### The sum of the first n odd numbers is always a square

I was staring at the checked pattern on the back of an airline seat the other day when I suddenly saw that the sum of the first n odd numbers is always a square. For example,
11 + 3 = 41 + 3 + 5 = 91 + 3 + 5 + 7 = 16

And, of course, it occurred to me that it would be nice to be able to prove it. There are lots of ways to do that. Firstly, this is just the sum of an arithmetic progression starting at a = 1 with a difference of d = 2. So the standard formula gives us:
sum_odd(n) = n(2a + (n-1)d)/2           = n(2 + (n-1)2)/2           = n(1 + n - 1)           = n^2

So, the sum of the first n odd numbers is n^2.

But using standard formulae is annoying, so how about trying a little induction.
sum_odd(1) = 1sum_odd(n+1) = sum_odd(n) + (2n + 1)             = n^2 + 2n + 1             = (n+1)^2

But back to the airline seat. Here's what I saw (I added the numbering, Lufthansa isn't kind enough to do that for you :-):

You can view the square as the sum of two simpler progressions (the sum of the first n numbers and the sum of the first n-1 numbers):
1 + 3 + 5 + 7 =1 + 2 + 3 + 4 +    1 + 2 + 3

And given that we know from Gauss the sum of the first n numbers if n(n+1)/2 we can easily calculate:
sum_odd(n) = sum(n) + sum(n-1)           = n(n+1)/2 + (n-1)n/2           = (n^2 + n + n^2 - n)/2           = n^2

What do you do on long flights?

Unknown said…
Just noticing that they agree on the first three values

1 = 1
1 + 3 = 4
1 + 3 + 5 = 9

is actually a valid proof. $\sum_{i=1}^n p(i)$, where $p(i)$ is a polynomial of degree $d$, is a polynomial of degree at most $d + 1$ (you can prove this by induction). Now $n^2$ and $\sum_{i=1}^n (2n -1)$ are both polynomials of degree at most 2, and they agree on three values, hence must be identical.
James said…
What do you do on long flights?
Watch movies...
turd said…
The sum of the first n odd numbers is = to n^2. Im in 7th grade and I know that
turd said…
the sum of the first n odd numbers is n^2. U ok sir im in 7th grade and i know that
turd said…
The sum of the first n odd numbers is = to n^2. Im in 7th grade and I know that
Unknown said…
Find the sum of first n odd numbers............1,3,5,7,.........2n-1........will you please give me the answer????? I am 10 std student
Unknown said…
Write a C program to take the lower limit and upper limit of the range as inputs and calculate!

i. Square sum of those interval numbers
ii. Show the odd numbers only within the interval
iii. Determine the average of those numbers.
Unknown said…
This comment has been removed by the author.
Unknown said…
Just to make it more fun
1=1^3
3+5=2^3
7+9+11=3^3

Proof isn't quite so pretty, but you can notice their average is always n^2
Unknown said…
Please answer: the sum of odd numbers from 10 to 60 is??
Unknown said…
25/2*(11+59)
Since 25 odds from 10 to 60
=875

### Your last name contains invalid characters

My last name is "Graham-Cumming". But here's a typical form response when I enter it:

Does the web site have any idea how rude it is to claim that my last name contains invalid characters? Clearly not. What they actually meant is: our web site will not accept that hyphen in your last name. But do they say that? No, of course not. They decide to shove in my face the claim that there's something wrong with my name.

There's nothing wrong with my name, just as there's nothing wrong with someone whose first name is Jean-Marie, or someone whose last name is O'Reilly.

What is wrong is that way this is being handled. If the system can't cope with non-letters and spaces it needs to say that. How about the following error message:

Our system is unable to process last names that contain non-letters, please replace them with spaces.

Don't blame me for having a last name that your system doesn't like, whose fault is that? Saying "Your last name …

### All the symmetrical watch faces (and code to generate them)

If you ever look at pictures of clocks and watches in advertising they are set to roughly 10:10 which is meant to be the most attractive (smiling!) position for the hands. They are actually set to 10:09.14 if the hands are truly symmetrical. CC BY 2.0image by Shinji
I wanted to know what all the possible symmetrical watch faces are and so I wrote some code using Processing. Here's the output (there's one watch face missing, 00:00 or 12:00, because it's very boring):

The key to writing this is to figure out the relationship between the hour and minute hands when the watch face is symmetrical. In an hour the minute hand moves through 360° and the hour hand moves through 30° (12 hours are shown on the watch face and 360/12 = 30).
The core loop inside the program is this:   for (int h = 0; h <= 12; h++) {
float m = (360-30*float(h))*2/13;
int s = round(60*(m-floor(m)));
int col = h%6;
int row = floor(h/6);
draw_clock((r+f)*(2*col+1), (r+f)*(row*2+1), r, h, floor(m…

### The Elevator Button Problem

User interface design is hard. It's hard because people perceive apparently simple things very differently. For example, take a look at this interface to an elevator:

From flickr

Now imagine the following situation. You are on the third floor of this building and you wish to go to the tenth. The elevator is on the fifth floor and there's an indicator that tells you where it is. Which button do you press?

Most people probably say: "press up" since they want to go up. Not long ago I watched someone do the opposite and questioned them about their behavior. They said: "well the elevator is on the fifth floor and I am on the third, so I want it to come down to me".

Much can be learnt about the design of user interfaces by considering this, apparently, simple interface. If you think about the elevator button problem you'll find that something so simple has hidden depths. How do people learn about elevator calling? What's the right amount of informati…