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


A quick puzzle for you — look at the first few square numbers:

1, 4, 9, 16, 25, 36, 49…

This is a companion discussion topic for the original entry at


cubes: 3x^2+3x+1


Jeff has posted the green part (distance to the next cube). Don’t forget the original x^3 (the blue part).

So (x+1)^3 = x^3 + 3x^2 + 3x + 1

You can use Pascal’s triangle or the binomial theorem to get the result for (x+1)^4, but at that point I usually stop using pebbles.


What software did you use to draw the pebbles?


going from 3 square to 6 square
x=3 dx=3
(2x + dx)x = (23 + 3) * 3 = 27
… whoa it works


Geometric explanation is intuitive. I also like the Calculus method. Thanks!




@Jeff, Dave: Right on. I like to visual 3 flat panels, 3 gutters, and 1 corner piece to fill in the gap. Pascal’s Triangle is another great tool, would be fun as another post!

@Duncan: I used PowerPoint 2007, my secret weapon :slight_smile:

@Akshay: Surprising, right?

@Harash, @hitoshi: Glad you liked it!


Nice article. That geometric approach though is taught to high-school students, I think.

Also,I’ve left you a message that kinda pertains to this topic, so you might wanna check it. It’s a pattern of squares. Original research, but it works and that’s all that matters. One spoonful of free pie too.

But no seriously, check it out.


Wow, another nice article. I love your thinking, I love the geometric consideration, as my Maths teachers says every time: “Don’t forget mathematics started with geometry”.
Keep posting!


@Sahil: Thanks, I got your message, really interesting patterns in there.


@nschoe: Thanks for the support :). Totally agree with your math teacher, we can often learn so much by drawing a simple diagram instead of doing everything in algebra.


The first thing I thought of was that squaring a number keeps, like you said, its “evenness” of the number. Also, subtracting an even and odd (or odd and even) produces an odd.

And since each number alternates between even and odd and your taking its square to keep its evenness, the difference stays odd.

This works without a power of 1 as well, but the difference between consecutive numbers will always be “1” (an odd).


@Joe: Ah, interesting insight on the subtraction!


A^2-B^2 = (A+B)*(A-B).
when A-B =1,A^2-B^2 = A+B = 2A-1 = 2B+1.


Nice and “better explained” than I’ve seen in other places.

A great follow up article would be to show how the sums of cubes is related to the volume of a pyramid. Perhaps even a separate article on how to derive the formula for the volume of a pyramid without using calculus.


@ne_akari: I like the manipulation – either twice the current plus 1, or twice the next minus 1.

@Sol: Thanks for dropping by! A cube article could be fun, I like these geometric manipulations (they feel more like play than work!).


Wow , thank you very Khalid for all your articles , I am learning a lot from you :slight_smile: .

Keep up the good work


@Seifeddine: Thanks for the encouragement! :slight_smile:


fun! i felt like a kid thinking about these things. :slight_smile: