Jumbles and anagrams

“GORAC”

I’ve always liked doing those Jumble puzzles in the paper. And I’ve found them to be very easy.

CARGO

“TIDEF”

I don’t really set out to do them, actually. I see them somewhere, and I just idly look at them, and do them in my head while I’m standing there.

FETID

“NIROPS”

I might see one sitting on a nearby table at a coffee shop, and look at it upside-down while I’m stirring sugar into my coffee.

PRISON

“SHOIBY”

And, really, in the time it takes to put a little sugar in and snag a few paper napkins, I’ll have the thing done. My mind just happens to work that way, so that they mostly seem to fall right out for me. Not always, but more than 90% of the time.

BOYISH

Anagrams, though, are different. Here, we just have jumbled-up letters that you have to un-mix and turn into a word. But anagrams are all words — two words or phrases that make sense, and that have the same letters but in different orders.

The words “stop”, “spot”, “pots”, “post”, and “tops” are a set of anagrams.

And back in January 2008, BoingBoing showed us that “subtext” and “buttsex” are anagrams — they both fit as solutions to the jumble “euttsxb”. Who’d have imagined?

The best anagrams, riffing off the subtext of “buttsex”, are ones that have some sort of connection to each other. Preferably an amusing connection.

For example, “woman Hitler” is an anagram of “mother-in-law”. (Yes: punctuation and capitalization usually don’t count, and can be altered to suit.) In a similar vein, “he bugs Gore” is an anagram of “George Bush”.

And, while we’re on political figures, I’ll note that humourist Dave Barry pointed out, in one of his long-ago columns, that “Spiro Agnew” can be anagrammed into “grow a penis”. (OK, there’s no connection there: it’s just funny.)

These take much more creativity than just solving jumble puzzles does. And some anagrams that members of the National Puzzler’s League have come up with are quite different, involved, and truly amazing.

On the simple side, “beneath Chopin” becomes “the piano bench”, “moon starer” is “astronomer”, and “the desert oasis” “does ease thirst”.

Moving a step up, we have these:
I’d trust to these men in a scrape  =  the United States Marine Corps
oh, what stunning memento  =  the Washington Monument
theater to harmonious people  =  the Metropolitan Opera House
gem which cleans in a trice  =  electric washing machine
North hit a snag in career  =  the Iran-Contra hearings

One of my favourites:
inhaled, contrary to ban  =  tetrahydrocannabinol

One of the truly amazing ones:
can tell thee Hester hath worn an “A” bitterly  =  “The Scarlet Letter”, by Nathaniel Hawthorne

And, finally, one that’s absolutely unbelievable:

In one of the Bard’s best-thought-of tragedies, our insistent hero, Hamlet, queries on two fronts about how life turns rotten.

...is an anagram of...

To be or not to be; that is the question. Whether ’tis nobler in the mind to suffer the slings and arrows of outrageous fortune.

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The Jumble words early in this post are from the Jumble puzzle that was in the newspaper on 13 Jan 2010.

Infinity plus one

John Tierney has included a series of amusing puzzles, recently, in his TierneyLab blog in the New York Times. This week’s has an aspect that I’d like to talk about a bit. Here’s a shortened version of the puzzle, removing the cutesy “Russell Crowe” stuff (go look at the original for the full version):

You arrive at a hotel with an infinite number of rooms, and no vacancy. Yes, despite the infinite number of rooms, they’re all full. Still, the desk clerk does some wangling, and finds a room for you, without throwing any existing guests out nor making then share rooms. How did he do it?

I like this one in part because it touches on the sort of stuff I talked about in October, when I covered countable and uncountable sets and the concepts of cardinality and infinity.

The answer that many readers got depends upon the axiom of choice and the assumption that the number of rooms is countably infinite, but that works for the puzzle — and one can make a more rigorous solution for a more general case. So, given those additional assumptions, here’s one way to free up a room:

1. Because the rooms are countable and we have the axiom of choice, we can number the rooms with consecutive natural numbers starting at 1: 1, 2, 3, 4, ...
2. Ask each guest to change rooms, moving to the room with the next higher number. The guest in room n will move to room n+1.
3. Put the new guest in the now-available room 1.

It’s an amusing little demonstration of an aspect of infinite sets. It should be clear that we can repeat this a countably infinite number of times, thus fitting a countably infinite number of new guests into our already-full hotel.

