To add them up quickly, I'll do 1+14 and 2+13 and 3+12, and so on, until 7+8. All these add up to 15. The seven pairs multiplied by 15 is 105, plus the final term of 15 is 120. 120 times 4 is 480. Then the 15 x 2, +2 = 32. 480+32=512!

One final note: every term that you add on (the one that goes up by 4) should end in 0 every 2nd, 7th, 12th, 17th and so on time that it is added. It should have gone up by 20 every time this happens. Also, Every term that is the total (that goes up quicker and quicker) should end in zero every time the term number ends in 4 or 9. Also, it goes up 100 more than the last time 5 terms ago. The first is 50, so 50 more than last one, which was 0. Then it's 200, 150 more than last. then 450, 250 more. Then 800, 350 more. We can predict 2 of the 5 numbers in each set this way, the 2nd and 4th in it.
The numbers up to 20 (every 5th is bolded): 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288, 338, 392, 450, 512, 578, 648, 722, 800, 882

Flame changes ink color!

I noticed another thing today... When I was burning a few scraps of paper with typed letters on them I noticed that in the ashes that remained, if you could still see the words, most of the words had turned orange! I tried it again, as I thought maybe it had been orange in the beginning and I just hadn't noticed... but it still turned orange! When you think of fire, you usually think of orange or yellow and that's the color of the fire from my lighter. I wonder, if I burned the paper with blue flame, would the letters have turned blue...?

Strange Exponent Sequence

This is a strange sequence involving exponents. It is as follows: You pick any positive number. Take the last digit. This is what you'll be working with. Look at the chart below. If the number is...

1: 1 Always one!
2: 2, 4, 8, 6 Repeating 4 digit even sequence.
3: 3, 9, 7, 1 Repeating 4 digit odd sequence.
4: 4, 6 Only two numbers.
5: 5 Always 5!
6: 6 Always 6, surprisingly for me.
7: 7, 9, 3, 1 Repeating 4 digit odd sequence.
8: 8, 4, 2, 6 Repeating 4 digit even sequence.
9: 9, 1 Only two numbers.
10: 0 Always ends in 0.

11: 1
12: 2, 4, 8, 6
13: 3, 9, 7, 1
14: 4, 6
15: 5

Notice something strange? I do! For one, it repeats after 10. That's why this is so about the last number. Also, for every number 1-10, no matter how many digits the cycle was, the digits in the cycle were always even if the original number was even and vice versa.

Now, after you got your numbers in the sequence, you know what the last numbers will be! Whatever your number is, it is the first number in the sequence. The next number in the sequence occurs when you multiply your number by itself, or to the exponent of two. The last number should be the next number in the sequence. After that, multiply it by itself again. The last number now should be the 3rd number in the sequence. After your sequence is done, it restarts.

You might need an example. Here is one:

1. You pick 7. The sequence is 7, 9, 3, 1!
2. 7x7=49.The 9 is the last number, and is the number in the sequence!
3. Multiply by 7 again. You end up with 343. The 3 fits!
4. 343x7=2,401, which fits the 1!
5. After the sequence is done, it repeats. Next is 16,807, which is the first number in the sequence!

Adding, Subtracting, Multiplying and Dividing Exponents

1. Same base, different exponents add the exponents together. Example is: 8^3 x 8^5=8^8.

2. Have base and exponent in parenthesid and then exponent, multiply exponents together. Example is: (2^3)^5=2^15.

3. Base A with exponent times base B with exponent divided by Base A with different exponent times base B with different exponent equalssubtract exponents from each other. Example is: 3^4 x 4^6 /3^7 x 4^5 = 4^1/3^3=4/3^3=4/27.

4. If base divided by different base with same exponents, then ????. Example is 2^3/3^3.