# Thread: This looks like a game.....

1. ....but it's not - it is deadly serious.

1. Ask a friend to choose number with more than 1 digit

3. Ask her to add again if the result is more that 1 digit

4. And so on until the result is a single digit

5. Ask her to subtract the single digit result from the original choice of number, and then repeatedly add the resulting digits until a single digit remains

6. The answer is always 9

Why? Customary spoiler follows

It's called modulo 9 arithmetic. You didn't peek did you? If you did, (naughty naughty) say why the restriction in 1. above is irrelevant

2.

3. 'Cuz you always wind up with a multiple of (Base - 1), where Base = the base of the number system being used, in this case 10. I could be wrong but I think this true for bases other than 10 as well.

4. o.O Confusion strikes the number theory layman

At first, I thought it might involve modulo 9 arithmetic, because it involves summing the digits of a number and the result being 9. But I'm not so sure now, since the process is different (involving subtracting a resulting value from the original), and that not all two digit numbers are multiples of 9.

This kind of math with summing the digits and finding patterns has always interested me. I've found many basic patterns, but as to this problem, I'm clueless :P Then again, I am not a number theorist, much less a mathematician.

Either I did it wrong, or I've found a counter example.

I chose the number 428, it has more than 1 digit (WOW! 3 digits to be exact!). I added its digits, 4+2+8 and I got 14. I have to add the digits again since it's not yet a single digit, 1+4 is 5. Now I subtract this 5 from my original number 428, until I get a single digit. This number, according to my calculations, is 8. Not 9. Did I do it wrong? (as a matter of fact, I think there are several more counter examples, but I'm probably doing it wrong)

5. Originally Posted by halorealm
o.O Confusion strikes the number theory layman

At first, I thought it might involve modulo 9 arithmetic, because it involves summing the digits of a number and the result being 9. But I'm not so sure now, since the process is different (involving subtracting a resulting value from the original), and that not all two digit numbers are multiples of 9.

This kind of math with summing the digits and finding patterns has always interested me. I've found many basic patterns, but as to this problem, I'm clueless :P Then again, I am not a number theorist, much less a mathematician.

Either I did it wrong, or I've found a counter example.

I chose the number 428, it has more than 1 digit (WOW! 3 digits to be exact!). I added its digits, 4+2+8 and I got 14. I have to add the digits again since it's not yet a single digit, 1+4 is 5. Now I subtract this 5 from my original number 428, until I get a single digit. This number, according to my calculations, is 8. Not 9. Did I do it wrong? (as a matter of fact, I think there are several more counter examples, but I'm probably doing it wrong)
You should have subtracted 5 from 428 to get 423, then add 4+2+3 which equals 9.

6. And it's related to casting out nines. If you look at it more closely, you're casting out nines until you get the remainder, then you're subtracting the remainder from the original number, and when you cast out nines again, you automatically end up with 9. Always.

7. I didn't understand the solution but the idea was fun.

8. It is called modulo 9 arithmetic. I apologize to anyone who find the following patronizing,likewise to those who find it hard going (just STARE at it a while!!) but it illustrates an important general point in mathematics.

If, for arbitrary integers and FIXED integer I say that I mean simply that and differ by the integer . That is , say. This is called a "congruence" and has the following properties.

(reflexivity)

(symmetry)

(transitivity)

These three properties define the modulo relation as an equivalence relation, and these crop up all over the place in mathematics (usually masquerading as isomorphisms or even equalities!).

So now let's set , and we easily see that any two "adjacent" members of the set can be written as . We will see the "adjacent" qualifier is redundant.

So. This set is called an "equivalence class" and it is customary to elect a class representative, and write although this choice is entirely arbitrary.

Likewise we will have that and so on up to the class . So that, say, for all and any

Right, so the number, say is really just shorthand for , and since we easily see then it is no great shock to learn that .

So that is in the same equivalence class as i.e.

Now by the definition of this particular sort of equivalence relation, by the addition rule I gave above, we must have that subtracting one member of an equivalence class from another member of the same class sends the result to the class i.e. of which is a member. So from the above we will have, no matter how many digits we start with, proceeding recursively we will always have the result to be 0 or 9.

Here's the shocker - this is no more (or less) than base 10 arithmetic!

9. Originally Posted by Chrisgorlitz
I didn't understand the solution but the idea was fun.
To make a long story short, in casting out nines, 10 is the same as 1 because 10–9=1. The same is true about 100: 100 is the same as 1 because 100–99=1. Also 1,000, and 10,000, etc. Thus, you can consider all the digits in a number as on par with each other, which is why you can add them together. So, whatever digits you have comprising a number, adding them together (reiteratively) is like casting out nines, and when you subtract that final single digit from the original number, and then perform the reinterative addition again, it will always result in an answer of 9.

It's like dividing the original number by 9, and then subtracting the remainder from the original number, and when you divide this new number by 9, you get no remainder (because you subtracted it out), but in casting out nines, "no remainder" results in a 9 for an answer. This is a quick explanation, but if you think about it, hopefully it will make sense.

10. Ah, so it is modulo 9 arithmetic! If I really understood the math, I could've gotten it. Oh well.

No intention of offense to above posters, but throwing out "mod" would seem other-worldly to someone who doesn't know what it is. I'll try making an explanation; any real experts are welcome to correct me if wrong.

So this is basically modular arithmetic, aka "clock" arithmetic (and you'll know why it's called that now). In normal counting, numbers can go up indefinitely (1, 2, 3 ... -> ∞), but what about counting time on a clock? You can count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ... but then what? It goes back to 1, instead of going to 13 (since there is no 13th hour on a clock obviously). Even though you've counted 13 spaces, you ended up at 1. Counting 14 spaces will get you to 2. Why? Because a clock's numbers don't just keep going like regular counting, they "wrap around" every 12 times.

In math, there's certain notation for this. So what would counting 13 spaces on a clock look like? . "mod 12" tells what number the counting wraps around. 13 mod 12 is the same thing as 1, because counting at 12 comes back and wraps around to 1 at 13. The weird is like the "equal to" symbol but with an extra line. We say this as "congruent to" or "equivalent to" when using clock arithmetic.

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