# Silly U.S. Map Puzzles #4a and 4b

First, for puzzle #4a, what are the meanings of the colors on this map?

For puzzle #4b, what do the colors mean on this second, similar map?

To find the answers, simply scroll down.

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Solution:

In the first map, consider the number of letters in the name of each state. Is this number prime or composite?

In the second map, consider the number of characters, rather than letters, in each state’s name. This number is different for states with two-word names, due to the single character, a blank space, needed to separate the two words. Again: prime, or composite?

# Silly U.S. Map Puzzle #3

What is represented by the colors on this map?

The answer may be found by scrolling down.

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Keep scrolling….

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Do any of the borders of this state contain squiggles? (Note: if you think New Mexico is the wrong color, check the part of that state which borders El Paso, Texas.)

# Silly U.S. Map Puzzle #2

What is represented by the colors on this map?

If you give up, you can scroll down to find the answer.

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Keep scrolling….

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Answer: the colors show whether the name of each state starts with a letter in the first, or second, half of the alphabet.

# Silly U.S. Map Puzzle #1

What is represented by the colors on this map?

If you decide to give up, you can scroll down for the answer . . . but, I promise, the solution to this puzzle is extremely simple.

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Keep scrolling, if you’re looking for the answer….

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The map shows how many words are in the name of each state.

# A Chiral Solution to the Zome Cryptocube Puzzle

This is my second solution to the Zome Cryptocube puzzle. In this puzzle, you start with a black cube, build a white, symmetrical, aethetically-pleasing geometrical structure which incorporates it, and then, finally, remove the cube. In addition, I added a rule of my own, this time around: I wanted a solution which is chiral; that is, it exists in left- and right-handed forms.

It took a long time, but I finally found such a chiral solution, one with tetrahedral symmetry. Above, it appears with the original black cube; below, you can see what it looks like without the black cube’s edges.

# The Second of Dave Smith’s “Bowtie” Polyhedral Discoveries, and Related Polyhedra

Dave Smith discovered the polyhedron in the last post here, shown below, with the faces hidden, to reveal how the edges appear on the back side of the figure, as it rotates. (Other views of it may be found here.)

So far, all of Smith’s “bowtie” polyhedral discoveries have been convex, and have had only two types of face: regular polygons, plus isosceles trapezoids with three equal edge lengths — a length which is in the golden ratio with that of the fourth side, which is the shorter base.

He also found another solid: the second of Smith’s polyhedral discoveries in the class of bowtie symmetrohedra. In it, each of the four pentagonal faces of the original discovery is augmented by a pentagonal pyramid which uses equilateral triangles as its lateral faces. Here is Smith’s original model of this figure, in which the trapezoids are invisible. (My guess is that these first models, pictures of which Dave e-mailed to me, were built with Polydrons, or perhaps Jovotoys.)

With Stella 4d (available here), the program I use to make all the rotating geometrical pictures on this blog, I was able to create a version (by modifying the one created by via collaboration between five people, as described in the last post) of this interesting icositetrahedron which shows all four trapezoidal faces, as well as the twenty triangles.

Here is another view: trapezoids rendered invisible again, and triangles in “rainbow color” mode.

It is difficult to find linkages between the tetrapentagonal octahedron Smith found, and other named polyhedra (meaning  I haven’t yet figured out how), but this is not the case with this interesting icositetrahedron Smith found. With some direct, Stella-aided polyhedron-manipulation, and a bit of research, I was able to find one of the Johnson solids which is isomorphic to Smith’s icositetrahedral discovery. In this figure (J90, the disphenocingulum), the trapezoids of this icositetrahedron are replaced by squares. In the pyramids, the triangles do retain regularity, but, to do so, the pentagonal base of each pyramid is forced to become noncoplanar. This can be difficult to see, however, for the now-skewed bases of these four pyramids are hidden inside the figure.

Both of these solids Smith found, so far (I am confident that more await discovery, by him or by others) are also golden polyhedra, in the sense that they have two edge lengths, and these edge lengths are in the golden ratio. The first such polyhedron I found was the golden icosahedron, but there are many more — for example, there is more than one way to distort the edge lengths of a tetrahedron to make golden tetrahedra.

To my knowledge, no ones knows how many golden polyhedra exist, for they have not been enumerated, nor has it even been proven, nor disproven, that their number is finite. At this point, we simply do not know . . . and that is a good way to define areas in mathematics in which new work remains to be done. A related definition is one for a mathematician: a creature who cannot resist a good puzzle.

