Problem 8 · Checkerboard coloring and construction · 5 points
Made by AI and not reviewed by a real person.
A 7×7 grid of unit squares loses its four corner squares and its center square. What is the greatest number of 1×2 dominoes you can cut from what remains, in either orientation, if you may leave some squares unused?
- 20
- 21
- 19
- 18
- 22
Show a hint
- Counting squares gives a tempting limit, but can the dominoes actually reach it? Try coloring the grid like a checkerboard.
- The corners and center all have the same color. After finding an upper bound, look for a way to cut that many dominoes.
Show the answer and solution
Answer: A — 20
A checkerboard coloring has 25 dark and 24 light squares. The five removed squares are all dark, leaving 20 dark and 24 light. Each domino covers one of each, so at most 20 can be cut; 22 is the trap from counting area alone. To reach 20, also leave unused the squares at (row, column) (2,1), (2,7), (6,1), and (6,7). The top and bottom two rows each become a tileable 2×5 rectangle. In the middle three rows, tile the 2×3 blocks in rows 3–4, columns 1–3 and rows 4–5, columns 5–7, and the 1×4 strips in row 3, columns 4–7 and row 5, columns 1–4.