SSC CGL Tier 1 2017 General Intelligence & Reasoning Questions with Solutions

Explanation

The umpire is a human official appointed to oversee a match, make decisions on the field, and ensure that the rules of the game are strictly followed. This role is distinct because it refers to a person acting in a judicial capacity, whereas the remaining items represent the physical, inanimate sporting equipment used by the players to participate in the game.

Explanation

In the letter cluster GIK, the letters follow an increasing alphabetical sequence with a constant difference of +2 between their positional values. Moving from G to I involves skipping one letter (H), and moving from I to K involves skipping one letter (J), represented numerically as 7 → 9 → 11. This increasing pattern makes it the odd one out because the other alternatives follow a decreasing alphabetical sequence where two is subtracted from the positional value of each preceding letter.

Explanation

The number 133 is the odd one out because it is not divisible by 11. According to the divisibility rule for 11, the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be 0 or a multiple of 11. For the number 133, the digits at the first and third positions sum to 1 + 3 = 4, and the digit at the second position is 3. The calculation (1 + 3) - 3 = 1 results in a value that is not zero or a multiple of 11, which distinguishes 133 from the other numbers in the set that all share the common property of being multiples of 11.

Explanation

The words in the given series follow a pattern based on the shifting position of the letter r. In the first word, rupees, r occupies the first position. In the second word, arena, r moves to the second position. In the third word, pursue, r is the third letter. In the fourth word, spare, r is in the fourth position. In the fifth word, separate, r is in the fifth position. Following this logical sequence, the next word in the series must have the letter r in the sixth position. The word pusher satisfies this pattern as the letter r is the sixth character in the word.

Explanation

The series follows a pattern where the positions of the letters in the English alphabet decrease by an increasing sequence of integers. The numerical positions of the given letters are U (21), T (20), R (18), O (15), and K (11). The differences between consecutive terms are calculated as follows: 21 - 20 = 1, 20 - 18 = 2, 18 - 15 = 3, and 15 - 11 = 4. Following this pattern of subtracting consecutive natural numbers, the next term is found by subtracting 5 from the position of K. Calculating 11 - 5 = 6 identifies the 6th letter of the alphabet, which is F.

Explanation

The series is an arithmetic progression where each term is increased by a common difference d. The difference between consecutive terms is calculated as: d = -5.75 - (-7.25) = 1.5 d = -4.25 - (-5.75) = 1.5

Following this pattern, the missing term is found by adding the common difference 1.5 to the preceding term -2.75: -2.75 + 1.5 = -1.25

This result is consistent with the final term of the series, as adding the common difference to the calculated value yields the next number: -1.25 + 1.5 = 0.25

Explanation

The numerical logic is based on replacing the symbols with specific mathematical operators: the symbol # represents division (÷) and the symbol × represents multiplication (×). These operations are executed from left to right as they share the same level of precedence in arithmetic.

Verification of the first condition: 3 ÷ 4 × 8 = 0.75 × 8 = 6

Verification of the second condition: 9 × 4 ÷ 3 = 36 ÷ 3 = 12

Applying this substitution to the target expression: 12 × 6 ÷ 24

First, calculate the product of the first two terms: 12 × 6 = 72

Then, divide the result by the final term: 72 ÷ 24 = 3

The result of the expression is 3.

Explanation

To determine the day of the week for Mitul's birthday, calculate the number of odd days between April 27 and September 28 of the same year. Odd days are the remainder obtained when the total number of days is divided by 7.

Number of days from April 27 to September 28: Remaining days in April = 30 - 27 = 3 Days in May = 31 Days in June = 30 Days in July = 31 Days in August = 31 Days in September = 28

Total days = 3 + 31 + 30 + 31 + 31 + 28 = 154

Now, calculate the odd days: Odd days = 154 7 Since 154 = 22 × 7 + 0, the remainder is 0.

A result of 0 odd days means the day of the week remains the same. Since Aarush's birthday on April 27 is a Thursday, Mitul's birthday on September 28 is also a Thursday.

Explanation

The total weight of all four boxes is 20 + 30 + 40 + 60 = 150 kg. To determine if a specific weight can be formed, we can either sum combinations or subtract excluded boxes from the total. For a total weight of 140 kg to be possible, the sum of the boxes not included in the combination must be 150 - 140 = 10 kg. However, the weights of the individual boxes are 20, 30, 40, and 60 kilograms. Since the minimum weight of any single box is 20 kg, there is no box or combination of boxes that totals 10 kg. Thus, it is mathematically impossible to reach a total of 140 kg using any combination of these four boxes where each is used only once.

