In the number series 312, 231, 295, ?, 282, 257, one term is missing. Choose the number that should replace the question mark to maintain the pattern.

Introduction / Context:
The series 312, 231, 295, ?, 282, 257 involves both increases and decreases. This suggests that the differences between consecutive terms may themselves follow a pattern, possibly involving squares with alternating signs. We must identify this rule and then find the missing number.

Given Data / Assumptions:

• Series: 312, 231, 295, ?, 282, 257.
• Exactly one term is missing between 295 and 282.
• The pattern seems to involve alternating positive and negative movements of varying magnitudes.

Concept / Approach:
We look at the known jumps between terms and see if they relate to square numbers. A common examination pattern uses successive squares with alternating signs, for example minus 9^2, plus 8^2, minus 7^2, plus 6^2, minus 5^2 and so on. We test whether this style fits the known terms.

Step-by-Step Solution:
1. Compute known differences: 231 - 312 = -81. 295 - 231 = +64. 282 - ? = ?. 257 - 282 = -25. 2. Recognise that 81 = 9^2 and 64 = 8^2, and 25 = 5^2. This hints at a pattern using squares of 9, 8, 7, 6, 5. 3. Suppose the sequence of operations is: -9^2, +8^2, -7^2, +6^2, -5^2. 4. Apply this from the first term 312: Step 1: 312 - 9^2 = 312 - 81 = 231 (matches the second term). Step 2: 231 + 8^2 = 231 + 64 = 295 (matches the third term). 5. Next, apply -7^2 = -49 to get the missing term: Missing term = 295 - 49 = 246. 6. Then apply +6^2 = +36: 246 + 36 = 282 (matches the fifth term). 7. Finally apply -5^2 = -25: 282 - 25 = 257 (matches the last term). 8. So the pattern using successive squares is perfectly consistent when the missing term is 246.

Verification / Alternative check:
Write out all operations clearly: 312 - 81 = 231. 231 + 64 = 295. 295 - 49 = 246. 246 + 36 = 282. 282 - 25 = 257. The square sequence used is 9^2, 8^2, 7^2, 6^2, 5^2 with alternating minus and plus signs, which is a neat and typical exam pattern.

Why Other Options Are Wrong:

• 210, 25, 279: Substituting any of these for the missing term breaks the square based pattern and does not yield exactly -7^2 and +6^2 operations between the surrounding terms.

Common Pitfalls:
Students may focus on approximate differences and miss the link to perfect squares. Another pitfall is to assume a constant difference or ratio. When differences are large and irregular, it is often beneficial to check for square or cube patterns, especially if values like 81, 64 and 25 appear.

Final Answer:
The number that should replace the question mark is 246.