Understanding Number Patterns in Logical Reasoning

This question asks us to identify the hidden pattern connecting the first number to the second number in each given pair and then apply that pattern to find the missing number.

Let's carefully examine each pair to uncover the logical rule governing the transformation:

Analyzing the Number Pairs and Discovering the Pattern

Pair 1: 42 → 26

• The first number is 42. Its digits are 4 and 2. The digits are different.
• The second number is 26.
• Let's consider the digits of 42 and their sum. The units digit is 2, the tens digit is 4, and the sum of the digits is \(4 + 2 = 6\).
• Notice that the result 26 is formed by the units digit (2) followed by the sum of the digits (6).

Pair 2: 71 → 78

• The first number is 71. Its digits are 7 and 1. The digits are different.
• The second number is 78.
• The units digit is 1, the tens digit is 7, and the sum of the digits is \(7 + 1 = 8\).
• Notice that the result 78 is formed by the tens digit (7) followed by the sum of the digits (8).

We have two examples with different digits, but they seem to follow slightly different rules (using the units digit vs. the tens digit). Let's look at the magnitude of the first numbers. 42 is less than 50, and 71 is greater than 50. This suggests the rule might depend on whether the number is below or above a certain threshold, like 50, when the digits are different.

Pair 3: 33 → 16

• The first number is 33. Its digits are 3 and 3. The digits are the same.
• The second number is 16.
• The sum of the digits is \(3 + 3 = 6\).
• The result 16 does not seem to directly combine the digits or their sum in the same way as the previous pairs. However, notice that the result (16) is 10 more than the sum of the digits (6). \(16 = 6 + 10\).

This suggests a separate rule applies when the digits of the first number are the same.

Formulating the Number Pattern Rules

1. Case 1: Digits are the same. The result is the sum of the digits plus 10.
2. Case 2: Digits are different AND the number is less than 50. The result is formed by taking the units digit and appending the sum of the digits.
3. Case 3: Digits are different AND the number is 50 or greater. The result is formed by taking the tens digit and appending the sum of the digits.

• 42 → 26: Digits are 4 and 2 (different). 42 is less than 50. Case 2 applies. Units digit is 2. Sum of digits is \(4+2=6\). Result: 2 followed by 6 is 26. Correct.
• 71 → 78: Digits are 7 and 1 (different). 71 is 50 or greater. Case 3 applies. Tens digit is 7. Sum of digits is \(7+1=8\). Result: 7 followed by 8 is 78. Correct.
• 33 → 16: Digits are 3 and 3 (same). Case 1 applies. Sum of digits is \(3+3=6\). Result: \(6 + 10 = 16\). Correct.

The pattern holds for all the given examples.

Applying the Pattern to Find the Missing Number

• The first number is 62. Its digits are 6 and 2. The digits are different.
• The number 62 is 50 or greater.
• Therefore, Case 3 applies.
• According to Case 3, the result is formed by taking the tens digit and appending the sum of the digits.
• The tens digit of 62 is 6.
• The sum of the digits of 62 is \(6 + 2 = 8\).
• The result is 6 followed by 8, which is 68.

The missing number is 68.

Revision Table: Summarizing the Number Pattern Logic

Additional Information on Number Series and Pattern Recognition

Number pattern questions are common in logical reasoning and quantitative aptitude tests. They require you to identify the underlying rule or sequence connecting the numbers. Patterns can involve basic arithmetic operations (addition, subtraction, multiplication, division), squares, cubes, prime numbers, or more complex relationships involving the digits of the numbers themselves, as seen in this example.

Strategies for solving number pattern problems often include:

• Looking for differences between consecutive terms.
• Looking for ratios between consecutive terms.
• Checking for squares, cubes, or triangular numbers.
• Analyzing the digits of the numbers (sum, product, difference, position).
• Considering alternating patterns or multiple simultaneous series.
• Formulating hypotheses and testing them against all the given examples.

Practice with various types of number patterns helps improve your ability to quickly spot the logical rule.