Solve this problem ---**Header:** NJATC Aptitude Test **Question Stem:** Calculate the Missing Number **Diagram Description:** A grey octagon divided into 8 equal segments by lines radiating from the center. Each segment contains a number, except one which contains a red question mark. The numbers in the segments, starting from the top-right and going clockwise, are: 85, 16, 36, 25, ?, 51, 93, 72. **Options:** A) 9 B) 39 C) 49 D) 98

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Welcome to this NJATC aptitude test problem. We have an octagon divided into 8 equal segments. Each segment contains a number: 85, 16, 36, 25, then a missing number marked with a question mark, followed by 51, 93, and 72. Our task is to find the pattern and determine what number should replace the question mark. Now let's look more carefully at these numbers. I notice that some of them are perfect squares. 16 equals 4 squared, 36 equals 6 squared, and 25 equals 5 squared. These three perfect squares appear consecutively in positions 2, 3, and 4 of our octagon. This suggests there might be a pattern involving perfect squares. Let's examine the multiple choice options. Option A is 9, which equals 3 squared. Option B is 39, which is not a perfect square. Option C is 49, which equals 7 squared. Option D is 98, which is also not a perfect square. If we assume the missing number is 49, then we would have the sequence of perfect squares with bases 4, 6, 5, and 7. Let's see if there's a pattern in these base numbers. Welcome to this NJATC aptitude test problem! We have an octagon divided into 8 equal segments, each containing a number except one which has a question mark. The numbers are 85, 16, 36, 25, question mark, 51, 93, and 72. Our goal is to find the pattern and determine what number should replace the question mark. Let's start by checking if these numbers are perfect squares. Looking at our numbers: 85 is not a perfect square, but 16 equals 4 squared, 36 equals 6 squared, and 25 equals 5 squared. However, 51, 93, and 72 are not perfect squares. So we have a mix - some are perfect squares and some aren't. This suggests we need to look more carefully at the arrangement to find the true pattern. Now let's try a different approach and look at the positions more systematically. If I arrange the numbers by their positions in the octagon, I can check for patterns in alternating positions. Looking at positions 2, 4, 6, and 8, I see 16, 25, 51, and 72. Some are perfect squares but not all. Let me try looking at this differently - maybe the pattern isn't just about alternating positions. Here's the key insight! If I focus on the positions that contain perfect squares, I find them at positions 2, 3, and 4, which are 16 equals 4 squared, 36 equals 6 squared, and 25 equals 5 squared. The missing number is at position 5. So I have the base numbers 4, 6, 5, and an unknown base. Let me analyze the pattern in these base numbers to find what the missing base should be. Now I can see the pattern clearly! Looking at the base numbers 4, 6, 5, and the unknown, the differences are: 4 plus 2 equals 6, then 6 minus 1 equals 5, and following the pattern, 5 plus 2 equals 7. So the pattern is plus 2, minus 1, plus 2. This means the missing base is 7, and therefore the missing number is 7 squared, which equals 49. The answer is C, 49. Now I can see the pattern clearly! Looking at the base numbers 4, 6, 5, and the unknown, the differences are: 4 plus 2 equals 6, then 6 minus 1 equals 5, and following the pattern, 5 plus 2 equals 7. So the pattern is plus 2, minus 1, plus 2. This means the missing base is 7, and therefore the missing number is 7 squared, which equals 49. The answer is C, 49.