The figure shows two 4-tile patterns. Either one or both of the patterns can be used any number of times and in any orientation to construct a new pattern. Which one of the options below cannot be constructed by using only these two 4-tile patterns assuming there are no overlaps among them?
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Each tile covers 4 cells; check which option's total cell count is not a multiple of 4.
Step 1: Identify the two given tiles. The figure shows two distinct 4-tile patterns (tetrominoes), each made of exactly 4 unit square cells joined edge to edge: the first is a 2x2 square block of 4 tiles, and the second is a straight 1x4 line of 4 tiles. Both tiles, in any rotation or reflection, always cover exactly 4 unit cells with no gaps and no overlaps within the tile itself.
Step 2: Set up the area condition for any constructed pattern. If a new pattern is built using k1 copies of the square tile and k2 copies of the line tile, with no overlaps and no gaps, then the total number of unit cells covered is exactly Total = 4*k1 + 4*k2 = 4*(k1 + k2). Since k1 and k2 are non-negative integers, the total area of any pattern built purely from these two tiles must be a multiple of 4. This is a necessary condition - any option whose total cell count is not a multiple of 4 can be eliminated immediately, regardless of shape.
Step 3: Compute the cell count of each option. Option (A) is a 4x2 rectangle = 8 cells. Option (B) is a 4x3 rectangle = 12 cells. Option (C) is a 5x3 rectangle = 15 cells. Option (D) is a 5x4 rectangle = 20 cells.
Step 4: Apply the divisibility-by-4 test. Check each: 8/4 = 2 (multiple of 4, passes). 12/4 = 3 (multiple of 4, passes). 15/4 = 3.75, that is, 15 = 4x3 + 3 (not a multiple of 4, fails). 20/4 = 5 (multiple of 4, passes). Only option (C) fails this necessary condition.
Step 5: Confirm the passing options can actually be tiled. Option (A), a 4x2 rectangle, can be built with two square tiles placed side by side, or with two line tiles laid horizontally. Option (B), a 4x3 rectangle, can be tiled using three horizontal line tiles of length 4 stacked in three rows. Option (D), a 5x4 rectangle, can be tiled using five vertical line tiles of length 4 placed side by side across the width of 5. So options (A), (B), and (D) are all genuinely constructible, not merely area-compatible.
Step 6: Conclude for option (C). Since 15 is not divisible by 4, it is mathematically impossible to exactly cover a region of 15 cells using only tiles of size 4 with no overlaps and no gaps, no matter how they are arranged or oriented. Therefore option (C), the 5x3 rectangle with 15 cells, cannot be constructed.
