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A square number is any number times itself.

For example, 1x1, 2x2, 3x3, 4x4, 5x5...

A triangular number is a number plus the numbers before (in a row,1+2+3+4+5) 1, 3, 6, 10, 15... are all triangular numbers.

Cube numbers are the same number times itself 3 times ex. 3x3x3=27

A Triangular Square Number is the square number of triangular numbers.

ex. 21 is the 6th triangular and 21x21=441

441 is a triangular square number!

Pattern.1

First, what is a square number minus a triangular number? The answer is another triangular number. You can use the formula s-t=s where s is a square number and t is a triangular number. So if you use the 15th square number, you also need to use the 15th triangular number.

Example.

The 15th square is 225, and the 15th triangular number is 120. So,

225-120=105

105 is the 14th triangular number! Mathematical magic!

Pattern. 2

Ever ask what is a square plus a triangle? The same rule applies(nth triangular number has to be nth square number)

So, 1+1=2, 3+4=7, 6+9=15… 2,7,15,26,40…there is a pattern, what you add is increasing by 3.

On the left side of the sums are the triangular numbers, and on the right side of the sums are the square numbers!Awesome!

Pattern.3

It also turns out that, if you multiply any square number by any one number there is always a number pattern. Ex. Square number x 5= 5, 20, 45, 80, 125… ,the number pattern is the number is increasing by 10 to what you add, and you start at +15.

Triangles are half of squares. Like square numbers if you multiply triangular numbers by a number there is also a number pattern. Ex. Triangular number x 5= 5, 15, 30, 50, 75... , the pattern is that the number pattern is increasing by 5 to what you add starting at +10.

Pattern. 4

What happens when you multiply a triangular number by a triangular number, with the rule that if it is 3rd triangular number it must be the 3rd square number?

0, 1, 9, 36, 100, 225… squares of triangular numbers have a pattern 1-0=1, 9-1=8, 36-9=27, 100-36=64, and 225-100=125 1, 8, 27, 64, 125… the difference between triangular square numbers and are what mathematicians call cube numbers.

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