Three of the following four are alike in a certain way and thus form a group. Which is the one that does NOT belong to that group?
NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 - Operations on 13 such as adding/deleting /multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed)

Question

Three of the following four are alike in a certain way and thus form a group. Which is the one that does NOT belong to that group?

NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 - Operations on 13 such as adding/deleting /multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed)

Prepp Team · Accepted Answer

Analyzing Number Patterns to Find the Odd One Out

This question asks us to identify which set of three numbers follows a different pattern compared to the others. We are given four options, each containing three numbers linked by hyphens. The rule is to perform operations on the whole numbers themselves, not on their individual digits.

Let's examine the relationship between the numbers in each option. We'll look for a consistent rule that connects the first number to the second, and the second number to the third.

Step-by-Step Analysis of Each Option

We will analyze each option as a set of three numbers, let's call them A, B, and C.

Option 1: 9 - 44 - 218
Option 2: 6 - 31 - 156
Option 3: 13 - 66 - 331
Option 4: 16 - 81 - 406

Let's try to find a pattern between the numbers in each set. A common approach in such problems is to look for multiplication, addition, or subtraction relationships.

Consider the relationship between the first number (A) and the second number (B):

Option 1: From 9 to 44. $9 \times 5 = 45$. $45 - 1 = 44$. So, B = $A \times 5 - 1$.
Option 2: From 6 to 31. $6 \times 5 = 30$. $30 + 1 = 31$. So, B = $A \times 5 + 1$.
Option 3: From 13 to 66. $13 \times 5 = 65$. $65 + 1 = 66$. So, B = $A \times 5 + 1$.
Option 4: From 16 to 81. $16 \times 5 = 80$. $80 + 1 = 81$. So, B = $A \times 5 + 1$.

From this, we see that options 2, 3, and 4 follow the pattern $B = A \times 5 + 1$, while option 1 follows $B = A \times 5 - 1$. This is a strong indication that option 1 might be the one that does not belong to the group.

Now, let's check the relationship between the second number (B) and the third number (C) for the options that followed the $A \times 5 + 1$ pattern (Options 2, 3, and 4):

Option 2: From 31 to 156. $31 \times 5 = 155$. $155 + 1 = 156$. So, C = $B \times 5 + 1$.
Option 3: From 66 to 331. $66 \times 5 = 330$. $330 + 1 = 331$. So, C = $B \times 5 + 1$.
Option 4: From 81 to 406. $81 \times 5 = 405$. $405 + 1 = 406$. So, C = $B \times 5 + 1$.

It is clear that options 2, 3, and 4 consistently follow the pattern where each subsequent number is obtained by multiplying the previous number by 5 and adding 1.

Let's verify if Option 1 follows this second part of the pattern (C = $B \times 5 + 1$) or a different one based on its first part (B = $A \times 5 - 1$).

Option 1: From 44 to 218.
- Using $B \times 5 + 1$: $44 \times 5 + 1 = 220 + 1 = 221$. This is not 218.
- Using $B \times 5 - 1$: $44 \times 5 - 1 = 220 - 1 = 219$. This is not 218.
- Let's try another operation based on the first part of the pattern ($A \times 5 - 1$). What if the second step is $B \times 5 - 2$? $44 \times 5 - 2 = 220 - 2 = 218$. This matches the third number in Option 1.

So, Option 1 follows the pattern: First number (A) $\xrightarrow{\times 5 - 1}$ Second number (B) $\xrightarrow{\times 5 - 2}$ Third number (C).

The other three options (2, 3, and 4) follow the pattern: First number (A) $\xrightarrow{\times 5 + 1}$ Second number (B) $\xrightarrow{\times 5 + 1}$ Third number (C).

Since Option 1 follows a different set of operations to relate its numbers compared to Options 2, 3, and 4, it is the one that does NOT belong to the group.

Summary of Number Patterns

Option	Numbers (A - B - C)	Pattern A to B	Pattern B to C
1	9 - 44 - 218	$9 \times 5 - 1 = 44$	$44 \times 5 - 2 = 218$
2	6 - 31 - 156	$6 \times 5 + 1 = 31$	$31 \times 5 + 1 = 156$
3	13 - 66 - 331	$13 \times 5 + 1 = 66$	$66 \times 5 + 1 = 331$
4	16 - 81 - 406	$16 \times 5 + 1 = 81$	$81 \times 5 + 1 = 406$

As the table shows, only Option 1 exhibits a different pattern for both steps compared to the consistent pattern observed in Options 2, 3, and 4.

Conclusion

The set of numbers 9 - 44 - 218 does not follow the same pattern as the other three sets. Therefore, it is the one that does NOT belong to that group.

Revision Table: Number Pattern Analysis

Concept	Description	Application in this Problem
Number Pattern	A sequence of numbers that follow a specific rule or set of rules.	Identifying the mathematical rule connecting the three numbers in each option.
Odd One Out	Identifying the element in a group that does not share the characteristic or pattern common to the others.	Finding the option where the number relationships differ from the rest.
Arithmetic Operations	Basic mathematical operations like addition, subtraction, multiplication, division.	Used to find the relationship between numbers (e.g., $A \times 5 + 1$).

Additional Information: Reasoning and Grouping

Questions like these are common in reasoning tests and aim to assess your ability to identify patterns and relationships between numbers or other elements. The key is to systematically test potential rules and check for consistency across the majority of the options provided. Once a pattern is found that applies to three out of four options, the option that does not fit that pattern is typically the 'odd one out'. In number series or grouping questions, common patterns involve arithmetic progressions, geometric progressions, differences, ratios, squares, cubes, prime numbers, or combinations of these operations. Sometimes, the pattern might involve the digits of the numbers, but the question explicitly disallowed that here.