### How to divide by 9 really, really fast (and why it works)

An interesting post on Hacker News shows tricks for dividing by 9. The fastest trick goes like this. Suppose you want to do 53876 / 9 in your head. You do the following:

1. The first digit is going to be the same as in the left hand side, so it's 5.

2. The second digit is the first digit plus the second digit: 5 + 3 = 8

3. The third digit is the previous answer plus the third digit: 8 + 8 = 16. Because that's bigger than 10 carry it to the digit above (which is now 9) and just take the rest. So it's 6.

4. The fourth digit is previous answer plus the fourth digit: 16 + 7 = 23. Again carry the tens to the preceding digit (which now becomes 8) and you're left with 3.

5. The last digit is the sum of all the digits: 23 + 6 = 29. For this final digit it's necessary to work out how many times 29 can be divided by 9 (it's 3). That gets added to the previous digit which becomes 6. The remainder (29 - 9 x 3 = 2) is the remainder of the whole calculation.

So, reading off the digits 53876 / 9 = 5986 remainder 2.

Why does this work? To show why I'll do a smaller example of a three digit number (with digits a, b, c) being divided by 9. The calculation abc/9 can be written as:
```(a x 100 + b x 10 + c) / 9

=  a x 100 + b x 10 + c
-------   ------   -
9         9     9

=  a x (11 + 1/9) + b x (1 + 1/9) + c/9

=  a x 11 + b x 1 + (a + b + c)
-----------
9

=  a x 10 + a x 1 + b x 1 + (a + b + c)
-----------
9

=  a x 10 + (a + b) + (a + b + c)
-----------
9
```
So, the first digit depends on a and any carry from (a + b), the second digit depends on (a + b) and any carry from the final term, and the only division left is the division of (a + b + c) by 9 and so that's where the remainder comes from.

Exactly the same pattern appears for longer numbers. Just be sure to carry; for very long numbers there could be a lot of carrying. Unknown said…
So applying this "trick" to divide 900 by 9 what happens ... the first digit is "9"?! @Unknown.

900/9

1. First digit is calculated as 9
2. Second digit is 9 + 0 = 9
3. Last digit is 9 + 0 = 9. Divide 9 into that gets 1 and add that to the second digit 9 + 1 = 10. Must carry the 10 so second digit is 0, first digit is 9 + 1 = 10. No remainder, so answer is 100. Unknown said…
so to apply this trick what would 28.008 divided by 9 be then plz explain quickly thanks for ur help bye Unknown said…
its very simple

consider 28008/9

1st digit is 2
2nd is 8+2=10 carry 1 so now 1st digit equals 2+1=3 and 0 remains in 2nd digit

3rd digit is 10+0 =10 so now 2nd digit becomes 1 and 3rd digit is zero

4th digit is 10+0 .so 3rd digit becomes 1 and sum of all numbers=18,18/9=2

3112

3112/1000=3.112 Unknown said…
What for a number like: 999586?
Can someone provide an explanation?

### 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…