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改变的状态 2

Altered States 2

专题
Discrete Math / 离散数学
难度
L7

题目详情

A little while ago we asked solvers to smoosh as many of the 50 U.S. states into a 5-by-5 grid as possible.

Now we’re at it again! Once more, your goal is to score as many points as possible by placing U.S. states in a 5-by-5 grid.

  • States can be spelled by making King’s moves from square to square. (See the example.)
  • This time around, the score for a state is its population in the 2020 U.S. Census. So, for example, CALIFORNIA scores 39,538,223 points.
  • In the true spirit of the puzzle’s title, you may “alter” the name of a state by at most one letter. “Altering” a state means replacing a letter with another letter. (So NEWPORK, NEWYORF, and NEWYORK would all score for NEWYORK, but NEWYRK and NEWYROK would not.)
  • If a state appears multiple times in your grid, it only scores once.

The 3-by-3 example above scores 32,913,047 points, for Illinois (Inlinois), Ohio, Utah (Atah), Iowa (Ioha), and Idaho (Ieaho).

To send in your entry, please render your grid as one unbroken 25-digit string by concatenating the rows. (The 3-by-3 grid from the example would be “thoainesl”.)

To qualify for the leaderboard, your entry must score at least half of the available points. (So: at least 165,379,868.)

Beyond that requirement, the local tourism authority has established awards for particularly special boards, using the following citations:

  • 20S: visits at least 20 states
  • 200M: scores at least 200 million
  • PA: visits Pennsylvania
  • M8: contains all 8 states that begin with an M
  • 4C: contains the “Four Corners” states
  • NOCAL: avoids California
  • C2C: contains a coast-to-coast1 chain

  1. “Coast-to-coast” means “East coast to West coast”. States in the chain must have positive joint border length (e.g. Arizona-to-Colorado would not count as a “step” in the chain); “East coast” states range from Florida (southernmost) to Maine (northernmost). ↩

解析

Original Explanation

We received over 500 valid boards for this month’s challenge. Above are some of our favorites, courtesy (in order) of Alex Jian, Cathan Gormley, David Spencer, Mark Rothery, Josh Perks, and Daan Oorts.

A table containing many of the entries we received1 can be viewed here. A few interesting points:

  • The most commonly used stated was Texas, narrowly beating out Ohio and Iowa.
  • Connecticut and Rhode Island were the only states to appear in exactly zero boards. (Does there exist a valid board that contains both of these, and which excludes Texas?)
  • The number of states appearing in valid boards ranged from 12 to 33. (Does there exist a valid board with fewer than 12? Or more than 33?)

We also received a handful of improvements to the original Altered States.

Thanks to everyone who sent in a valid board!

  1. We tried to limit this doc to one board per submitter. ↩