Surprising Patterns in the Square Numbers (1, 4, 9, 16…)


for some thing before i haven’t seen this site, i am expecting [(n-1)2]+1 to be seen

anyway, i used that formula (that i invented or discovered by myself if someone had it) to get the difference before







Triangular numbers anyone?

It wasn’t 'till much later that I learned about triangular numbers and the ‘Handshake Problem’ but I most certainly remember experiencing my first triangular number. This gave me a great experience for the way math can teach us about growth and how deceptive our intuition can be without reasoned calculation.

One particular kenpo class our instructor had a great idea for sparring. Everyone lineup. The first student takes his turn:
-first student spars 2nd until 3 points are scored (everyone else do 10 pushups)
-regardless of winner 1st student spars 3rd student (drop & give me 10 everyone else)
-continue until the 1st student spars last student
Now the second student gets his turn.
Everyone gets a turn!

If memory serves there were 12 of us that day. Bonus question: How many sparring matches? How many push ups? At an average 3 minutes per sparring match (with 1 round of 10 pushups being completed sooner) how long were we there?


Thank you ,you help me a lots.I have stear on the web for thirty hours and still cannot think it instuitional.Could you tell me why the length of the arc equal to the"x" of e^xi ?forgive my Chinglish


HI,the words:
’‘We apply i units of growth in infinitely small increments, each pushing us at a 90-degree angle. There is no “faster and faster” rotation - instead, we crawl along the perimeter a distance of |i| = 1 (magnitude of i).’’

I still cannot understand.If I think e^1 on the same way,then I get e^1=e(intuitively the length is e),but we get 1 radan(the length of arc is 1)from 'e^i '.
best regard


very good


Hi Dyron, great question (this is probably the wrong post though!).

When we have e^i, the base e is not the size or length: it’s the type of growth we have (growing continuously).

The exponent, i, has a size of 1, and modifies our growth to be a rotation. e^i is one radian around the circle, e^(2i) is 2 radians, and so on.


I can waiting for to the school


Did anybody else notice that the sum of squares is always 0,1,4,7,9?


yes COLIN…i have noticed that…

moreover the sum digits in the difference between any two successive cube will
EITHER ‘1’ OR ‘7’…


You might add something on percents, fractions and ratios. My daughter always had trouble. What was obvious to me was not to her. She needed a more visual approach. Such things as (but not limited to):
percents to/from fractions and decimal numbers
30% discount followed by a 20% discount on $100 is $56 not $50.
Doing the above problem as $100 * .7 * .8 = $56
A 50% decrease followed by a 50% increase yields 75% of the original value.
Adding and multiplying fractions


What are the connections between
1, 1
2, 2
3, 4
4, 8
5, 16
6, 32
7, 64
8, 128
9, 256
10, 512


if x,y,z are three successive numbers then,the cube of the middle number (y) is related to its neighbours as,


                              [THIS HOLDS GOOD FOR ANY NUMBER]


6(x-1)+12 I figured this out on my free time without seeing this page. Simple.


Delighted to have come across this article and interesting comments.
My 3rd grader just recently figured out the difference to the next square, which he thought of as n + (n +1), and I worked it out algebraically with him to show that it works for all cases. He will love the related observations above (though the calculus section will wait for another day :slight_smile: ).


@seanoduill: Awesome, glad to hear you and your child enjoyed it!




Dear Kalid,
Great job Sir!
Your posts have had a great impact on the way I looked at Mathematics.
I thought math was just about remembering theorems and applying them in Problems and frankly speaking, the education system in India forced me to believe the same.
A genuine thanks!