1.) Jo announces: "I have more than 999 books." Jean says: "No, Jo!
You have fewer than 1000 books." Mary says: "Jo has at least 1 book."
Only one of these statements is true. How many books does Jo own?

2.) A stranger comes to a mountain inn and tells the owner tht he must
stay & wait for a friend who will come sometime within the next 21
days. Unfortunately, he says, he has no money. He produces, however, a
chain of exactly 21 gold links and convinces the innkeeper that th
elinks are of the purest gold. He offers to give in payment 1 link per
day. The innkeeper agrees that a gold link is easily worth a day's
food and lodging, but only if the link is in tact. A cut link, he
says, is nowhere nearly as valuable as an intact link. He does not
want to end up with 21 cut links. The visitor then works out an
arrangement where, giving the innkeeper one more link each day, he
will cut the minimum number of links. What is the smallest number of
links he must cut? (Note: "Swapping" may take place; for example, on
the 10th day the guest may exchange a string of 10 links for the 9
links the innkeeper is holding).

3.)Nine men and two boys, trekking through the jungle, need to cross a
river. The have a small inflatable boat and it's easy enough to row it
across the river. The boat, however, can hold no more than one man and
two boys. How can they all get acrss? (Hint: Suppose there was only
one man and two boys).

============
 PROBLEM #1
============

Jo has zero books. 

If the first statement (Jo has more than 999 books) were true, then
the third statement (Jo has at least one book) would be true. This
cannot be, since there can be only one true statement.

If the third statement (Jo has at least one book) were true, the
second statement (Jo has fewer than 1000 books) would also be true.
This cannot be, since there can be only one true statement.

Thus we are left with the second statement (Jo has fewer than 1000
books), which can be true only if Jo has zero books.

============
 PROBLEM #2
============

You can do this with two cuts: cut the fourth link and the tenth link. 

Your 21-link chain now looks like this, with 0 representing intact
links, and C representing cut links:

000 C 00000 C 00000000000

Here's how you can arrange your payments to the innkeeper:

On the first day. give the innkeeper one of the cut links.

On the second day, let the innkeeper keep the cut link, and give him
the other cut link, so that he now holds the two cut links.

On the third day, take back both of the cut links, and give the
innkeeper the first segment of the chain, which contains three links.

On the fourth day, let the innkeeper keep the 3-link segment, and add
to it one of the cut links.

On the fifth day, take back everything from the innkeeper and
substitute the 5-link segment of the chain.

On the sixth day, let the innkeeper keep the 5-link segment, and add
one of the cut links.

On the seventh day, let the innkeeper keep the 5-link segment and the
cut link, and give him the other cut link.

On the eighth day, let the innkeeper keep the 5-link segment; swap the
3-link segment for the two cut links.

On the ninth day, let the innkeeper keep the 5-link and 3-link
segments, and add one cut link.

On the tenth day, let the innkeeper keep the 5-link and 3-link
segments and the cut link, and add the other cut link.

On the eleventh day, take everything back and give the innkeeper the
11-link segment.

On the twelfth day, let the innkeeper keep the 11-link segment, and
add one cut link.

On the thirteenth day, let the innkeeper keep the 11-link segment and
the cut lnk, and add the other cut link.

On the fourteenth day, let the innkeeper keep the 11-link segment;
take back both cut links, and give the innkeeper the 3-link segment.

On the fifteenth day, let the innkeeper keep the 11-link segment and
the 3-link segment, and add one cut link.

On the sixteenth day, let the innkeeper keep the 11-link segment. Take
back the 3-link segment and the cut link, and give the innkeeper the
5-link segment.

On the seventeeth day, let the innkeeper keep the 11-link segment and
the 5-link segment, and give the innkeeper one cut link.

On the eighteenth day, let the innkeeper keep the 11-link segment, the
5-link segment, and the cut link, and give him the other cut link.

On the nineteenth day, let the innkeeper keep the 11-link segment and
the five-link segment; take back both cut links, and give him the
3-link segment.

On the twentieth day, let the innkeeper keep the 11-link segment, the
five-link segment, and the 3-link segment, and add one cut link.

On the twenty-first day, let the innkeeper keep the 11-link segment,
the five-link segment, the 3-link segment, and the cut link; add the
other cut link. The innkeeper now has the entire chain.

============
 PROBLEM #3
============

2 boys go across.

1 boy comes back.

1 man goes across.

The other boy comes back.

Repeat this till all 9 of the men have gone across. Then both boys
come across together, and everyone has crossed the river.

======================================================================

If anything is unclear, or if I have misunderstood any of these
puzzles, please request clarification; I'll gladly offer further
assistance before you rate my answer.

Best regards,
pinkfreud

1. 

Mary's statement is saying Jo's books >= 1.
Jo is saying Jo's books > 999
Jean is saying Jo's books < 1000

If Jo's is true, then Mary's must also be true, and vice versa, so
since only one is true, the must both be false
So Jean must be telling the truth.
So books < 1000
and books < 1.

So Jo has 0 books.

------------------

3.
 
2 boys ride across
1 comes back.
1 man rides over.
The other boy comes back.
Then the process is repeated until all 9 men are across, then both
boys ride together, and everyone is across.

Couldn't figure out number 2. 

-Sachit

(1)  If Jo's is the truth, then Mary's is true, so it can't be either
Jo's  or Mary's statement.
If Jean's statement is true, Jo must have zero books, so that Mary's
statement is not true.
Answer: Jean's statement is the only true one.

(2)  The stranger gives the chain to the inn owner, (on deposit), and
the chain is cut on the day the stranger's friend arrives, according
to the days the stranger has stayed at the inn.  The non-paid part of
the chain is returned to the stranger.

(3)  One man and two boys row across the river.  One, (or two), of the
boys row back across the river , and one man and the boy(s) row across
the river.  This is repeated seven times.

For #3, Is it one man OR two boys or one man AND two boys?
If the former, the previous comment is correct. If the latter, just
have one boy do all the paddling.

For #2, he can make 4 cuts: 1 link, 2 links, 4 links, 8 links and 2
links. As I read it, he can't get a full 21 days anyway because he
*has* to cut. Binary-wise, up to 15 days, and then he can get 17.