Category Archives: Asked & Answered

Asked and Answered: 13.0

Last year, I told you about Spelling Bee, a word game that appears daily in the New York Times.  The point of the game is to form as many words as possible, using any of the seven letters provided, any number of times, as long as the center letter is used at least once.  It is a fairly undemanding diversion which, along with my cup of coffee, helps get me going in the morning.  Plus, it gives me something to kvetch about to my friend Eric, who also plays.

Eric and I often compare notes and complaints about the Bee of the Day.  My usual beef is about the exotic foodstuffs (e.g., BOBA, CALLALOO, GHEE) included in the answer list, and Eric (former chemistry professor) and I both gripe about the chemical words, such as NICOTINIC and PROPANOL, that of course should be accepted by the Bee but are not.

Nonetheless, we were pleasantly surprised by a recent Bee in which BORON (Element 5) and CARBON (Element 6) were among the answers.  This got me wondering: what is the greatest number of element names one can generate from a set of seven different letters?

My first step toward an answer was to create a spreadsheet to count the number of times each letter of the alphabet appears in the list of element names.  Note: I decided to limit the number of elements in my list to the first 100, i.e., from HYDROGEN to FERMIUM.  Elements 101 and beyond — all man-made — are unfamiliar even to Eric and me.  One of those is darmstadtium (Element 110) of which only a few atoms have ever been produced. So it’s not like I’m disrespecting Nature by excluding darmstadtium and its ilk.

Anyway, back to my spreadsheet.  I found that the consonants appearing most often in my 100-element list were M (50), N (36), R (33), L (22) and T (22).  And as you might guess, the most common vowels were I (56) and U (50).  After spending about 15 minutes playing around with frequently-appearing letters, I was able to find two different seven-letter sets which “contain” the names of four elements.  (Before I reveal, would you like to try?)

The first set I found was EIO/BMNR. (I’ll refer to these sets by their vowels/consonants). This set spells the elements BORON, BROMINE, IRON and NEON.  And my second set was AEIO/DNR, which spells IODINE, IRON, NEON and RADON.  Interesting, but…

I was, of course, not satisfied.  Humankind needed to know: are there other seven-letter sets that spell out four element names?  And more importantly, are there seven-letter sets that spell out five (or more) element names?  I did not yet have these important answers.

So, for humankind’s sake, I was obliged to resort to brute-force computation, employing the only modern programming language I know — PHP.  I am familiar with PHP because it is the language used by WordPress, the platform for this and millions of other blogs.  And though I have a PHP reference manual, most times when I want to write code for a new task, I just do an internet search — 99 percent of the time someone has already done the thing that I want to do and has provided functions and/or code for it.

And that is (mostly) how I wrote a PHP program to print out all the seven-letter sets that spell out four or more element names.  I’ll spare you the details, but suffice it to say that before now I had no idea there were PHP functions like count_chars (finds the number of unique letters in a word), array_intersect (lists the items that two sets have in common) and implode (combines a set of individual letters into a single word).  Those functions served me well here, but they (like many other PHP functions) are so special-purpose that I can’t imagine any programmer having good command of them all.

In any case, I ultimately wrote a program that evaluated all 213,333 of the seven-letter sets containing one or more vowels and one or more consonants found in the element list. Without further delay, here are the results.

SEVEN-LETTER SETS WHICH CONTAIN FOUR ELEMENT NAMES:

AEIO/DNR IODINE
NEON
IRON
RADON
AIU/BDMR BARIUM
RADIUM
IRIDIUM
RUBIDIUM
AIU/CDMR CADMIUM
IRIDIUM
CURIUM
RADIUM
AIU/DMNR INDIUM
RADIUM
IRIDIUM
URANIUM
AIU/LMNT ALUMINUM
TIN
TANTALUM
TITANIUM
EIO/BDNR BORON
IRON
IODINE
NEON
EIO/BMNR BORON
IRON
BROMINE
NEON
EIO/BNRT BORON
NEON
IRON
TIN
EIO/BNRX BORON
NEON
IRON
XENON
EIO/DNRT IODINE
NEON
IRON
TIN
EIO/DNRX IODINE
NEON
IRON
XENON
EIO/DNTX IODINE
TIN
NEON
XENON
EIO/GNRT IRON
NITROGEN
NEON
TIN
EIO/NRTX IRON
TIN
NEON
XENON
EIU/HLMT HELIUM
LUTETIUM
LITHIUM
THULIUM

SEVEN-LETTER SETS WHICH CONTAIN FIVE ELEMENT NAMES:

None.  Zero.  Not-a-single-one-ium.