# My First Solution to the Zome Cryptocube Puzzle, with Special Guest Appearances by Jynx the Kitten

Last month, in a special Christmas promotion, the Zometool company (www.zometool.com) briefly sold a new kit (which will return later) — a fascinating game, or puzzle, called the “Cryptocube.” Zome usually comes in a variety of colors, with each color having mathematical significance, but the Cryptocube is produced in black and white, which actually (in my opinion) makes it a better puzzle. Here’s how the Crypocube challenge works:  you use the black parts to make a simple cube, and then use the smaller white parts to invent a structure which incorporates the cube, is symmetrical, is attractive, and can survive having the twelve black cube-edges removed, leaving only the cube’s eight black vertices in place. I had a lot of fun making my first Cryptocube, and photographed it from several angles.

If this was built using standard Zome colors, the round white figure inside the cube, a rhombic triacontahedron, would be red, and the pieces outside the cube, as well as those joining the rhombic triacontahedron to the cube (from inside the cube), would be yellow.

It isn’t only humans who like Zome, by the way. Jynx the Kitten had to get in on this!

Jynx quickly became distracted from the Cryptocube by another puzzle, though: he wanted to figure out how to pull down the red sheet I had attached to the wall, as a photographic backdrop for the Cryptocube. Jynx takes his feline duties as an agent of entropy quite seriously.

As usually happens, Jynx won (in his never-ending struggle to interfere with whatever I’m doing, in this case by pulling the sheet down) and it took me quite a while to get the red sheet back up, in order to take kitten-free pictures of my Cryptocube solution, after removal of the black cube’s edges.

Here’s the view from another angle.

The Cryptocube will be back, available on the Zometool website, later in 2015. In the meantime, I have advice for anyone not yet familiar with Zome, but who wants to try the Cryptocube when it returns: go ahead and get some Zome now, at the link above, in the standard colors (red, blue, and yellow, plus green in advanced kits), and have fun building things with it over the next few months. The reason to do this, before attempting to solve the Crypocube, is simple: the colors help you learn how the Zome system works, which is important before trying to solve a Zome puzzle without these colors visible. After gaining some familiarity with the differing shapes of the red, blue, yellow, and green pieces, working with them in white becomes much easier.

On a related note, Zome was recommended by Time magazine, using the words “Zometool will make your kids smarter,” as one of the 14 best toys of 2013. I give Zome my own strong, personal recommendation as well, and, as a teacher who uses my own Zome collection in class, for instructional purposes, I can attest that Time‘s 2013 statement about Zome is absolutely correct. Zome is definitely a winner!

# North American Geographical Oddity

You’re standing on the mainland of North America — not on an island. From where you are, you can travel due East, and you’ll come to the Pacific Ocean. If you travel due West, however, you will come to the Atlantic Ocean. What’s more, this is true for a relatively large percentage of locations in the country where you are located — a greater percentage than would be the case for any other country on the North American continent, if there even are others.

In what country are you standing?

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You’re in Panama! Now, before anyone protests that Panama is in Central America, not North America, let me point out that Central America is part of the North American continent, just as Europe and India are part of the Eurasian continent. (Yes, I looked them up.)

# A Number Theory Puzzle Involving Primes, Perfect Numbers, and “Paraperfect” Numbers

There is something about the definitions of prime and perfect numbers that always struck me as rather odd. Prime numbers are those which no factors other than themselves, and one. Perfect numbers, on the other hand, equal the sum of all of their factors, excluding themselves, but including one. The first two examples of perfect numbers are 6 (which equals 1 + 2 + 3) and 28 (which equals 1 + 2 + 4 + 7 + 14). Perfect numbers are far more rare than primes.

The thing I find annoying is the exclusion of one, as a factor, from one of these definitions, but not the other. I therefore decided to give a name to a new type of number:  one which equals the sum of its factors, excluding itself AND the number one. The first name I thought of, “exceptional numbers,” turns out to have already been taken, so I thought of another, and called these numbers “paraperfect numbers” instead.

Having done that, it was time to start searching for them. I have a reasonably fast mental calculator, but it didn’t take long to figure out that I wasn’t up to this task, so I wrote this program to search for paraperfect numbers:

It’s written in BASIC, an archaic computer language I learned in high school, and, as you can see, I am a horribly sloppy programmer. A better programmer would have written a program for the same purpose, but with only about half this length. Be that as it may, though, the program does work. I’ve had it running for a few minutes now.

It’s gotten past 22,000 — and has found no paraperfect numbers at all. This is not what I expected . . . and now I am wondering if any exist. Right now, of course, “no paraperfect numbers exist” is a mere conjecture. If I can prove it, it will be a theorem. However, I don’t know nearly enough about number theory to write such a proof.

I could use some help. If anyone does find a paraperfect number, please leave a comment on this post identifying your find. If anyone can prove — or simply explain to me — why there are no paraperfect numbers, if that is the case, please let me know that as well.

