You’re hosting a wedding at your home next summer, and the happy couple asked you to decorate the four gardens on the grounds with — let me check my notes — ah, fresh gridflowers. Gridflowers are planted in square plots. Each autumn, they cast seeds to all neighboring squares, including diagonals. The next spring, the old flowers are gone. A fresh flower will grow only in the spots that had exactly two neighboring flowers the year before (see examples below). And true to its name, a gridflower won’t grow beyond a given grid plot.
But the couple doesn’t want hand-planted gridflowers. They want flowers that grew naturally from the spread of the previous year’s flowers.
Therefore, you must plant your four gardens by hand this summer, so that next summer, each one contains a desired number of gridflowers.
How can you achieve the following number of flowers for each garden next year?
• Exactly eight flowers for a 3 × 3 grid
• Exactly 12 flowers for a 4 × 4 grid
• At least 17 flowers for a 5 × 5 grid
• At least 24 flowers for a 6 × 6 grid
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