Math Is Fun Forum

If you have a multiplication table, pick any 4 by 4 square and add the numbers on each diagonal. For e.g

24 32 40 48
27 36 45 54
30 40 50 60
33 44 55 66

The numbers in diagonal 1 would add up to (24 + 36 + 50 + 66) = 176
and in diagonal 2, (33 + 40 + 45 + 48) = 166

I've noticed that D_1 = D_2 + 10

1.How would I prove that this statement will always be true?

2.How can I predict what will happen if I pick a 5 by 5 square, a 6 by 6 square... an n by n square, and add all the numbers on the diagonal?

I think I've pretty much solved q.1, using algebra, although I'm not sure if it's correct (it should be uploaded as an image) but i'm having most trouble with question 2.

So far for it I've come up with:
1 by 1 square, 0 difference
2 by 2 square, 1 difference
3 by 3 square, 4 difference
4 by 4 square, 10 difference
5 by 5 square, 20 difference
6 by 6 square, 35 difference...
n by n square, ? difference?

We have a problem where we have to prove that

is true no matter what the value of x.

Actually, to be precise, we were given the following information:

Think of a number...Add one...Double it...Take away 3...Add the number you first thought of...Add 7... Divide by 3... Take away the number you first thought of... You should be left with 2. Explain why this works.

So I try to solve for x:

This proves that the value of x does not matter since it will be eliminated anyway.

Some of my friends, however, think that I must leave the right side equaled to 2 for it to be correct:

Does it really matter?

This is a little problem I thought up while I was bored, I can't seem to solve it, but you have a go: