سوال ۱۱۱ کنکور ارشد علوم کامپیوتر ۱۴۰۴
به چند طریق میتوان از ۶۴ خانه یک جدول ۸×۸ سه خانه انتخاب کرد، بهطوریکه یکی از این سهخانه با هر دو خانه انتخاب شده دیگر ضلع مجاور داشته باشد؟
پاسخ صحیح: گزینهٔ B — ۲۹۲
We must count sets of 3 cells in which every chosen cell is edge-adjacent to at least one other chosen cell. For exactly 3 cells, this condition forces the three cells to form a connected polyomino of size 3 — i.e. a tromino. There are exactly two tromino shapes: the straight (I) tromino and the bent (L) tromino.
Straight trominoes ( strips).
A horizontal strip occupies 3 consecutive cells in a row. In an -column row there are positions, over rows:
By symmetry the vertical strips also give . So
L-trominoes (bent shape).
Every L-tromino fits inside a block and uses exactly of its cells. The number of blocks in the board is
and each block yields distinct L-trominoes (one for each omitted corner). Since the omitted corner is uniquely determined by the block, no L-tromino is double-counted:
Total.
This matches option 2.
Pitfall: Do not forget the straight trominoes — counting only the L-shaped ones gives option 1, a deliberate trap. Conversely, the bent shapes must be counted via blocks, not by trying to place an "L" directly.