## Saturday, April 26, 2008

### Some puzzles for fun

1) Homophones are words such as main and mane that are spelled differently but pronounced the same. Recently I decided that the coolest homophones are surely those that contain their own homophones. (Maybe call them "isophones"?) An example would be buss, which contains its homophone bus. How many examples like this can you think of?

2) A quickie. Starting with the ten-letter word STREAMBEDS, remove a letter to yield a nine-letter word. Then remove another letter to yield an eight-letter word. Continue in this way until you have a one-letter word.

3) The same as above, starting with INSOLATING.

4) Say that a letter is "special" if it can be used together with the letters A, F, and W to form a four-letter word. (You are allowed to rearrange the letters.) Find all of the letters that are "special" in this way. Then rearrange the special letters to form a new word.

5) The same as above, using the letters I, C, and L in place of the letters A, F, and W.

6) (Some math knowledge is required for this one.) I'm thinking of a game with a curious feature. The probability of winning the game once is exactly the same as the probability of losing the game a million times in a row. Are the chances of winning this game one in a million, greater than one in a million, or less than one in a million?

(Math wizards may wish to show that if "a million" is replaced by a large number N, then the probability of winning is given approximately by W(N)/N, where W is the Lambert W-function.)

7) This one is based on the Paradox of the Liar (background here). The classic paradox is the sentence "This sentence is false." The sentence is paradoxical because we cannot assign it a truth value that respects its meaning. Also well-known is the paradoxical pair of sentences, ("The next sentence is true", "The previous sentence is false.") Again, there is no way to assign truth values to the sentences in a way that respects their meaning.

One day a couple of years ago, it occurred to me that these two famous examples can be seen as instances of a general pattern. We consider an ordered n-tuple of sentences (S1, S2, ..., Sn) in which each sentence asserts the truth or falsehood of the next. The last sentence closes the loop by asserting the truth or falsehood of the first. Under what circumstances is such a collection of sentences paradoxical?

As a warmup exercise, you might want to verify that the following pair of sentences is not paradoxical: ("The next sentence is false", "The previous sentence is false.") There is no paradox here because we can read the first sentence as true and the second sentence as false; this reading respects the meaning of the two sentences.

(For another take on this, see my vanity-published puzzle book Word Puzzles for the Seriously Smart.)