On a chess board, in how many different ways can 6 consecutive squares be chosen on the diagonals along a straight path ?

(a) 4
(b) 6 ✓ UPSC's answer
(c) 8
(d) 12

Why the answer is (b)

• A standard chessboard is an 8x8 grid, and a 'straight path' along a diagonal implies a sequence of squares where both row and column indices change by 1 in each step.

• For a diagonal of length $L$, the number of ways to choose 6 consecutive squares is $L - 6 + 1 = L - 5$.

• The main diagonals (from corner to corner) have a length of 8. There are two such diagonals (one from top-left to bottom-right, and one from top-right to bottom-left).

• For each main diagonal of length 8, the number of ways is $8 - 5 = 3$.

• Since there are two main diagonals, the total number of ways is $3 + 3 = 6$.

• Therefore, the correct option is (b).

Why the other options are wrong

(a) 4: Option (a) is incorrect because it underestimates the count, likely considering only one main diagonal or miscalculating the combinations.
(c) 8: Option (c) is incorrect because it overestimates the count, possibly including diagonals of length 7 which cannot accommodate 6 consecutive squares in the required manner or double-counting.
(d) 12: Option (d) is incorrect because it significantly overestimates the count, likely confusing the problem with choosing any 6 squares or considering all possible diagonal segments regardless of length constraints.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.