Make A Hundred!

First note that all the numbers summed together equal 156, so the two numbers not used in the sum to 100 add to 56.

These numbers must have an average of $56 \div 2 = 28 .$ This means one of the numbers must be larger than 28, that is, either 29 or 31.

Since 29 would require 27 to make a sum of 56 and that's not on the list, 31 must be one of the two. $56 - 31 = 25$ is the other one.

The difference between 31 and 25 is 6.

8 numbers overall, 4 or them are odd. That means that we will use either 2 or 4 odd numbers and likewise 4 or 2 even numbers. If we use all 4 even numbers, their sum is 50. Immediately we find 21 and 29 to also add up to 50. As such, 31-25=6.

$\text{Sum of all numbers = 100} \\ \text{Sum of given numbers = 6 + 8 + 14 + 21 + 22 + 25 + 29 + 31 = 156} \\ \text{Sum of numbers that don't sum 100 = } 156 - 100 = 56 \\ \text{Looking at unit digit of given numbers, only 1 + 5 = 6, so two of possible numbers are (25, 21) , (25, 31)} \\ 25 + 21 = 46 \\ \boxed{35 + 21 = 56} \\ \therefore 35 - 21 = 6$

The sum of the numbers is 156, the only two numbers that add up to 56 is 31 and 25, and the difference between those two numbers is 6

The sum of all numbers is 156 .. then 156 -100 = 56 ..... then find the two numbers with sum = 56 ....

The sun of 6, 14, 29, 21, 8, 22 meets our conditoon of adding to 100, and so 31-25 = 6