Image: Cloe Liane Shasha
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!
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13 Comments
Add CommentDuring the last ice age the sea level on Earth was over 300 feet lower. That means a bunch of beach front property from 15,000 years ago is now sea bed. Has this been a disaster, or is it what has been happening here for millions of years.? The Hawaiian Islands sink into the sea and erode away as they migrate west past the magma vent that formed them. The Outer Banks are , and have always been in flux for their entire existence. The light house on Cape Hatteras had to be moved to keep the Island from migrating out from under it. New Orleans is below sea level for crying out load! I bought property in the Coachella Valley. I am counting on a beach front view of the Sea of Cortez, unless another Ice Age occurs first. It has all happened before and will again. "Man made Global Warming" is a political wedge to separate you from your freedom and money in the name of science. If you think "we" can stop ( or even slow) CO2 production while China brings on line a new coal fired power plant every week for the foreseeable future, pass that burning root my way. China has over 280 coal fire plants in the Q as we speak. Now think about what India is doing. Get a clue!
Reply | Report Abuse | Link to this"Experts" tell us CO2 caused the planet to start warming at the beginning of the Industrial Revolution. In fact the CO2 load from industry in 1870 was an insignificant % of what is produced today by any one of the countries that make up the G 8. If the 'all knowing' are correct, the current day effects of CO2 would be exponential by comparison. They never address that we are about 12,000 years down wind of the last major Ice Age. They also never mention the Mini Ice Age that gripped the Earth from approx. 1350 to 1860 AD. This was a well documented period of famine and plague. All the archaeologic evidence is clear. Plant and animal populations (especially Man) boom during warm periods on Earth and die back as the ice creeps toward the equator. It is really hard to find food in a 100 foot deep ice field.
So buy an efficient car because smog sucks. Recycle because there are many benefits besides reducing CO2. Plant a tree for the shade and their beauty. Think and learn something about the Causes you support before letting Idiots like All Gore brain wash you. By the way, which CO2 reduction conference did Big Al fly his private jet to this week?
Oh yeah! do you know the difference between a carbon credit and Santa Claus...... nothing, they are both make believe and a good way to separate people from their money.
Ignore the troll and his illogic and poor spelling.
Reply | Report Abuse | Link to thisName calling without one cogent argument too support your view. Me thinks it is your brain that has shriveled like a raisin. PS the word you were looking for is illogicalness. You just made up a word, I suspect just like you make up a lot of your facts. Oh, I forgot. You did not sight ANY facts.
Reply | Report Abuse | Link to thisI concurred with "Not Duped".
Reply | Report Abuse | Link to thisTake a look at any map anywhere over a two hundred year period, and you will not find one shoreline anywhere that retains is shoreline characteristics. Unless its totally made of the most solid stone, most shorelines are ebbed away by errosion.
But for Whit Island, ask yourself, is the water rising or the land falling? Because the water levels aren't rising in comparison to other shorelines common to the same body of water.
So again, misinformation by global climate change morons.
I believe that a regular heptagon would be the best solution.
Reply | Report Abuse | Link to thisThe area would be about 29.071 km ^ 2
Peter Cooper
plcooper@mindspring.com
I believe that a regular heptagon would be the b
Reply | Report Abuse | Link to thisOkay, smarty, it's "to support", "methinks", "site", and illogic is a word: http://www.thefreedictionary.com/illogic
Reply | Report Abuse | Link to thisAnd if you want science and facts, start here: http://mediamatters.org/items/200901060014
Reply | Report Abuse | Link to thisHey both of you.... it's "cite" if you are talking about quoting things like facts.
Reply | Report Abuse | Link to thisHa, oh yeah. Duh.
Reply | Report Abuse | Link to thisStill not a fact to be heard from the shriveled one. In any debate format, you would be described as the looser. Probably a term you hear often. The tide has turned, the Emperor has no clothes and the avalanche of scientists speaking out against the "party line" is deafening. Please excuse my less then stellar typing and attention to spelling. I spent most of my time in lectures and labs where scientific thought and exploration was more important then tie-die and protest. You may call me Doctor Not Duped!
Reply | Report Abuse | Link to thisFor problem 2, it's fairly simple to get a score of 10.5, assuming you're allowed to build walls that have no purpose when they're first built.
Reply | Report Abuse | Link to this_ _ _
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Build one of the small squares each year, and use the extra material to begin construction on the larger part. You end up with a 3 * 3.5 rectangle.
I can get a Multi-Count score of 83. No luck doing better than that so far.
Reply | Report Abuse | Link to thisTo get 83 you spend the first 5 years making a 5x5 square and the next two years building a 2x4 rectangle along one side. You get to count the 5x5 three times (after 5, 6, and 7 years) and the 2x4 once for a total of 83.