2021 CSAT — Q64

This item has a contested official key — both readings are documented; it is excluded from readiness scoring.

A run of $6$ consecutive squares must lie on a diagonal of length $\ge 6$. On an $8\times 8$ board, the diagonals in one direction have lengths $1,2,\dots,8,\dots,2,1$; those of length $\ge 6$ are the lengths $6,7,8,7,6$ (five diagonals per direction). The number of $6$-windows on a diagonal of length $L$ is $L-5$, so per direction:

and over both diagonal directions the full count is $9\times 2 = 18$ — which is not among the options.

Restricted reading (the two principal/main diagonals only): each main diagonal has $8$ squares, giving $8-5 = 3$ windows; the two main diagonals give $3 + 3 = 6$. This is the only clean reading that lands on an offered option.

Where the official $12$ comes from (and why it is wrong): index the $\backslash$-diagonals by $d=\text{row}-\text{col}\in\{-7,\dots,7\}$, length $=8-|d|$. Those of length $\ge 6$ are $d=0$ (len $8$), $d=\pm 1$ (len $7$, two of them), $d=\pm 2$ (len $6$, two of them). Windows: $3+2+2+1+1 = 9$ per direction, $\times 2 = 18$. The figure $12$ is reproducible only by counting each length $6,7,8$ once per direction — i.e. taking just the three diagonals from one corner up to the main diagonal, $1+2+3 = 6$, and doubling to $12$ — which silently drops the symmetric length-$6$ and length-$7$ diagonals on the far side of the main diagonal. That is an enumeration error, not a reading: $12$ corresponds to no correct geometric count.

• Our blind-solve under the natural “the diagonals” $=$ the two principal diagonals reading gives (b) 6.
• The published UPSC key is (d) 12, reproducible only via the under-count above; the rigorous full-board count is 18 (not offered), and the clean principal-diagonal count is 6.

Because the rigorous full enumeration ($18$) is not an option and the official key ($12$) is reachable only through a demonstrable under-count, this item is marked contested; correct_key records the most defensible option-yielding reading ($6$).

Answer (our reading): (b) 6 — contested vs official (d) 12.

The correct method is a window count: a length-$L$ diagonal holds $L - 5$ runs of $6$. The ambiguity that makes this contested is the phrase “the diagonals” — read as all diagonals (both directions) it is $18$; read as the two principal diagonals it is $6$; neither equals the official $12$. Under the clock and negative marking, an item whose rigorous count ($18$) isn’t even offered is a flag to mark and move on, not to force-fit. We surface the defect rather than bend the math.