# MATHEMATICAL PUZZLES FOR FUN AND ENLIGHTENMENT

PUZZLE #1: THREE MEN FACING A WALL (an easy puzzle to get started with)

Three men form a line facing a wall. While their eyes are closed, a hat is placed on each of their heads. The men know that the hats were selected from a barrel that originally contained 3 white hats and 2 black hats. The man at the back of the line, who can see only the hats on the two men ahead of him, says that he does not know what color hat he is wearing. The man in the middle, who can see only the hat of the man at the front of the line, then says that he does not know what color hat he is wearing either. Finally, the man in the front, who can see no hats, says that he knows what color hat he is wearing.

PUZZLE #2: BAGS OF GOLD (a slightly harder puzzle than the first one)

All 10 Sheiks in the Sultan's dominion have delivered their annual tribute, bags of one ounce gold coins. The Sultan has learned through an informant that one of the Sheiks has cheated him by shaving off a tenth of an ounce of gold from each coin in the bag he gave the Sultan. However, the informant didn't know which Sheik was the culprit. The Sultan wants to identify the cheater while the public executioner is still in town, but the Sultan has only an old drug store scale available to weigh coins. The scale is very accurate, but it requires the deposit of a U.S. penny for each weighing, and the Sultan has only one penny.

How can the Sultan determine which of the Sheiks have cheated him, using the drug store scale to provide a single weight reading? Click here for the answer.

PUZZLE #3: THE DOOR TO FREEDOM (an example of a popular type of logic puzzle)

You have been knocked unconscious and kidnapped. You wake up in a bare room that contains two identical doors, each guarded by a grim silent guard. A voice on a loudspeaker explains that one of these doors leads to freedom, the other to instant death. You are told that one of the guards will always tell the truth and the other will always lie, but you are not informed which guard is the liar. You will be permitted to select a guard and ask him only one question that requires a yes or no answer. After receiving the answer to this one question, you must select a door and meet your fate.

PUZZLE #4: SNOW PLOW PROBLEM (you need fist year calculus to solve this problem; believe it or not, the information provided below is sufficient)

It starts snowing some time before 10:00 in the morning, and it continues to snow all day at a constant rate. A snowplow leaves the county garage and begins to plow the main highway at 10:00 am. At 11:00 am a second plow leaves from the same place to plow the same road. At noon, a third snow plow leaves from the same place to plow again the same road. Each plow is identical, and each moves at a rate inversely proportional to the depth of the snow. Some time in the afternoon, the 2nd plow catches up to the first one. At the same moment, the third plow also catches up to the other two.

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PROBLEM #1 SOLUTION

The last man would only know which color hat he was wearing if both of the men ahead of him where wearing black hats, in which case all the hats he couldn't see, including the one on his head, would be white. Since he didn't know his hat color, the other men were not both wearing black hats. The second man would know his hat color only if the man in front of the line was wearing black, in which case the second man would have to have a white hat (i.e. not both black). Since the second man did not know his hat color, the first man knew he was wearing white.

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PROBLEM #2 SOLUTION

Assign each Sheik a number from 1 to 10. From Sheik #1's bag take one coin, from #2's two coins, etc. Place all 55 coins that were selected on the scale and obtain the weight. If the total weight is 1/10 ounce short of 55 ounces, bag #1 has the light coins; if the weight is 2/10 ounces short, bag #2 has the light coins; etc.

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PROBLEM #3 SOLUTION

Ask either guard, "If I asked the other guy if this door you are guarding is the door to freedom, would he say yes?" If you get a "no" answer, it is safe to use the door. Here's why:

• You might have talked to the LIAR guarding the door to FREEDOM. If so, you would get a "no" answer, since the other guard, the truth-teller, would say yes, so the liar you are talking to would say no.
• If you were talking to the TRUTH-TELLER guarding the door to FREEDOM, you would also get a "no" answer, since the other guard, the liar would say "no" and the guard you were talking to would truthfully report this to you.
• If you were talking to the LIAR guarding the door to DEATH, you would get a "yes" answer.
• If you were talking to the TRUTH-TELLER guarding the door to DEATH, you would also get a "yes" answer.
There are no other possibilities.

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