Problem 11 · 2013 Math Kangaroo
Medium
Counting & Probability
Geometry & Measurement
careful-countingcasework
Anne plays “sink the ship” with a friend on a 5×5 grid. She has already drawn in a 1×1 ship and a 2×2 ship (see picture). She must also draw a rectangular 3×1 ship. Ships may be neither directly nor diagonally adjacent to one another. How many possible positions are there for the 3×1 ship?
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Answer: E — 8
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Hint 1 of 3
Shade every cell that touches an existing ship, even at a corner, as off-limits.
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Hint 2 of 3
On the cells that remain, count the straight runs of three (across and down) where the new ship fits.
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Hint 3 of 3
Each free column tall enough holds several vertical placements, so check the open columns carefully.
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Approach: block the buffer cells, then count straight runs of three free cells
- Each existing ship needs a one-cell gap on every side (including diagonals), so shade those buffer cells as forbidden.
- After shading, the two right-hand columns stay completely open from top to bottom, plus the top two rows have a free stretch of three cells.
- Each open length-5 column holds 3 vertical placements (rows 1-3, 2-4, 3-5), giving 3 + 3 = 6 vertical positions.
- The two open top rows each hold one horizontal 3-in-a-row, adding 2 more, for 6 + 2 = 8.
- So there are 8 possible positions.
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