Explanation

The word BLUE contains the letter E, which is not present in the source word HUMBLING. Since a required letter is missing from the available set of letters H, U, M, B, L, I, N, and G, the word BLUE cannot be formed.

Explanation

The coding follows the logic of opposite pairs, where the sum of the numerical positions of a letter and its corresponding code is 27. In the word FOREIGN, the letters F (6), O (15), R (18), E (5), I (9), G (7), and N (14) are paired with codes U (21), L (12), I (9), V (22), R (18), T (20), and M (13) because 6 + 21 = 27, 15 + 12 = 27, 18 + 9 = 27, 5 + 22 = 27, 9 + 18 = 27, 7 + 20 = 27, and 14 + 13 = 27. Applying this logic to SAP, S is the 19th letter, so its code is the 27 - 19 = 8th letter, which is H. A is the 1st letter, so its code is the 27 - 1 = 26th letter, which is Z. P is the 16th letter, so its code is the 27 - 16 = 11th letter, which is K.

Explanation

The problem requires substituting the given operators into the expression 25 + 10 ÷ 240 × 16 based on the provided code language. Replacing ‘+’ with ‘×’, ‘÷’ with ‘-’, and ‘×’ with ‘÷’, the modified mathematical expression becomes:

25 × 10 - 240 ÷ 16

Applying the BODMAS rule, division is performed first:

240 ÷ 16 = 15

Substituting this back into the expression:

25 × 10 - 15

Next, multiplication is performed:

250 - 15

Finally, performing the subtraction yields the final result:

250 - 15 = 235

Explanation

The operation A#B follows the mathematical pattern of the arithmetic mean of the two numbers. The relationship can be defined by the formula:

A#B = (A + B)/(2)

Applying this logic to the provided examples: For 9#3: (9 + 3)/(2) = ¹²⁄₂ = 6 For 15#3: (15 + 3)/(2) = ¹⁸⁄₂ = 9 For 60#4: (60 + 4)/(2) = ⁶⁴⁄₂ = 32

To find the value of 27#3, we apply the same formula: 27#3 = (27 + 3)/(2) = ³⁰⁄₂ = 15

Explanation

The pattern in the given table follows a column-wise logic where the product of the numbers in the second and third rows results in the number in the first row.

In the second column, the calculation is 23 × 13 = 299. In the third column, the calculation is 29 × 17 = 493.

Following the same logic for the first column, let the missing number be x. The relationship is x × 7 = 91. To find the value of x, we divide 91 by 7, which gives x = 13.

Explanation

The movement can be tracked using a coordinate system starting at (0, 0). The initial 5 km North displacement reaches (0, 5). The subsequent 4 km East turn results in the position (4, 5). Turning South and cycling 5 km brings the position to (4, 0), as the North and South distances are equal and cancel each other out. From this point, facing South, a right turn indicates a move towards the West. Cycling 6 km West from (4, 0) results in a final position of 4 - 6 = -2 on the horizontal axis. This displacement of 2 km in the negative x-direction means she is 2 km West with reference to her starting position.

Explanation

According to the rules of syllogism, the statement All rectangles are triangles is a universal affirmative proposition (A-type). This statement can be logically converted into a particular affirmative proposition (I-type), which states Some triangles are rectangles. Since all members of the rectangle group are contained within the triangle group, there is an existential overlap where those specific triangles are indeed rectangles. This confirms that Conclusion II follows directly from Statement 2. In contrast, the relationship between squares and triangles is not definitively established; while squares cannot be rectangles, they may or may not overlap with the remaining portion of the triangle set, making Conclusion I uncertain.

Explanation

The pattern in the grid shows that the number in the fourth row of each column is derived from the product of the numbers in the first three rows of the same column plus a constant of 3. The general formula for this operation is R₄ = (R₁ × R₂ × R₃) + 3.

For the first column, the calculation is (7 × 2 × 8) + 3 = 112 + 3 = 115. For the second column, the calculation is (6 × 5 × 9) + 3 = 270 + 3 = 273. Applying the same logic to the third column to find the missing number, the calculation is (3 × 1 × 4) + 3 = 12 + 3 = 15.

