Magic C

For today's question, I took another Yahoo Answers great question to give you the opportunity to solve a nice question. Here it comes:

The digits from 1 to 5 should be placed on a C shaped form like this:

x-x
x
x-x

(Each x represents a different digit)
The two horizontal rows (x-x) and the vertical row with three x's should have the same sum of digits.
In how many ways can this be done (reflections don't count as different), and why there are no other possibilities?

We can see at first that in order to fill in the digits, we need to find two pairs of digits that add up to the same number (the two x-x). There are three numbers that work like that: 5 (1+4, 2+3), 6 (2+4, 1+5), and 7 (3+4, 2+5).

However, 5 is impossible as a sum, since it leaves the digit 5 outside, which means it'll be in the middle x, creating a sum definitely greater than 5. We are left with only 6 and 7 as sums. After some playing with arrangements, we can see that the two solutions are:

For 6:
1-5
3
2-4

For 7:
2-5
1
4-3

There are no other possibilities because all other sums (except 5) have a maximum of one pair that works for them, and we already showed why 5 doesn't work.

Now it's your turn. Send in the riddles!

Have a great weekend,
Nadav

nadavs