## 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.)

Anonymous said...

knew
new

does two & to count?

JasonZimba said...

Yes, the basic idea is to remove one or more letters from a word to produce a homophone of it. (Or to add one or more letters to a word to produce a homophone of it.) So TWO works because striking the W leads to a homophone, TO. For that matter, TOO and TO are also examples!

Anonymous said...

belle
bell

know
no

knot
not

bee
be

Tambu said...

Streambeds
Streambed
Steamed
Teamed
Tamed
Tame
Tam
Am
A

Reid said...

From the group...

6. probability of winning = probability of losing 1,000,000 times in a row

1/2 > (1-1/2)^1,000,000

1/1,000,000 < (1-1/1,000,000)^1,000,000

Since the right side is a polynomial function of the left side, it's continuous, and the two sides must have been equal when 1/1,000,000 < lhs < 1/2

So the probability of winning is greater than 1/1,000,000

danimal said...

Nice to see you back on your blogspot! Tried puzzles 1-3.

For #1, most of the pairs I found actually were also anagrams of each other:

meet and mete
break and brake
hide and hied
pride and pried
great and great

Maybe the term "isophone" should be used for these pairs, since they have this relationship ("sounds like" and "contains") in both directions?

Also found :
banned-band
maize-maze

A triple play!:
Scent - sent - cent

JasonZimba said...

scent-sent-cent is nice!

Here is the list I had in my notebook when I posted the puzzle:

buss/bus
canvass/canvas
cell/cel
bee/be
bann/ban
too/to
fore/for
ore/or
jamb/jam
lamb/lam
plumb/plum
tore/tor
torte/tort
inn/in
two/to
too/to
whole/hole
llama/lama
butt/but
knew/new
whit/wit
know/no