The Predetermined Sum Puzzle

Date: 02/02/2001 at 01:43:54
From: Tanya
Subject: Can't really describe the equation...

Hi. A friend of mine gave me a math problem to solve and I solved it. 
My problem is that I have to explain the logic of it, and I can't 
figure it out. 

He asked me to pick any 5 digits from 0-9, and wrote them down in a 
line. He asked me to pick 5 more. He then picked 5 of his own, I 
picked 5 more, and he finished the problem with 5 more digits of his 
own. I'll put an example below.

When I had picked the 5 digits on the first line, but no other digits 
had been chosen, he wrote down what would be the sum of all 5 lines at 
the bottom. Here is the example: 

       2 3 4 5 6  <-- what I picked      
       3 5 2 6 7  <-- what I picked
       6 4 7 3 2  <-- what he picked
       1 2 3 4 5  <-- what I picked
       8 7 6 5 4  <-- what he picked
     -----------
     2 2 3 4 5 4  <-- the sum

The problem is; how did he know what the sum would be before the other 
four lines had been picked?

Now, I understand that he knew the answer from the first line by 
adding a 2 in front and subtracting 2 from the last digit. I also know 
that whatever 5 digits I chose next, he made them all equal to 9 when 
he chose his 5 digits for the third line. So the 2nd and 3rd lines 
equal 99999 and the 4th and 5th lines equal 99999 as well when you add 
them. 

I also understand that the answer won't be over 200,000 because you 
can only go up to 279,999, so you add the 2 in front for the answer 
and subtract 2 from the last digit because it is 18 plus the digit. I 
just don't understand why this works with 9 and not any other numbers.

I have to explain to him in detail how I figured out everything. If I 
get this right, I get a free car. My car broke and he was going to 
help me buy a used car, but he gave me this problem and said that if I 
solve it he will pay for it. He even told me I could ask teachers and 
friends and look on the Internet because he didn't think I would 
figure it out. I know it's not too hard, I just need to figure out the 
logic behind it. I would greatly appreciate your help.

Date: 02/02/2001 at 08:35:07
From: Doctor Rick
Subject: Re: Can't really describe the equation...

Hi, Tanya. You've presented the problem very clearly - good work; now 
maybe I can help you get that car.

Let's add the second and third lines in your example, and also the 
fourth and fifth lines. Now we have:

       2 3 4 5 6 <-- what I picked      
       9 9 9 9 9 <-- sum of my pick and his pick
       9 9 9 9 9 <-- sum of my pick and his pick
     -----------
     2 2 3 4 5 4 <-- the sum

Again, let's add the two middle lines: 99999 + 99999.

       2 3 4 5 6 <-- what I picked      
     1 9 9 9 9 8 <-- sum of four picks
     -----------
     2 2 3 4 5 4 <-- the sum

That number I just found, 199998, is 200000 - 2. Thus, starting from 
your initial pick, you can first add 200000 (which puts a 2 in the 
left column), then subtract 2 from the result.

What would happen if he made the digits in your pick and his pick add 
to 8? Then the sum of your pick and his pick would be 88888, and the 
sum of four picks would be twice this, or 177776. He could predict the 
sum ahead of time by adding 177776 to your number, just as he added 
199998 to your number in the real trick. The only difference is that 
it's much easier to add 199998 in your head than to add 177776. (There 
is another problem: if your digit is 9, he can't put down a number 
that will make it add to 8.)

What mathematical principle lies behind this trick? It's the 
associative property of addition. This principle says that, when you 
have a list of numbers to add like this, you can group them any way 
you want (adding first these two, then those two ...) and you'll 
always get the same result in the end. That's why I could choose to 
add the second and third numbers first, etc.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/