Explanation

Let B be the set of boys, H be the set of hardworking individuals, and I be the set of intelligent individuals. The first statement, some boys are hardworking, indicates that the intersection of B and H is not empty (B ∩ H ≠ ∅). The second statement, no intelligent are boys, establishes that the sets I and B are disjoint (I ∩ B = ∅).

Since there is a common region between B and H, any individual in that specific region is both a boy and hardworking. According to the second statement, any individual who is a boy cannot be intelligent. Therefore, the individuals in the intersection B ∩ H are hardworking but cannot be intelligent. This logically necessitates that some hardworking individuals are not intelligent, which validates conclusion I. Because the statements provide no information regarding the relationship between the set of intelligent individuals and the hardworking individuals who are not boys, no other conclusion can be established as a certainty.

Explanation

The relationship in the analogy is based on a deficiency disease and the specific nutrient required to prevent it. Goiter is a condition characterized by the enlargement of the thyroid gland, which occurs due to a lack of iodine in the diet. Similarly, anemia is a condition where the body lacks enough healthy red blood cells to carry adequate oxygen to the body's tissues. The most common cause of this condition is a deficiency of iron, which is a vital component of hemoglobin needed for oxygen transport in the blood.

Explanation

The relationship is determined by the numerical positions of the letters in the English alphabet (A = 1, B = 2, ..., Z = 26).

For the first pair DE : O: The numerical position of D is 4 and E is 5. Adding the positions: 4 + 5 = 9. Applying the transformation: (9 × 2) - 3 = 18 - 3 = 15. The 15th letter of the alphabet is O.

Applying the same logic to the second pair AF : ?: The numerical position of A is 1 and F is 6. Adding the positions: 1 + 6 = 7. Applying the same transformation: (7 × 2) - 3 = 14 - 3 = 11. The 11th letter of the alphabet is K.

Explanation

The relationship between the numbers is based on the sum of the product of the digits and the sum of the digits of the first number. For the first pair, the calculation is (5 × 6) + (5 + 6) = 30 + 11 = 41. Applying the same logic to the second pair with the number 94, we get (9 × 4) + (9 + 4) = 36 + 13 = 49.

Explanation

Huge is an adjective used to describe the size, scale, or magnitude of an object, indicating its physical dimensions. It characterizes the quality or extent of an entity. This distinguishes it from the other words in the group, which are used to specify the spatial orientation or relative position of an object in a physical environment.

Explanation

In the English alphabet, two letters are considered opposites if the sum of their numerical positions is 27. For the pairs MN, NM, and OL, the positions satisfy this condition: M(13) + N(14) = 27, N(14) + M(13) = 27, and O(15) + L(12) = 27. However, for the pair ET, the numerical position of E is 5 and T is 20. The sum of their positions is 5 + 20 = 25, which does not equal 27. Consequently, ET is the only pair that does not consist of opposite letters.

Explanation

In the numbers 253, 473, and 143, the sum of the first and the third digits equals the middle digit. Specifically, for 253, 2 + 3 = 5; for 473, 4 + 3 = 7; and for 143, 1 + 3 = 4. This pattern also indicates that these three numbers are multiples of 11. In contrast, for the number 633, the sum of the first and third digits is 6 + 3 = 9, which is not equal to the middle digit 3. Consequently, 633 is the only number in the set that is not divisible by 11.

Explanation

The words are arranged by comparing each letter from left to right. All five words share the first four letters Flin. Comparing the fifth letter of each word, Flinchers (5) and Flinching (1) have c, Flinptes (3) has p, and Flintlock (2) and Flintlocks (4) have t. In alphabetical order, c comes before p, and p comes before t.

Comparing Flinchers (5) and Flinching (1), the sixth letter is e in Flinchers and i in Flinching. Since e precedes i, Flinchers comes first, followed by Flinching. Next is Flinptes (3) because its fifth letter p follows c but precedes t. Finally, comparing Flintlock (2) and Flintlocks (4), Flintlock is a shorter word that forms the prefix of Flintlocks, meaning it appears first in the dictionary. This results in the sequence 5, 1, 3, 2, 4.

