On sports day, 12 children are placed in each column, forming 98 columns. Later they are asked to stand in concentric circles so that the innermost circle has 1 student, the next circle has 2 students, the third has 3 students, and so on. How many such concentric circles will be formed?

Introduction / Context:
This puzzle combines arithmetic with the concept of triangular numbers. We are first told how many children there are in total by describing them in columns. Then they are reorganized into concentric circles, with each successive circle having one more student than the previous one. The pattern 1, 2, 3, 4, ... is a sequence whose partial sums give triangular numbers. The question asks how many circles are formed, which is equivalent to finding which triangular number equals the total number of students.

Given Data / Assumptions:
- There are 12 children in each of 98 columns.
- Total number of students is therefore 12 × 98.
- In the circle arrangement, the first (innermost) circle has 1 student, the second has 2, the third has 3, and so on up to k circles.

Concept / Approach:
The total number of children in k such circles is 1 + 2 + 3 + ... + k, which is the k-th triangular number. This sum equals k(k + 1) / 2. We first compute the total number of students from the column information, then set this total equal to k(k + 1) / 2 and solve the resulting quadratic equation for k. The integer solution for k is the required number of circles.

Step-by-Step Solution:
Step 1: Compute total number of students in columns: total N = 12 × 98. Step 2: 12 × 98 = 12 × (100 - 2) = 1200 - 24 = 1176 students. Step 3: In the circle arrangement, the total number of students equals 1 + 2 + 3 + ... + k, which is k(k + 1) / 2. Step 4: Set k(k + 1) / 2 = 1176 and multiply both sides by 2: k(k + 1) = 2352. Step 5: We need integer k such that k^2 + k - 2352 = 0. Try k = 48: 48 × 49 = 2352, which satisfies the equation. Step 6: Therefore k = 48 circles are formed.

Verification / Alternative check:
Check by summation formula: for k = 48, the total number of students is 48 × 49 / 2 = 24 × 49 = 1176. This matches the total number of children previously computed from the column arrangement, confirming that all students are exactly used and no students are left out or missing. Thus the pattern of 1, 2, 3, ..., 48 circles correctly accounts for every child.

Why Other Options Are Wrong:
Options 44, 46, 49: For each of these values of k, k(k + 1) / 2 does not equal 1176. Either too few or too many students would be accounted for, so they cannot be correct.
Option None of these: Not correct because 48 matches the requirements exactly.

Common Pitfalls:
One common mistake is to forget or misapply the triangular number formula and attempt to add numbers manually, which is error-prone for large totals. Another error is to assume that 1176 is close to 50 × 50 / 2 = 1250 and then guess nearby values without checking the exact product. Using the precise equation k(k + 1) = 2352 and testing exact factor pairs avoids such guessing and ensures the correct answer.

Final Answer:
The children will form 48 concentric circles.