# Math… amazing

Every so often someone will forward me one of these “amazing!” math tricks, and I will of course feel compelled to explain just how outrageously simple the math in them actually is. The latest one going around is even simpler and more obvious than most, and yet people still seem impressed by it:

Take the last two digits of the year you were born, add your age this year, and it will add up to 111. Amazing!

I have to say, I’m kind of amazed that it’s not gobsmackingly obvious to absolutely everyone who can add and subtract two digits. But so many people will do anything to avoid arithmetic, so it seems to have that “magic wand” quality pretty readily.

So OK. Say someone were to send you an email that said “The year you were born plus your age this year equals 2011 – but only this year! Amazing, huh?” Wouldn’t you find that obvious? Now, 2000–1900=100, and you were born in the 1900s (we assume no one under 12 years old got the email), and it’s 2011 now…

Put it another way: if you subtract 1900 from everything, as though 1900 were the year 0, this year would be the year 111; and if you start with the last two digits of your birth year, you’re subtracting 1900, so…

There are some really cool number tricks out there. But you don’t too often see them being passed around in emails, because different people have different definitions of “cool”.

All of this is explainable with simple algebra on the basis that a two-digit number cen be represented as ten times a one-digit number plus another one-digit number, e.g., 49=(4×10)+9.

So for any number 10x+y (e.g., 40+9, where x=4 and y=9), the reverse will be 10y+x (e.g., 90+4), meaning if you add 10x+y (the original number) to it you get 11x+11y (e.g., 40+9+90+4=44+99), and if you subtract the reverse you get 9x–9y (e.g., (40+9)–(90+4)=40+9–90–4), and if you subtract the sum of the digits (x+y, e.g., 4+9) you get 9x (because 10x+y–(x+y)=9x, e.g., 40+9–(4+9)=36). And of course 10x+y+10y+x=11x+11y=(x+y)×11.

So assuming a person of a normal adult age, you can say

1. Take your age (e.g., 49).
2. Add the digits together (e.g., 4+9=13).
3. Subtract that from your age (e.g., 49–13=36).
4. Add the numbers of the resulting number together (e.g., 3+6).

Of course, you want to gussy this up with something fancy. Add in some other calculations to distract. Instead of step 5, maybe say

5. Multiply by the last two digits of the year.
6. The answer is 99. This always works!! But it will only work this year!!! And not again for a hundred years!!!! OMG it’s amazing tell all your friends!!!!11

or, if you think they can handle the math (!), say

6. Divide the result by the sum of the numbers in your age (the number in step 2).
7. Multiply this by the number from step 4.
8. The result is the answer to the question “Who’s the greatest hockey player of all time?”!!! OMG Gretzky rules!!! Number 99 forever!!!!

Even this is pretty straightforward for people who like to think about numbers. But there aren’t that many of us. Anyone who graduated from high school is officially able to figure this sort of thing out easily. But as long as people think math is hard and mystifying…

I suppose you could argue that the general “Numbers! Oh noooooes!” attitude people tend to have in our culture allows them actually to have fun with simple things like this, but it deprives them of the much greater fun they can have with more complex number problems, and it makes them easy marks for misleading advertising, misleading politicians, and so on. And generally vulnerable to making dumb mistakes. There’s a classic Dilbert cartoon (two of them, in fact) illustrating this – see http://search.dilbert.com/comic/40%25%20Sick.

### 3 responses to “Math… amazing”

1. Chris West

I enjoyed these little math tricks. I used to tell people of the cool things you can do with a little arithmetic and I used to try to explain that it all has to do with the fact that we count in base-10 but that is usually where everyone loses interest haha. On the other hand, I did recently write this little post to catch the general eye (especially since it talks about having the computer or your cell phone being able to read your mind haha): http://cwestblog.com/2015/10/26/symbolic-mind-meld/

2. Nikki

Can you please explain why if I add a birthday say for example 12/23/2003 in the form of 1+2+2+3+2+0+0+3=13 if i add the digits I’m left with from 13 1+3=4
However if I add the birthday 12+23+2003=2038 And then I reduce the sum 2038 in the same format 2+0+3+8=13 I also end with 13 which in the same way could be reduced 1+3=4.

My feeble mind is not able to sort out an explanation for this….maybe one too many joints of something.

Thanks.

• sesquiotic

If you add them as whole numbers rather than as individual digits, what you’re really doing is adding the digits in the 1’s column (2+3+3), the digits in the 10’s column (1+2+0), and then adding the sums of those plus the digit in the 100’s column (0) plus the digit in the 1000’s column (2). You’re adding all the individual digits, just in a specific order.

In the case where digits in one column add up to more than 10, here’s what happens:
Let’s say you’re adding 87+56. Adding individually would be 8+7+5+6=26; 2+6=8. If you first add them as 87+56=143, 1+4+3=8, what you’re doing is adding the 1’s (7+6=13) and then taking that 1 from 13 and adding it to the 10’s (8+5+1=14) and then adding that to the 3 from the 1’s (14+3=17=8). The effect, no matter what you do, is that when you add individual digits and they make more than 10, the amount in the 10’s column ends up getting added back to them. The order in which it gets added back makes no difference:
8+7=15, 5+6=11, 11+15=26 or (1+1=2) and (1+5=6), 2+6=8
8+6=14, 5+7=12, 12+14=26 or (1+1=2) and (2+4=6), 2+6=8

So the two principles at operation here are:
1) The order in which you add numbers makes no difference. This is a universal principle in arithmetic.
2) Any time the sum adds a 1 in the 10’s column, you just add that 1 back in the 1’s column. This is a rule specific to this kind of addition.

Since the order in which you add makes no difference, the same number of 1’s will be added in the 10’s column no matter how you do it. Think of numbers as poker chips that you’re putting in stacks of 10. (If you make a stack of 10, you take the remainder and start a new stack. Not sure if you play poker but this is standard chip handling. 🙂 ) You can try as many different ways as you want of piling them and it will still make a new stack of 10 exactly as many times. In this kind of addition, you get an extra chip for every stack of 10 you make. So you will always end up with the same number no matter the order you do it in.