Deane Does Some Math

By Deane Barker on June 27, 2005

In an attempt to get back into shape, I’ve started running. While I run, my mind has a tendency to wander, and when the going gets tough, I start doing math in my head to take my mind off the raging inferno in my lungs.

The other day, I came up with a formula. I was squaring integers in sequence, and I figured out that the distance between squares of consecutive integers is derivable. Put another way, if you know the square of one integer, there’s a pattern — a simple trick — to finding the square of the next integer.

If you know this:

14 x 14 = 196

Simply double 14, add 1, and add to the previous result to get the square of the next integer — 15.

(14 x 14) + (14 x 2) + 1 = 225 = 15 x 15

This can be represented by:

x2 = (x-1)2 + 2(x - 1) + 1


x2 + 2x + 1 = (x + 1)2

It works on small numbers:

(1 x 1) + (1 x 2) + 1 = 4 = 2 x 2

And bigger numbers:

(165 x 165) + (165 x 2) + 1 = 27,556 = 166 x 166

Now, I’m not naive enough to think that I’ve stumbled on Fermat’s Theorem or anything. I’m fully aware this is simple math that has no-doubt been documented thousands of times.

But does it have a name? Is there some kind of principle here that I inadvertantly stumbled on? I’m just curious. If anyone has a mathmetician friend, send him this link.



  1. Draw at a square of 10 x 10 dots, then add dots to get 11 x 11 — then you will see the equation (two extra lines with 10 dots plus one dot in the corner)!

    See also (not the same — but same graphical approach was used by Gauss):

    “At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately. His teacher, Buttner, and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 101.”

  2. Actually, the general formula is (a+b)^2 = a^2 + 2ab + b^2 There’s no principle here other than simple calculous: (a+b)^2 = (a+b)(a+b) = a(a+b) + b(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2

  3. Indeed, the equation “(a + b)^2 = a^2 + 2ab + b^2” is something we all probably learned in algebra (not quite calculus). It’s how you, for instance, complete the square to solve for x. Given that it’s just an upshot of the way parenthetical expressions are multiplied, I don’t think there’s a special name for it.

    It never occurred to me to consider it in the manner that Deane did (i.e., sequential squares of integers), though. Great insight. I like EU’s graphical representation, too.

  4. Dude…seriously, get yourself to the bike path for your runs…there is plenty of “scenery” there to take your mind off math. Einstein would still be working on the atom if would have run down there. I’m worried that you are ascending to a new level of geekdom.

  5. In math terminology, this is a general solution to the quadratic: y^2=(x+1)^2 Expanding: y^2=x^2+2x+1 Solving for y: y=(x^2+2x+1)^1/2 What you have observed is that to obtain the square of x+1, find x^2 and add 2x+1. In a similar way, it can be said to obtain the square of x+2, find x^2 and add 4x+4. Please don’t add me to the “list” – at least I answered your original question.

  6. THE SECRET LIVES OF NUMBERS “Humanity‚Äôs fascination with numbers is ancient and complex. Our present relationship with numbers reveals both a highly developed tool and a highly developed user, working together to measure, create, and predict both ourselves and the world around us. But like every symbiotic couple, the tool we would like to believe is separate from us (and thus objective) is actually an intricate reflection of our thoughts, interests, and capabilities. One intriguing result of this symbiosis is that the numeric system we use to describe patterns, is actually used in a patterned fashion to describe. ”

  7. You are not alone, my friend. I independendantly thought along the same lines many years ago, but I may have onle been walking on a city street. Never the less, my observation was that 14^2 +14 + 15 = 15^2

    166^2 + 166 + 167 = 167^2. The ‘square of dots’ analogy is the best explanation I have seen. Question: Will I be getting hit, or doing the hitting?

    I am really afraid to move up to running, I don’t wish to start on cubing of numbers!

  8. If you really want to have something to ponder while running, try and prove the following:

    i ^ i = e ^ (-pi/2)

    Yes, that’s imaginary(1) squared equals constant(e) raised to the negative pi/2 power.

    Very proveable.

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