Explanation

The series is completed by dividing the letters into four groups of four letters each. Each group contains a permutation of the set of letters {x, y, z, a}. By placing the letters from option a) into the gaps, we obtain:

Group 1 (positions 1-4): x y z a Group 2 (positions 5-8): a x y z Group 3 (positions 9-12): x y z a Group 4 (positions 13-16): a z y x

The first gap (position 1) is filled with x, the second (position 6) with x, the third (position 8) with z, the fourth (position 11) with z, and the fifth (position 16) with x. This corresponds to the sequence xxzzx. In this arrangement, Group 1 and Group 3 are identical (xyza), Group 2 is a right cyclic shift of Group 1 (axyz), and Group 4 is the reverse of Group 1 (azyx). This satisfies the common pattern in such series where each subset contains the same set of characters.

Explanation

The series follows a pattern of alternating multiplication by the factors 2 and 3. Starting from the first term, the sequence is generated as follows: 19 × 2 = 38 38 × 3 = 114 114 × 2 = 228 228 × 3 = 684 684 × 2 = 1368 By applying the multiplier 3 to the second term 38, the missing third term is determined to be 114. This result is verified by the next step in the pattern, where multiplying 114 by 2 correctly yields the fourth term, 228.

Explanation

Based on the statement that P's height is more than only one student, P must be the fourth tallest person in a group of five students (1 > 2 > 3 > P > 5). The conditions provided state that Q is taller than both S and P, and S is also taller than P. Furthermore, Q is not taller than R, which means R is taller than Q. Combining these relationships, we establish the descending order of R > Q > S > P. Since R, Q, and S are all taller than P, they must occupy the top three positions in the class. The only remaining student, T, must be the individual shorter than P to satisfy the condition that P is taller than only one person. This results in the final height hierarchy: R > Q > S > P > T. Consequently, R is the student with the maximum height.

Explanation

Shweta is 27th from the left end. Since Palak is 7th to the right of Shweta, Palak's position from the left end is L = 27 + 7 = 34.

To find the position from the right end (R) when the total number of girls (N) and the position from the left end (L) are known, use the formula R = (N + 1) - L.

Substituting the given values: R = (74 + 1) - 34 R = 75 - 34 R = 41

Explanation

The word TALE can be formed using the letters of the word CAPITULATE because all its constituent letters are present in the source word. Specifically, CAPITULATE contains the letters T, A, L, and E, providing the necessary components to construct the word TALE without requiring any letters not found in the original string.

Explanation

The code is based on the concept of opposite letters in the English alphabet, where the sum of the numerical positions of the original letter and its coded letter is 27. For the word BIOLOGY, the mapping is B (2) → Y (25), I (9) → R (18), O (15) → L (12), L (12) → O (15), O (15) → L (12), G (7) → T (20), and Y (25) → B (2).

Applying the same logic to the word PHYSICS: P (16) → 27 - 16 = 11 (K) H (8) → 27 - 8 = 19 (S) Y (25) → 27 - 25 = 2 (B) S (19) → 27 - 19 = 8 (H) I (9) → 27 - 9 = 18 (R) C (3) → 27 - 3 = 24 (X) S (19) → 27 - 19 = 8 (H)

Following this sequence, the word PHYSICS is written as KSBHRXH.

Explanation

By interchanging the + and - signs in the given equation 6 + 8 ÷ 4 - 4 = 8, we get:

6 - 8 ÷ 4 + 4 = 8

Following the BODMAS rule, we perform the division operation first:

6 - 2 + 4 = 8

Then, performing the addition and subtraction from left to right:

4 + 4 = 8

8 = 8

The equality holds true after the interchange of the + and - signs.

Explanation

The logic followed in these equations is that the result is the square root of the sum of the three given numbers.

In the first equation, the sum is 4 + 9 + 3 = 16. Taking the square root gives √(16) = 4.

In the second equation, the sum is 5 + 3 + 1 = 9. Taking the square root gives √9 = 3.

Following the same pattern for the third equation, the sum is 9 + 9 + 7 = 25. Taking the square root gives √(25) = 5.

Explanation

In syllogism, a conclusion is valid only if it is definitely true in all possible scenarios represented by the statements. Statement I indicates that the set of cups is entirely contained within the set of plates. Statement II indicates that there is an intersection between the set of plates and the set of glasses.

