There are ten numbers adding up to 200. If you could take out five numbers that add up to 100, then automatically, the other five must also add up to 100. Ten numbers add up to 200, means that the average value is 20. Five numbers add up to 100, also anaverage of 40. Once you pick one number to remove (n), then the other four must have an average of (100 - n)/4 It would be nice if you could pick out a number divisible by 4... and there is not one. --- Another way is to look at "modulo 4" arithmetic. A "modulo" is the remainder, when you divide a number by the base. Here, we make the base 4. Every number in the list is congruentto 2 modulo 4. Meaning that when you divide each number by 4, you get a remainder of 2. Modulo arithmetic has a few simple rules, one of them being that if you add two numbers, the modulo of the sum will be equal to the sum of the individual modulo. For example 14 18 = 32 14 is congruent to 2 (mod 4) 18 is congruent to 2 (mod 4) therefore their sum must be congruent to 4 (mod 4) which is thesame as 0 (mod 4) because 4 would leave a remainder of 0, when divided by 4 32 divided by 4 = 8, with a remainder of 0. If you add 2 to itself an even number of times (like 10 times), the final number will be divisible by 4. 200 is congruent to 0 (mod 4): it is divisible by 4. If you take an ODD number of numbers, all congruent to 2, their sum CANNOT be divisible by 4. For example, take the last three numbers: 30 34 38 = 102 102 is congruent to 2 (mod 4), as promised by the sum of three numbers congruent to 2 (mod 4). Yet, 100 is divisible by 4. Therefore, it is impossible to make these numbers add up to 100, by taking five of them (or three or seven or any odd combination). The ONLY way to get 100 is to take the sum of an EVEN number of numbers.