Squares!

Well, by squares, I obviously mean square numbers, right?

Welcome back to your favorite site! I'm going to do a rather short post today, as I can't really think of much to do.

Here's something I found out a couple of days ago.

For any integer, then x^2 is either a multiple of 4 or one more than a multiple of 4.
Proof: This theory depends on whether that integer is odd or even. Let's represent the set of any integer as N.
odd– (2N+1)^2=4(N^2)+4(N)+1=4(N^2+N)+1=4(N)+1
even– (2N)^2=4(N^2)=4(N)
As one can see, an odd integer squared is one more than four times another integer, and an even integer squared is four times another integer. Isn't that weird?

Another odd thing is that the sums of the first few cubes is always the square of the sum of their base numbers. For example,
1^3 + 2^3 = (1+2)^2
1^3 + 2^3 + 3^3 = (1+2+3)^2
1^3 + 2^3 + 3^3 + 4^3 = (1+2+3+4)^2

Speaking of shape--ish numbers, every square is the sum of two consecutive triangles. I wonder if the sum of two consecutive squares is a pentagonal? Or if you can figure out hyper-cubic numbers? Probably you can-- I'll update that later.

The difference between perfect squares is always odd. (n+1)^2 - n^2 = 2n+1, so if n is an integer, the difference is odd (and the squares are perfect squares).

There's a weird pattern in the squares of all of the integers from 969 to 1000. What is it? (Look closely; no, it's not the thousands digits)

An easy way to approximate a square root without a calculator is by:
1) guessing the root
2) dividing your guess into the original
3) finding the mean of the result and the guess
4) and repeating steps 2 to 4 with what you get.

All square numbers end with the digits 1, 4, 6, 9, 25, or 00.

Stay coolio,
John

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