The average of 25 results is 18. The average of first twelve of them is 14 and…

For any set of results, the sum of all values equals the count of values multiplied by their average. When a full set is split into non-overlapping subgroups, the sum of each subgroup equals its own count multiplied by its own average, and all the subgroup sums must add up to the overall sum. This lets you isolate one unknown value by subtracting the known subgroup sums from the overall total.

1. Total sum of all 25 results = 25 x 18 = 450.

2. Sum of the first twelve results = 12 x 14 = 168.

3. Sum of the last twelve results = 12 x 17 = 204.

4. The first twelve and the last twelve results together make up 24 of the 25 results, leaving only the thirteenth result uncovered.

5. Thirteenth result = Total sum - (sum of first twelve + sum of last twelve) = 450 - (168 + 204) = 450 - 372 = 78.

Cross-check: 168 + 204 + 78 = 450, which matches 25 x 18. A second check using deviations from the overall average: the first-twelve average (14) is 4 below the overall average (18), a shortfall of 12 x 4 = 48; the last-twelve average (17) is 1 below the overall average, a shortfall of 12 x 1 = 12. The combined shortfall of 60 from these 24 results must be exactly offset by the thirteenth result, so it equals 18 + 60 = 78, confirming the answer.