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

Instead of hunting for a multiplication rule right away, we can first use the built-in symmetry of each row, and then confirm it with a growth pattern along the second position of each row.

1. Look at every completed row: Row 3 (2, 4, 2) reads the same forwards and backwards. Row 5 (2, 16, 64, 16, 2) also reads the same forwards and backwards. This tells us every row in this triangle is a palindrome, symmetric about its center.
2. Row 4 is 2, 8, ?, 2, a 4-number row. Symmetry means the 1st number matches the 4th (2 = 2, already true), and the 2nd number must match the 3rd. Since the 2nd number is 8, the missing 3rd number must also be 8.
3. Row 6 is 2, 32, 1024, ?, ?, 2, a 6-number row. Symmetry means position 1 matches position 6 (2 = 2), position 2 matches position 5, and position 3 matches position 4. Since position 2 is 32, the missing position 5 must also be 32. Since position 3 is 1024, the missing position 4 must also be 1024.
4. To double check this symmetry assumption is the right rule, look at the second entry of each row: Row 2 = 2, Row 3 = 4, Row 4 = 8, Row 5 = 16, Row 6 = 32. Each is exactly double the previous row's second entry, a clean doubling pattern that supports the internal consistency of the triangle.
5. Also verify 1024 independently: it sits below Row 5's 16 and 64. Since 16 x 64 = 1024, the multiplicative rule confirms 1024 is correctly placed, which in turn validates using symmetry to find its mirror partner.

So the missing numbers are 8 (Row 4), 1024 and 32 (Row 6), giving the set {8, 1024, 32}, which is option (2). \[\boxed{\{8,\ 1024,\ 32\}}\]