The program is still running, and has now passed 25,000 without a paraperfect-number find. I guess I’ll leave it running for a while. Any help with this puzzle would be much appreciated.

[Later:  see the comments for the rest of the story on these elusive numbers.]

# My Favorite Passage from the Bible, and How One Atheist Thinks We Just Might Use It to Avoid Extinction.

You may already know I am an atheist, and may be unaware that some of us have favorite passages from the Bible which were not selected for purposes of ridicule, nor of criticism of the Bible, nor because of dislike of any religion. This is my favorite passage because it contains excellent advice. I do not need “faith,” as that word is commonly understood, nor a literal belief in the devil, to recognize, and appreciate, good advice.

What’s not to like about self-control? Or being alert? Those things can keep us all alive. They are important. I used to only cite the first sentence here as my favorite part of the Bible, but have decided to include two complete verses, for context, and elaboration through metaphor, as I interpret this passage. I see no reason not to.

Atheists (only capitalized at the beginning of a sentence, by the way) don’t have denominations, nor creeds, and there are as many different types of atheist as there are atheists. Atheism isn’t a religion — the word simply describes existence without religion. Everyone is born an atheist, albeit an unconscious one. Also, those who remain, or return to, atheism, change, during the course of our lives, just as theists do. The only people who do not change are the dead.

In defiance of stereotype, we are not all angry and bitter, although some of us, it must be admitted, are. (I used to be far more bitter than I am now, although I am working hard to change that.) Many of us even believe in non-theistic ideas which make absolutely no sense, such as, for example, 9/11 conspiracy theories. We only have one thing in common: we lack belief in deities. You almost certainly lack belief in at least some deities, ones which others fervently believe in. If you are a theist, well, atheists just take things a bit further than you — that’s all. We don’t all hate theists, and (thankfully) not all theists hate us. The ability to respectfully disagree is at least one of the keys to peaceful coexistence. Universal agreement among humans simply will not happen (and would be horribly boring, anyway), until the death of the penultimate person, at least. Even if there is a “last person alive” scenario in the (hopefully very distant) future, this unknown last human being will still have internal disagreements, and will almost certainly disagree with remembered ideas of the dead. In fact, given human nature, and history, such a disagreement might even be the cause of the next-to-last person’s death, at the hands of the last man, or woman, ever to live.

I do not want homo sapiens to end this way.  I’d like us to continue, for many generations, until evolution, and speciation, replace us with successor species, a long time from now — still people, but different, in ways we cannot now know, and, hopefully, people who have long ago learned to live without constantly killing each other.  Isn’t it about time we left this nasty habit called “war” behind, along with murder, rape, and the rest of the litany of human horror?

I’m a big fan of John Lennon, but I’d far rather imagine no war than “imagine no religion,” and I no longer accept the idea, common among atheists, that the second is a prerequisite for the first.

Since we have, as a species, figured out several ways to self-destruct, we cannot afford to wait for evolution to “teach” us how to coexist peacefully.  Evolution is far more efficient at destruction than creation, after all, being a random process.  Far more species have gone extinct than exist today, and the process of evolution simply does not care whether we live or die.  Entropy happens.  It took 3.85 billion years of natural selection to get here, and we will not get a second chance to get it right.

We must figure out effective ways to live with our differences now.  I do not mean that we should somehow erase our differences, for I have no desire to live in a world of clones of myself, and I doubt you want to live in your version of such a world, either.  We do, however, need to come to terms, as a world-wide society, with the inescapable fact that people are different.  We have a right to be different, it’s good that we are, and the fact that we vary so much is certainly is no excuse for killing, nor even hating, anyone.

There is another part of human nature that is on our side in our struggle for survival, and this is the hopeful part of this essay. We are good at figuring things out. We actually enjoy trying our best to solve puzzles. We pay hard-earned money for them constantly! Some of us absolutely obsess over single problems, for days — or years — at a time. Well, this is the best, most important problem we have ever faced, with the highest stakes imaginable:  how to avoid our own extinction. The world isn’t a casino with no exit, though.  It has been mostly a game of chance, so far — and we’ve been lucky to have made it to the present.  However, it doesn’t have to be the way it has been, with us stumbling through history, like drunk monkeys in a minefield — which we pretty much are, right now.

We have minds, and it’s time to use them. We can stop playing roulette, especially the Russian variety, and sit down at the table to play chess, instead. We can figure this out.

If this Big Problem isn’t solved soon, though, there may not be a long wait for extinction.  It could very well be later than you think.  Therefore, I encourage everyone to, in the words of the Bible, “Be self-controlled and alert.” That’s a good place to start.