Step 1: Understanding the Concept:
A "solid square" means the soldiers form a perfect square. We need to find the largest perfect square less than or equal to 36581. The remainder is the soldiers left over.
Step 2: Detailed Explanation:
We find the square root of 36581 using the long division method.
1. Grouping: 3, 65, 81.
2. $1^2 = 1$. Remainder 2. Bring down 65.
3. New divisor $2x$. $29 \times 9 = 261$. Remainder 4. Bring down 81.
4. New divisor $38x$. $381 \times 1 = 381$.
Step 3: Calculation:
$36581 - (191 \times 191)$
$191^2 = 36481$.
Left over = $36581 - 36481 = 100$.
Self-Correction: Let's re-calculate $191^2$. $191 \times 191 = 36481$.
$36581 - 36481 = 100$.
Wait, re-checking the long division. If the remainder is 100, then (c) is correct. Let's re-verify the square near 36581. $191^2 = 36481$. $192^2 = 36864$ (too big). So 100 are left.
Step 4: Final Answer:
There are 100 soldiers left over. Thus, the correct option is (c).