So there you have it.  There are 15 different seven-letter sets which can be arranged to spell four element names, but there are no seven-letter sets that will spell five element names, at least not with respect to the first one hundred elements.

If some nerd ever uses this edition of Asked and Answered to win a bar bet, I will expect due credit, if not a beer.

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Map Challenge II

Asked and Answered 12

As a followup to last year’s Map Challenge post, I thought I would pose a few geo-nerd problems inspired by the Four Corners in the southwestern U.S.  As most of you know, there is only one spot in the U.S. — about 12 miles north of the Sinclair gas station in Beclabito, NM — where four states meet at a single point.*  But this begs the question: what about the near-misses?  Shouldn’t those places get a little bit of nerd-love too?

I will call these areas Four-State Circles — circles that include at least some part of four different states, other than the aforementioned Four Corners.  Here are my challenges:

  1.  What and where in the U.S. is the smallest Four-State Circle (and its runner-up)?
  2.  Similarly, what about the smallest Five-State Circle (and its runner-up)?
  3.  Finally, the smallest Six-State Circle (and its runner-up)?

I am sure you’ll all agree that we can safely stop at six states without feeling like we have shortchanged the topic.  So let’s go exploring — and here’s a map to help.

Four-State Circles

When scanning the map for potential four-state solutions, the key is to focus on regions where two nearby states don’t share a border.  This suggests a closer look at places like the Oklahoma panhandle and the contorted path of the Mississippi River south of Illinois.

But it turns out that both the winner and the runner-up for the smallest Four-State Circle lie east of the Appalachians.  Surprisingly, neither circle involves Rhode Island, and both include my home state of Pennsylvania.

THE RUNNER-UP

Zoom to 39.594ɴ, 78.350ᴡ, 100 feet from Pepper Lane in Doe Gully, West Virginia.  Next draw a circle of 8.83 miles radius around that point.  This circle touches four states: from north to south, Pennsylvania, Maryland, West Virginia and Virginia.  This may be the runner-up Four-State Circle, but I believe it is the shortest path that crosses four states, a subtly different problem.

THE WINNER

Now shift your focus 200 miles eastward to 39.676ɴ, 75.666 in Delaware, across the street from an industrial park in Christiana.  A 6.55-mile radius circle around this point is mostly Delaware, with slivers of New Jersey, Maryland and Pennsylvania.  This would be a conflicted place to be a sports fan: 7 miles from three different states with major league sports teams, but not one to call your own.

Five-State Circles

The Five-State Circles were more painstaking to sort through.  It’s not like one can just enlarge a Four-State Circle a few miles in order to pick up a fifth state.  And what’s worse, all of the obvious candidates were so similar in size that some very careful plotting was needed to find the winner.  Here are the results:

THE RUNNER-UP

Take U.S. 60 toward Grayridge, Missouri, then head a couple of miles south, and you will find — besides a lot of farmland — the center of the runner-up Five-State Circle.  This 28.8-mile-radius circle takes in parts of Kentucky, Tennessee and Missouri, plus razor-thin slices of Arkansas and Illinois.  S0, Grayridge, population 122, you are now on the map.  Welcome to internet fame.

THE WINNER

Here is where the Oklahoma panhandle gets its chance to shine.  A 27.1-mile radius circle centered on 36.892ɴ, 102.514, just north of Boise City, touches parts of five states: New Mexico, Colorado, Kansas, Oklahoma and Texas.  Although the center of the circle is now just a pile of rubble next to U.S. 385, just wait until some enterprising Okie gets wind of this and builds a McDonalds there.

Six-State Circles

There are two attractive candidates here — the area around Baltimore, MD, and the area around Springfield, MA.  Only one of these, the smallest Six-State Circle, can be crowned the winner — the other will be the runner-up who, a la Miss USA, will assume the title if for any reason the winner cannot fulfill its duties.  So, may I have the envelope please…

THE RUNNER-UP

Cold Bottom Road crosses Piney Creek and I-83 at 39.563ɴ, 76.665, just north of the National Lacrosse Hall of Fame in Sparks, Maryland.  The six states within 58.9 miles of this spot are: Maryland, Pennsylvania, Delaware, New Jersey, West Virginia and Virginia.  If Washington, D.C., were a state, this would be the winning Seven-State Circle but sorry — rules are rules.