This configuration allows for a possibility where the part of the plates that are glasses has no overlap with the cups. Since the intersection between glasses and plates could exist entirely outside the circle representing cups, it cannot be stated with certainty that some glasses are cups. Furthermore, since a partial relationship cannot be established, a universal relationship where all glasses are cups is also not logically necessary. Because neither conclusion is guaranteed by the premises, neither conclusion follows.

Explanation

The relationship between the first pair of words, Vacant and Empty, is based on synonymy, as both terms describe a state of being unoccupied or containing nothing. To maintain this analogy, the second pair must also consist of synonyms. Dearth is defined as a scarcity or lack of something, particularly when there is an inadequate supply to meet demand. Therefore, Scarcity is the synonym that correctly completes the relationship.

Explanation

The pattern established in the first set involves shifting each letter forward by an increasing number of positions. Specifically, the first letter is moved forward by 1, the second by 2, the third by 3, and the fourth by 4. In the transformation HMPU to IOSY, H becomes I (+1), M becomes O (+2), P becomes S (+3), and U becomes Y (+4). Applying this same rule to GMRF, G shifts to H (+1), M shifts to O (+2), R shifts to U (+3), and F shifts to J (+4), resulting in the sequence HOUJ.

Explanation

The numerical analogy follows the pattern n² + 1 for consecutive integers. For the first pair, 50 = 7² + 1 and 65 = (7 + 1)² + 1 = 8² + 1. For the second pair, the given number is 122 = 11² + 1. Applying the same logic, the related number is (11 + 1)² + 1 = 12² + 1 = 144 + 1 = 145.

Explanation

Jupiter is the fifth planet from the Sun and the largest planet in our solar system, classified as a gas giant. A black hole is a region of spacetime where gravity is so strong that nothing, including light, can escape from it. Jupiter lacks the necessary mass and density to be categorized as a black hole, making this pairing logically and scientifically incorrect compared to the other pairs which correctly identify the classification of the celestial bodies listed.

Explanation

The numerical positions of the letters in the English alphabet determine the relationship within each pair. For the pairs KQ, DJ, and RX, the difference between the positions of the two letters is exactly 6. Specifically, K is 11 and Q is 17 (17 - 11 = 6), D is 4 and J is 10 (10 - 4 = 6), and R is 18 and X is 24 (24 - 18 = 6). In the case of SZ, S is the 19th letter and Z is the 26th letter. The difference between their positions is 26 - 19 = 7. Because this pair has a difference of 7 while all others maintain a difference of 6, it is the odd alternative.

Explanation

The number 876 is not a perfect square. While the other values provided in the set are all squares of integers, 876 does not satisfy this mathematical condition. Evaluating the value of 876 relative to the sequence of integer squares, 29² = 841 and 30² = 900. Since 876 lies between these two consecutive squares, it is not the square of any integer. In contrast, the other numbers in the set are the squares of 19, 21, and 28.

Explanation

To arrange words in dictionary order, they are compared character by character from left to right. All five words begin with the letter H. Comparing the second letters, Harmony (a) comes first. Next are the words beginning with e, which are Helm and Herbivorous; since l comes before r in the alphabet, Helm precedes Herbivorous. Finally, among the words beginning with o, Honour precedes House because the third letter n comes before u. Following this alphabetical logic, the correct sequence is Harmony (2), Helm (5), Herbivorous (1), Honour (4), and House (3).

Explanation

The given series follows a pattern where each subsequent letter is determined by adding 2 to the position of the preceding letter in the English alphabet.

A = 1 C = 1 + 2 = 3 E = 3 + 2 = 5 G = 5 + 2 = 7

The next term in the sequence is calculated by adding 2 to the position of G: 7 + 2 = 9. The 9th letter of the English alphabet is I.

Explanation

The series follows a pattern based on the addition of a cubic value plus one to each preceding term. The difference between consecutive terms increases according to the formula n³ + 1, where n represents the position of the gap starting from 1.

The first difference is 4 - 2 = 2, which can be expressed as 1³ + 1. The second difference is 13 - 4 = 9, which can be expressed as 2³ + 1. The third difference is 41 - 13 = 28, which can be expressed as 3³ + 1. The fourth difference is 106 - 41 = 65, which can be expressed as 4³ + 1.

Following this pattern, the next difference must be 5³ + 1. Calculating this value gives 125 + 1 = 126. To find the missing number in the series, we add this difference to the last known term: 106 + 126 = 232.

