## Number tiles

Hello,

I have this problem that's bugging me. It goes like this:

So you take the numbers 1,2,3 and 4. The other numbers from 5 to 20 are placed in a 4 x 4 square:

5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20

Apparently, all the numbers in this square can be attained using the numbers 1,2,3 and 4 once only through adding, subtracting, multiplying and dividing. For example, 9 = (4 x 2) + 1 and 20 = 4 ( 3 + 2 ).

Why is this? Why can every number in the square be obtained in this way?

Secondly, the square also works if you take another row out. For example, you take out 5,6,7 and 8 and the square is:

1 2 3 4
9 10 11 12
13 14 15 16
17 18 19 20

Every number in the square can be obtained through adding, subtracting, multiplying and dividing 5,6,7 and 8 where each numbers is used only once. For example, 4 = (8-5)+(6-7) and 20 = 5+7+8.

Why is this true? Why is it true if you take another row out?

I had a go at this myself and didn't get anywhere. I messed around with mod 4 and noticed that if you rewrite the question using mod 4 you get:

You take out the numbers 1,2,3 and 0 mod 4. The square is:

1 2 3 0
1 2 3 0
1 2 3 0
1 2 3 0

So it seems obvious that it would be true. You can get 0,1,2,3 mod 4 from 0,1,2,3 mod 4. However, the numbers are different in each row so it's not a full explanation.

I also noticed that you would need to multiply and add to get prime numbers. Unfortunately, there doesn't seem to be a way to figure out how many primes will be in a square. I found this a problem when it came to making a "general square".

I'd appreciate any ides anyone has.

Thanks.