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


Hey , I think I found something strange , (88) - (77) = 15 which is not a prime number , obviously.


@ash: Thanks for dropping by :). Yeah, when math is more about exploration / puzzle solving it can be pretty fun.

@Seifeddine: Ah, actually the differences just need to be odd (2n + 1), not prime.


Shame on me :frowning: , I should have read your article more carefully .


Wonderful approaches to regular problems.


I have a neat problem:
Suppose that we extend definitions of evenness and oddness to the entire set of real numbers by the following:
The set of odd numbers is exactly the set of all differences (n+1)^a-n^a for positive rational a and integral n.
A number is even iff it is twice an odd number.
All other real numbers are neither even nor odd.

Is the set of even numbers still disjoint from the set of odd numbers? (i.e. is an odd number twice another odd number?)

Prove that the set of real numbers neither even nor odd has the same cardinality as the set of real numbers.


Amazing job of finding these patterns. I know finding patterns was an easy way for me to learn mathmatics when I was younger.


@Tracy: Thanks! Yes, I think the heart (and fun) of math is really about finding and describing patterns.


really awesome insights into difference of squares but the main place this seems to apply is summations of Squares, how would you connect 2n+1 to n/6(n+1)(2n+1) i can see the 2n+1 in the formula but the n/6(n+1) part still doesn’t make sense to me


I like the posts on this. Am wondering if anyone can help my daughter and I figure out how to validate the pattern that we found in our squares. We found that, as long as the number beig squared is divisible by 3, that the sum of the numbers in the answer of the square is equal to 9. For example, 3 sq is divisible by 3 and the total of the answer (9) = 9. If we move onto 6 sq (also divisible by 3), the answer (36) = 9. And so on and so on…can anyone help explain this pattern to us?


@Sean: Great question. It turns out that if the sum of the digits of a number add up to 9 (or a multiple of 9, like 18, etc.) then that number is divisible by nine.

if the number being squared is divisible by 3 (so it’s 3*n), then the square is (9 * n^2) which is divisible by 9, and therefore falls into the pattern :). I’d like to do a post on why the digits need to add up this way, but here’s one insight:

If we start with 9, clearly the digits (the only digit!) adds to 9. Whenever we add 9, we’re really doing (10 - 1) which means “increase the tens digit by 1, and decrease the ones digit by 1”. This keeps the sum of digits in balance, so we should expect that the sum of digits always equals 9 as before. (For example, 18 means we changed 09 to 18). Of course, once you get to 90 and add (10-1) you are really only able to “fill” the ones digit and get 99 (which then increases the total sum to 18 – but, you kept the sum of digits divisible by 9). Hope this helps!


ur website is sooo awesome i never get maths question but now i do thanks


@Kelly: Awesome, glad it helped!


My question is that i don’t enough knowledge aboout even and odd number. Tells me that some examples and also difinitions of both numbers.

Muhammad Faisal


I was really surprised by the calculus method. It is cool.


Wo! really fantastic


i didnt like it


really , good.
add some more good patterns.

go ahead


i am for square patterns like[25sq=625={2*3}hundreds+25

                                                                             S.NITHEESH KUMAR


very iteresting
lurved it!!!


can we get any multiplication square patterns???i have not found any one!!!1foolish thing