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The people most threatened by global warming are those who live in low-lying areas near the seashore. In this puzzle, you are going to try to protect as much of the fictitious (but representative) Whit Island as possible. Whit at its highest point is only 10 feet above sea level but the previous government had ignored the issue until an unusual storm swept several houses into the sea—and at the next election, the government, as well.
The new government is drawing up plans and soliciting advice. The entire island is a 7- by 7-kilometer square whose edges lie precisely north-south or east-west. This yields a perimeter of 28 kilometers. During the dry season, the islanders can build 4 kilometers worth of wall. Walls are built from straight segments either east-west or north-south. Once built, a wall can't be moved. In the rainy season, the islanders can't build.
The most efficient idea would be to build a dike around the entire island, which would take seven years. But until completed, the wall would protect nobody, because there would be no land enclosed by a wall. The political reality requires that:
a) During the first year, the island must succeed in enclosing 1 square kilometer. This is called the First Year Rule.
b) At the end of every year, the government must arrange to enclose at least as much new land as was newly protected the year before. This is called the Growing Rule.
The people of the island are mathematically well trained, so in a town meeting the majority have specified that any plan should maximize the sum of the areas enclosed after each year. For example, if 1 square kilometer is enclosed by the end of the first year and 2.5 kilometers by the end of the second year, then the "score" for the first two years is 1 + 2.5 = 3.5. This method counts the first year's enclosure multiple times on the grounds that an enclosure gives value no matter when it was built. So, we call that method the Multi-Count score.
Warm-up:
Can you find a plan that would give a Multi-Count score of 3.5 after two years? (Hint: This means that the first year, the plan encloses 1 square kilometer and the second year an additional 1.5 square kilometers.)
Solution to Warm-Up
Now here are problems for you.
1. Can you achieve a Multi-Count score of 14 or better after four years?
The minority at the town meeting demand to be heard and propose another scoring method called Single-Count. (According to Single-Count scoring, the solution to the warm-up problem would give a score of 1 + 1.5 = 2.5.) This approach evaluates a plan by summing the newly enclosed land for each year. They agree with the majority, however, to uphold the First Year and Growing Rules.
Solution to Problem #1
2. Can you find a plan that would give a Single-Count score of 9 after five years, but satisfy the First Year and Growing Rules?
Solution to Problem #2
Here is a challenge whose solution I don't know:
Using the Multi-Count score, but ignoring the First Year and Growing Rules, find a strategy that gives the greatest score after seven years, the term of the new government.
Post a comment with the solution to that one!