Understanding the Number Pattern

The question asks us to identify the number that replaces the question mark (?) in the given sequence:

\(16 \quad 27 \quad 72 \quad 58 \quad 71 \quad 21 \quad ? \quad 12\)

Let's examine the sequence to find the underlying pattern.

Analyzing the Pattern Elements

Upon careful observation of the numbers, we can identify a prominent pattern involving digit reversal for certain terms in the sequence.

• Consider the numbers at position 2 and position 3: 27 and 72. The number 72 is the digit reversal of 27.
• Consider the numbers at position 6 and position 8: 21 and 12. The number 12 is the digit reversal of 21.

This suggests a pattern where certain terms are obtained by reversing the digits of a preceding term. Specifically, the term at position 3 is the reverse of the term at position 2, and the term at position 8 is the reverse of the term at position 6.

\(27 \xrightarrow{\text{Reverse digits}} 72 \quad (\text{Position } 2 \to \text{Position } 3)\)

\(21 \xrightarrow{\text{Reverse digits}} 12 \quad (\text{Position } 6 \to \text{Position } 8)\)

Identifying the Remaining Terms

The terms involved in the observed digit reversal pattern are at positions 2, 3, 6, and 8. The sequence has a total of 8 positions. The terms not involved in this specific reversal pattern are at positions 1, 4, 5, and 7.

These remaining terms are: 16, 58, 71, ?. Let's analyze these terms separately to find a pattern among them.

\(16 \quad 58 \quad 71 \quad ?\)

Analyzing the Remaining Sequence

Let's look at the digits of the known numbers in this remaining sequence:

• 16 has digits 1 and 6.
• 58 has digits 5 and 8.
• 71 has digits 7 and 1.

Concatenating the digits of these numbers in order, we get the sequence of digits: \(1, 6, 5, 8, 7, 1\).

Now, let's consider the full original sequence with the missing number as ?: \(16 \quad 27 \quad 72 \quad 58 \quad 71 \quad 21 \quad ? \quad 12\).

The pattern identified for the reversal pairs covers positions 2, 3, 6, and 8. The remaining positions are 1, 4, 5, and 7. The numbers at these positions are 16, 58, 71, and the missing number (?).

The digits from the known terms in this group (16, 58, 71) form the sequence: \(1, 6, 5, 8, 7, 1\).

A closer look at this digit sequence reveals a potential pattern. If we consider this sequence of digits, the number of digits is 6. The next number in the original sequence is the 7th term (?).

Let's hypothesize that the pattern for the missing term is determined by this sequence of digits. The sequence of digits from 16, 58, and 71 ends with the digit 1.

\(1, 6, 5, 8, 7, 1\)

If we assume the pattern for the missing number is to simply use the last digit from this concatenated sequence of digits as the next term, then the missing number would be 1.

Let's test this hypothesis. If ? = 1, the sequence of remaining terms is 16, 58, 71, 1. The sequence of digits from these terms is \(1, 6, 5, 8, 7, 1, 1\). This digit sequence shows the last digit (1) being repeated.

Confirming the Pattern

The overall pattern seems to be a combination of two interleaved ideas:

1. Digit reversal between specific pairs of terms (Positions 2 & 3, Positions 6 & 8).
2. A pattern on the digits of the remaining terms (Positions 1, 4, 5, 7). The digits of the numbers at positions 1, 4, and 5 (16, 58, 71) are concatenated to form a digit sequence (1, 6, 5, 8, 7, 1). The missing number at position 7 is a single digit formed by repeating the last digit of this sequence (1).

Thus, the missing number is 1.

Conclusion

Following the pattern where Term 3 is the digit reversal of Term 2 (27 → 72), and Term 8 is the digit reversal of Term 6 (21 → 12), the remaining terms are 16, 58, 71, and ?. By concatenating the digits of 16, 58, and 71, we get the digit sequence 1, 6, 5, 8, 7, 1. The pattern for the missing term is that it is the single digit obtained by repeating the last digit of this sequence. The last digit is 1, so the missing number is 1.

Digits from remaining terms (1, 4, 5): 1, 6, 5, 8, 7, 1. Next digit (for position 7) is 1 (repeat last digit). The missing number is 1.

Revision Table: Key Pattern Elements

Additional Information: Logical Reasoning Patterns

Number pattern questions in logical reasoning often involve various types of patterns. Some common types include:

• Arithmetic Progression: Adding or subtracting a constant value.
• Geometric Progression: Multiplying or dividing by a constant value.
• Differences/Ratios of Differences: Looking for patterns in the differences or ratios between consecutive terms.
• Digit-based Patterns: Patterns involving the digits of the numbers, such as sum of digits, product of digits, digit reversal, or sequences formed by digits.
• Alternating Patterns: Two or more independent patterns interleaved within the same series.
• Positional Patterns: Patterns related to the position number of the terms.
• Combination Patterns: A combination of different types of patterns.

Solving number pattern questions requires careful observation, trial and error with different potential rules, and systematic analysis of the sequence and its properties.