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# Any help with this proof question? watch

1. Any help with this question?

"How can you prove this:
You have 4 numbers, out of the choice of 1,2,3. How can you prove that any combination will have a sum divisible by 4?

E.g. {1,2,2,2} 2+2 = 4 or {3,1,1,2} 3+1 = 4 "
2. To be honest, after ruling out the "if you have at least two 2s you're done" case, it's probably going to be quickest to enumerate the other options and give examples (as you have started doing).
3. (Original post by thomoski2)
Any help with this question?
I think DFranklin's is by far the simplest solution - there are only 7 (correction 9) remaining combinations to check.

For something more involved.
Add the first and second digits. Is the result Odd or Even?
Add tthe third and fourth. Again is it Odd or Even?

Now investigate the possible combinations:

<expletive deleted> editor just deleted all my nice spoilers. Not going to type it out again.
4. Okay, I'll give this a try, thanks
5. Going through this method, I think i'm going to lose marks if i use this. Are there any other methods to try?
6. (Original post by thomoski2)
Going through this method, I think i'm going to lose marks if i use this. Are there any other methods to try?
We've suggested two. Some background to where the question is from and what it's for might suggest something else.
7. The exhaustive method will most likely lose me marks, so I want to try and avoid that if possible. I missed the other method, sorry, but I've never heard of the method, and this is for coursework, so I can't really use it. Do you think there's anything else I could try, even a point in the right direction? (Sorry to be a pain)
8. Why do you think the exhaustive method will lose you marks?
9. (Original post by thomoski2)
The exhaustive method will most likely lose me marks, so I want to try and avoid that if possible. I missed the other method, sorry, but I've never heard of the method, and this is for coursework, so I can't really use it. Do you think there's anything else I could try, even a point in the right direction? (Sorry to be a pain)
It only said prove it. It never said prove it elegantly. In a British Maths Olympiad exam (which is pretty much entirely proving stuff), they don't care how ugly your proof is. If you just bash away at every possible case, you can get full marks.

That said, start by eliminating the possibility of two 2s. This means at least three of the numbers must be 1s or 3s. Now, you can't have both 1s and 3s together because they'd add to make a 4, so now you've basically reduced it to four cases (two involving a 2, two without a 2). That's hardly an ugly proof.
10. (Original post by TheMindGarage)
It only said prove it. It never said prove it elegantly. In a British Maths Olympiad exam (which is pretty much entirely proving stuff), they don't care how ugly your proof is. If you just bash away at every possible case, you can get full marks.

That said, start by eliminating the possibility of two 2s. This means at least three of the numbers must be 1s or 3s. Now, you can't have both 1s and 3s together because they'd add to make a 4, so now you've basically reduced it to four cases (two involving a 2, two without a 2). That's hardly an ugly proof.
Sorry, i should've elaborated further, its a small sub section of my coursework, so i need a fairly concise answer, that was the issue
11. (Original post by thomoski2)
Sorry, i should've elaborated further, its a small sub section of my coursework, so i need a fairly concise answer, that was the issue
Most concise solution will be what I suggested. (The *reason* I said "just list them" was the realisation that a list with items like {1,1,1,3}: 1+3=4 was so concise that any argument with more than a couple of sentences would work out longer).

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