Question

The numbers below are arranged in a triangular pattern that follows a hidden rule. Find the missing numbers marked with (?).

2

2 2

2   4   2

2   8   ?   2

2   16   64   16   2

2   32   1024   ?   ?   2

• A. 16, 32, 64
• B. 8, 1024, 32
• C. 24, 1024, 64
• D. 16, 32, 128

Hint:

Each inner number equals the product of the two numbers diagonally above it.

Answer 1

The Correct Option is B

Step 1: Look for a pattern using the edges of the triangle.
In every row, the first and last numbers are always 2. This tells us the border of the triangle is fixed, and the pattern of interest lies in how each inner number is generated from the row above it.
Step 2: Compare each inner number with the two numbers diagonally above it.
Row 3 is 2, 4, 2. Notice that 4 = 2 x 2, the product of the two numbers diagonally above it in Row 2 (2, 2). Row 4 is 2, 8, ?, 2. The second number, 8, equals 2 x 4, the product of the two numbers diagonally above it in Row 3 (2 and 4). Row 5 is 2, 16, 64, 16, 2. The second number, 16, equals 2 x 8 from Row 4's first two numbers.
Step 3: State the rule clearly.
Just like Pascal's triangle adds the two numbers above to get the entry below, this triangle multiplies the two numbers diagonally above to get the entry below. Every entry, except the border 2's, equals the product of the two numbers immediately above it, one to the upper-left and one to the upper-right.
Step 4: Find the missing number in Row 4.
Row 4's third entry sits below Row 3's second and third numbers, which are 4 and 2. So the missing value = 4 x 2 = 8.
Step 5: Verify using Row 5.
With Row 4 now complete as 2, 8, 8, 2, check Row 5: the middle number 64 should equal 8 x 8, Row 4's middle two numbers, = 64. This matches the given Row 5 exactly, confirming the rule and the value found for Row 4.
Step 6: Find the missing numbers in Row 6.
Row 6 is 2, 32, 1024, ?, ?, 2, sitting below Row 5's 2, 16, 64, 16, 2. First, check the given 1024: it should equal Row 5's second and third numbers multiplied, 16 x 64 = 1024. This matches, confirming the rule holds for Row 6 too. The first missing number, Row 6's fourth entry, sits below Row 5's third and fourth numbers: 64 x 16 = 1024. The second missing number, Row 6's fifth entry, sits below Row 5's fourth and fifth numbers: 16 x 2 = 32.
Step 7: Collect the missing values.
The missing values are: Row 4's third entry = 8, Row 6's fourth entry = 1024, Row 6's fifth entry = 32. Together, these are {8, 1024, 32}, which matches option (2) exactly.