Back in 1971, these three people go to a hotel and to share a room. The manager tells them the room is $30, so they each give him $10. When they get to their hotel room, the manager drops by to explain that a special $25 rate is in place, and returns $5 in ones to them. They each take a dollar, and give back the remaining $2 to the manager for his honesty.

Now, if each of the three people paid $10 and got $1 back, they paid $9 a piece, or a total of $27 (3 × 9). The manager gets $2, so that adds up to $29.

Where is the other dollar?

The issue here is how the money is counted. When you subtract $1 from each persons $10, you have accounted only for the $3 returned by the manager. When you multiply each persons $9 (10 - 1) you get $27, whichs accounts for the $3 returned by the manager, but is $2 more than the $25 paid to the hotel. The $2 is, of course, with the manager, so if you subtract the $2 from the $27, you get $25, which accurately accounts for the money.

In order to understand this process, you must first have some basic understanding of algebraic concepts. Consider the following equation: (X+Y)(X-Y) = X(X-Y) + Y(X-Y) = X²-XY+YX-Y² = X²-Y²

Step		Process
Let X=Y		X=Y
Multiply both sides by X		X2=XY
Subtract Y2		X2-Y2=XY-Y2
Factor out		(X+Y)(X-Y)=Y(X-Y)
Divide by (X-Y)		X+Y=Y
Replace Y with X		X+X=X
Simplify		2X=X
Divide by X		2=1

As in any theorum, each step must be valid. In this case, the error occurs when you divide by (X - Y). Since it is stated that X = Y, then X - Y = 0. Dividing by zero produced an undefined quantity, thus invalidating the theorum.

We learned early in school that we can't divide by zero

Think of three words ending in gry.
Angry and hungry are two of them.
There are only three words in the English language.
What is the third word?,
The word is something everyone uses every day.
If you have read carefully,
I have already told you what it is.

What is the word?

This is the ultimate example of misdirection. Here you are led to believe that the answer is a word that ends in GRY. Although there are some obscure words that do (e.g. augry, a variant of augury), the issue is different. On the third line, you are told there are three words in "The English Language", however, the author of this puzzle conveniently left out the quotes. The quotes would have shown us that the focus of the puzzle had changed from the -GRY words to the phrase "The English Language".

The third word in "The English Language" is "Language", duh!

Count all the squares on a standard checkerboard. (English, not Chinese).

What is the amount?

Yes, there are 64 single color squares on a checkerboard, but there are also:

• 49 groups of 4 squares
• 36 groups of 9 squares
• 25 groups of 16 squares
• 16 groups of 25 squares
• 9 groups of 36 squares
• 4 groups of 49 squares
• 1 group of 64 squares

That makes 204 squares on a checkerboard. Games Magazine recently published a puzzle in which the answer was found by knowing how many squares on a checkerboard. The editors conceded that the question was ambiguous.