Explanation

To determine the person with the maximum height, we can represent the five individuals in a sequence from position 1 (top) to position 5 (bottom). According to the problem, Hitesh is at the third place.

The condition that Bharat is between Nitin and Hitesh implies two possible arrangements for these three individuals: Nitin is at the 1st position, Bharat is at the 2nd position, and Hitesh is at the 3rd position. Hitesh is at the 3rd position, Bharat is at the 4th position, and Nitin is at the 5th position.

The problem specifically states that Nitin is not at the bottom, which rules out the 5th position. Therefore, the first arrangement must be correct: Nitin is at the 1st position and Bharat is at the 2nd position.

In a height-based arrangement starting from the top, the 1st position represents the individual with the maximum height. Since Nitin occupies the 1st position, he has the maximum height among the group.

Explanation

To determine the relationship between Q and S, the segments of the expression P + Q R - S are analyzed based on the given definitions. The part Q R indicates that Q is the brother of R. The part R - S indicates that R is the mother of S. Because Q is the brother of S's mother R, Q is the maternal uncle of S. In standard kinship terminology, Q is the uncle of S.

Explanation

The word Bland cannot be formed from the letters in Abandoned because it requires the letter L. The word Abandoned contains only the letters A, B, N, D, O, and E. Since the letter L is entirely absent from the base word, any word containing that letter cannot be constructed.

Explanation

The coding scheme follows two steps: first, reversing the order of the letters in the word, and second, adding a decreasing sequence of integers to the numerical positions of these letters, starting from the length of the word down to 1. For the word REMOTE, which has 6 letters, the reversed form is ETOMER. Applying the shifts gives E(5) + 6 = 11 (K), T(20) + 5 = 25 (Y), O(15) + 4 = 19 (S), M(13) + 3 = 16 (P), E(5) + 2 = 7 (G), and R(18) + 1 = 19 (S), resulting in the code KYSPGS. Similarly, for the 5-letter word BRAND, the reversed form is DNARB. The shifts are D(4) + 5 = 9 (I), N(14) + 4 = 18 (R), A(1) + 3 = 4 (D), R(18) + 2 = 20 (T), and B(2) + 1 = 3 (C), resulting in IRDTC. For the word MOBILE, which also has 6 letters, the reversed order is ELIBOM. The corresponding transformations are E(5) + 6 = 11 (K), L(12) + 5 = 17 (Q), I(9) + 4 = 13 (M), B(2) + 3 = 5 (E), O(15) + 2 = 17 (Q), and M(13) + 1 = 14 (N). Combining these letters yields the final code KQMEQN.

Explanation

Based on the operator definitions provided, we substitute the symbols in the expression 12 - 6 + 28 × 3 ÷ 9 as follows: - denotes ÷, + denotes -, × denotes +, and ÷ denotes ×.

Substituting these into the given expression: 12 ÷ 6 - 28 + 3 × 9

Following the BODMAS rule (Division, Multiplication, Addition, Subtraction):

First, perform the division: 12 ÷ 6 = 2 The expression becomes: 2 - 28 + 3 × 9

Next, perform the multiplication: 3 × 9 = 27 The expression becomes: 2 - 28 + 27

Finally, perform addition and subtraction: 2 + 27 - 28 = 29 - 28 = 1

Explanation

The numerical logic followed in the given sequences is that the middle number is the square of the difference between the first and the third numbers. This can be expressed by the formula Middle = (First - Third)².

In the first instance, 19 - 13 = 6, and 6² = 36. In the second instance, 37 - 28 = 9, and 9² = 81.

Applying this same logic to the target sequence where the first number is 43 and the third number is 38, we calculate the difference as 43 - 38 = 5. The value of A is the square of this difference: 5² = 25.

Explanation

The numerical grid follows a pattern where the product of the four numbers in every pair of adjacent columns remains constant. By dividing the six columns into three distinct pairs, the constant value can be determined.

In the first pair comprising columns 1 and 2, the product of the four integers is 3 × 2 × 7 × 4 = 168 . In the second pair comprising columns 3 and 4, the product is 8 × 1 × 3 × 7 = 168 .

Given that the product for each set of two columns is 168, the third pair comprising columns 5 and 6 must satisfy the same condition. Setting up the calculation for the final pair: 1 × 2 × 6 × ? = 168 12 × ? = 168 ? = 168 / 12 ? = 14