Understanding the Number Pair Pattern

The problem asks us to identify the number pair that does not follow the same mathematical operation rule as the others. We are given four pairs of numbers, and the second number in each pair is related to the first number by a specific operation. This operation is consistent across all pairs except one.

The given number pairs are:

• (125, 91)
• (64, 61)
• (27, 37)
• (8, 20)

Identifying the Base Number

Let's examine the first number in each pair. We can observe that these numbers are perfect cubes:

• \(125 = 5^3\). The base number is \(n = 5\).
• \(64 = 4^3\). The base number is \(n = 4\).
• \(27 = 3^3\). The base number is \(n = 3\).
• \(8 = 2^3\). The base number is \(n = 2\).

Let the first number in a pair be \(N_1\) and the second number be \(N_2\). We see that \(N_1 = n^3\), where \(n\) is the base.

Finding the Operation Rule

We need to determine the mathematical operation that relates the base number (\(n\)) to the second number (\(N_2\)) for the majority of the pairs. Let's list the pairs along with their base numbers and second numbers:

We need to find a relationship \(N_2 = f(n)\) that holds for three of these pairs. Let's consider the relationship for the first three pairs:

• When \(n=5\), \(N_2=91\).
• When \(n=4\), \(N_2=61\).
• When \(n=3\), \(N_2=37\).

Let's examine the difference between consecutive \(N_2\) values as \(n\) decreases:

• From \(n=5\) to \(n=4\): \(N_2\) changes from 91 to 61. Difference = \(91 - 61 = 30\).
• From \(n=4\) to \(n=3\): \(N_2\) changes from 61 to 37. Difference = \(61 - 37 = 24\).

The differences (30 and 24) are not constant. The difference between these differences is \(30 - 24 = 6\), which is constant. This pattern in differences suggests a quadratic relationship between \(N_2\) and \(n\), of the form \(N_2 = An^2 + Bn + C\).

We can use the points \((n, N_2)\) for \(n=3, 4, 5\) to find the values of A, B, and C:

1. For \(n=3\): \(A(3)^2 + B(3) + C = 37 \implies 9A + 3B + C = 37\) (Equation 1)
2. For \(n=4\): \(A(4)^2 + B(4) + C = 61 \implies 16A + 4B + C = 61\) (Equation 2)
3. For \(n=5\): \(A(5)^2 + B(5) + C = 91 \implies 25A + 5B + C = 91\) (Equation 3)

Subtracting Equation 1 from Equation 2:

\((16A + 4B + C) - (9A + 3B + C) = 61 - 37\)

\(7A + B = 24\) (Equation 4)

Subtracting Equation 2 from Equation 3:

\((25A + 5B + C) - (16A + 4B + C) = 91 - 61\)

\(9A + B = 30\) (Equation 5)

Now, subtract Equation 4 from Equation 5:

\((9A + B) - (7A + B) = 30 - 24\)

\(2A = 6 \implies A = 3\)

Substitute \(A=3\) into Equation 4:

\(7(3) + B = 24 \implies 21 + B = 24 \implies B = 3\)

Substitute \(A=3\) and \(B=3\) into Equation 1:

\(9(3) + 3(3) + C = 37 \implies 27 + 9 + C = 37 \implies 36 + C = 37 \implies C = 1\)

Thus, the mathematical operation relating the base \(n\) to the second number \(N_2\) is \(N_2 = 3n^2 + 3n + 1\), where \(N_1 = n^3\).

Checking the Operation with All Pairs

Let's test this rule for each of the given pairs:

• Pair (125, 91): \(N_1 = 125 = 5^3\), so \(n=5\). According to the rule, \(N_2 = 3(5^2) + 3(5) + 1 = 3(25) + 15 + 1 = 75 + 15 + 1 = 91\). This matches the given second number.
• Pair (64, 61): \(N_1 = 64 = 4^3\), so \(n=4\). According to the rule, \(N_2 = 3(4^2) + 3(4) + 1 = 3(16) + 12 + 1 = 48 + 12 + 1 = 61\). This matches the given second number.
• Pair (27, 37): \(N_1 = 27 = 3^3\), so \(n=3\). According to the rule, \(N_2 = 3(3^2) + 3(3) + 1 = 3(9) + 9 + 1 = 27 + 9 + 1 = 37\). This matches the given second number.
• Pair (8, 20): \(N_1 = 8 = 2^3\), so \(n=2\). According to the rule, \(N_2 = 3(2^2) + 3(2) + 1 = 3(4) + 6 + 1 = 12 + 6 + 1 = 19\). The given second number is 20.

The pair (8, 20) does not follow the determined rule \(N_2 = 3n^2 + 3n + 1\), as the rule predicts 19 while the given value is 20.

Identifying the Odd Pair

The mathematical operation \(N_2 = 3n^2 + 3n + 1\), where the first number \(N_1\) is \(n^3\), holds true for the number pairs (125, 91), (64, 61), and (27, 37). The number pair (8, 20) is the one that does not conform to this rule.

Therefore, the odd number pair is (8, 20).

Revision Table: Number Patterns and Relationships

Additional Information: Techniques for Finding Number Patterns

Finding patterns in numbers is a fundamental skill in many aptitude tests and mathematical puzzles. Several techniques can be employed:

• Difference Method: Calculate the difference between consecutive terms. If the first differences are constant, the pattern is linear (arithmetic progression). If the second differences are constant, the pattern is quadratic. Higher-order differences indicate polynomial relationships of higher degrees.
• Ratio Method: Calculate the ratio between consecutive terms. If the ratio is constant, the pattern is geometric.
• Relating to Position: Sometimes the pattern relates the number to its position in the sequence (1st, 2nd, 3rd, ...). For example, the \(k^{th}\) term might be \(k^2\) or \(2k+1\).
• Relating Numbers within Pairs: As seen in this problem, the relationship might exist between the numbers within a pair itself, often involving powers or multiple operations on one of the numbers or its components (like the base of a power).
• Prime Numbers, Squares, Cubes: Check if the numbers are related to prime numbers, perfect squares, or perfect cubes.
• Combinations: Many patterns involve a combination of operations, such as multiplying by a number and then adding another, or alternating operations.

When analyzing number pairs like this, it's crucial to first look for obvious characteristics of the numbers (like being squares, cubes, etc.) and then systematically test potential relationships between the numbers in the pair or derived values like the base of a power.