THE WINNER

The center of the smallest Six-State Circle is American Legion Post 452 in Chicopee, MA, at 42.1456ɴ, 72.6140.  (The post meets on the fourth Thursday of each month at 17:30, so snap to it.)  This 43.2-mile radius circle covers parts of Massachusetts, Connecticut and Vermont plus thin slices of New York, New Hampshire and Rhode Island.  And yes, Rhode Island is a state, not a plantation.

So there you have it, asked and — exhaustively — answered.  But for those who need even more bar-bet material, here’s one last bit of U.S. state-border trivia.  The number of places where three (or four) states meet is (a) 55, (b) 68 or (c) 92.  Answer below.

_______________

* The four states which meet at one spot are Arizona, Utah, Colorado and New Mexico.  And by my count, there are 55 places in the U.S. where three or four states meet at a point.
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Map Challenge!

Asked & Answered: 10

I have always been a map nerd.  When I was a kid, I drew street maps of imaginary cities with improbably curvy roadways and myriad underpasses.  One of my fictional streets would invariably be named “Standard Avenue” or “Standard Road,” being that my father worked at Rockwell-Standard and my imagination had its limits.  When I was a little older, I biked down to city hall and bought a street map of our town (for one dollar), tacked it up on my bedroom wall, and then went about exploring all the streets I could pedal to on my mono-speed bicycle. 

I also remember inventing a map game to play with the younger kids who lived next door.  I appropriated a fold-out world map from one of our National Geographic magazines, drew a grid on the map to create “spaces” for tokens to move about, and then marked various squares on the map as “targets.”  The game required players to draw “target” cards and navigate from one target to the next via the roll of the dice.  I think I made up other special rules or cards to introduce an element of chance/excitement into an otherwise mundane undertaking, but hey, I was like, eleven years old?

Anyway, map-nerdiness is the only thing I likely have in common with Jeopardy! champ Ken Jennings, who in fact wrote a book titled “Maphead” which I really must read one day. I don’t know whether “Maphead” deals with challenges like the following, but here goes.

I recently came across a post heralding the longest straight overland line in the continental U.S, which I call the Nandor Line in honor of its discoverer.  The Nandor Line, 2,802 miles long, extends from Ocean Creek in Washington State to Ocean Drive in Jupiter, Florida.  In theory, one could walk the Nandor Line without crossing an ocean, gulf or foreign land.  (Here, we must stipulate that Idaho does not constitute a foreign land.)

As best I can tell, the Nandor Line passes through 12 states: Washington, Idaho, Montana, Wyoming, Nebraska, Kansas, Missouri, Arkansas, Mississippi, Alabama, Georgia, Florida. But there are close calls with a number of other states.  Naturally, this begs the question, what overland line passes over the greatest number of U.S. states, regardless of its length?

I have an answer… but I am not certain if it is the answer.  As with all scientific endeavors, my answer will be the answer until someone else comes up with a better one.  Here is how I came up with mine.

I decided that my solution — as well as those of any challengers — should be expressed in terms of a direct flight from Airport A to Airport B.  This allows the proposed path to be accurately mapped and checked via the website GPS Visualizer.

I also decided that flying over the Great Lakes is allowed, as long as the flight path does not cross the border with Canada.  Sticklers for full-overland solutions may object, but then how would they propose we cross the Mississippi River?

Anyway, here’s my solution (click to zoom): New Bedford, MA (EWB) to Fullerton, CA (FUL), a 2597-mile flight over 17 U.S. states: Massachusetts, Rhode Island, Connecticut, New York, Pennsylvania, Ohio, Indiana, Illinois, Iowa, Missouri, Nebraska, Kansas, Colorado, Utah, Arizona, Nevada, California. 

Although the illustration may suggest otherwise, the flight path is “straight” insofar as it represents the shortest distance between the two airports.  Several state flyovers are very close calls, and the flight path over Lake Erie comes within yards of the Canadian border.  Nonetheless!  I challenge like-minded map nerds to come up with an 18-state solution, or a 17-state solution that is strictly overland (excepting rivers).

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