Unfortunately, some people will understand it imperfectly and turn it into some attempt to do finite arithmetic on “infinity”, as commenter PolarBearNJ does (errors as in the original, but emphasis mine):

Well, this time I did some reading before answering...the puzzle is really a question of proving the following: n = n+1. Where N = infinity. How do we prove infinity plus one?

Some solutions suggest it means doubling up the guests temporarily, adding one guest to the first room, and the 2nd person leaves for the 2nd room, repeating this infinite room shifting...

Another solutions suggests it means moving the guest in the first room to the Hallway, and then moving into the 2nd room, forcing the 2nd guest into the Hallway, repeated to infinity...

Apparently, there is no answer that adequately proves n = n+1...but I would like to hear from the Math Wizards.

The problem with PolarBearNJ’s characterization is that “infinity”, as he’s thinking about it, is not a number; it’s a concept. It doesn’t make sense to talk about “infinity plus one” in the same way as “two plus three is five”. And not all “infinities” are the same, as we saw in my October series of posts.

But when we talk about cardinal numbers and the cardinality of sets, it becomes very clear, and it’s easy to see that the mapping f(n) := n + 1 defines a one-to-one correspondence between the set of natural numbers and the set of natural numbers except 1. Both sets have cardinality .

And we can define “addition” on cardinal numbers:
Let a be the cardinality of set A, and b be the cardinality of set B. Define a + b to be the cardinality of the set A ∪ B.

So if N is the set of natural numbers and M is the set of natural numbers except 1, our mapping above (f(n) := n + 1) shows that |M| = |N|. But since N = M ∪ {1}, that means that |M| = |M| + |{1}|, or = + 1.

Which is sort of what PolarBearNJ was getting at, but not strictly the same thing. We should avoid talking about “infinity” as though it were a number.

What colour is my hat?

In his TierneyLab blog in the New York Times, John Tierney recently posted an old puzzle, of a kind of which I’m not fond. It goes like this:

Three wise men are told to stand in a straight line, one in front of the other. A hat is put on each of their heads. They are told that each of these hats was selected from a group of five hats: two black hats and three white hats. The first man, standing at the front of the line, can’t see either of the men behind him or their hats. The second man, in the middle, can see only the first man and his hat. The last man, at the rear, can see both other men and their hats.

None of the men can see the hat on his own head. They are asked to deduce its color. Some time goes by as the wise men ponder the puzzle in silence. Finally the first one, at the front of the line, makes an announcement: “My hat is white.”

He is correct. How did he come to this conclusion?

The answer depends, as is conveyed in this version of the story by calling them “wise men”, on the assumption that the men are all ideal subjects for the puzzle — they they all use perfect reasoning:

1. If the third man, who can see both other hats, say two black hats, he would know that his own hat was white. Otherwise, he could not be sure — he’d have a 2/3 chance of having a white hat if he saw one black and one white in front of him, and a 2/3 chance of having a black hat if he saw two whites.
2. Because the third man said nothing after being given a moment to reason it, the others can assume that they do not both have black hats.
3. Now if the second man sees that the first has a black hat, he knows that his own is white, and should say so, since his reasoning is also perfect.
4. Because the second man also said nothing, the first man knows that he has a white hat.

I don’t like these kinds of puzzles because they require assumptions. You have to assume that the men are all perfect, that the second man, for instance, would understand the information the third man’s silence conveys. You have to assume that they have the right sense of how long to wait before making an inference from the others’ silence. What if the first man announced that his hat was white just as it dawned on the second man that the first man’s black hat and the third’s silence told him that his was white?

People’s opinions of these sorts of puzzles differ. Some feel that they show an ability to “think outside the box”. Some, like me, just consider them trick questions. Sometimes I get them, and sometimes I don’t... but I always dislike them, either way.

Like this one, which I remember from an old episode of All in the Family:

Archie: Edith, did I tell you about the guy who bowled three hundred and one?

Edith: Whaaaat?

Archie: Yeah, I know a guy who bowled three hundred and one.

Edith: But Archie, the most you can get is three hundred.

Archie: Well, this fella bowled three hundred and one.

Edith: Oh, no, Archie, that can’t be.

Archie: You don’t think so. Edith?

Edith: No, I don’t. It’s impossible.

Archie: Well... did you ever see anyone who bowled three hundred... and lost?