How can you determine whether the sum of several numbers, such as 13 + 45 + 24 + 17, is even or odd without actually calculating the sum?

If we add all the evens, their sum must even, so ignore those.
Pair off as many odds as can be paired, the sum so far is still even.
If left with one more odd the sum becomes odd; otherwise it remains even

Example: 13 + 14 + 20 + 31 + 22 + 45 + 12 + 24 + 17 + 11
Paired odds are (13 + 31) + (45 + 17)
We were left with one more odd, namely 11 so total was odd.

wanszeto expressed this first

The method is simply to count the number of odd numbers.
If the number of odd numbers is odd, the sum is odd.
If the number of odd numbers is even, the sum is even.

For example: 13 + 45 + 24 + 17
There are 3 odd numbers (13, 45 and 17), and thus the sum is odd.

Just count the number, N, say, of odd numbers in your complete number list that are odd. If N is odd, the sum is odd. If N is even, The sum is even.;