Puzzles and Problems

Here is a list of great problems and puzzles. Some famous, some just fun to figure out. Many take advantage of popular math mistakes. Back to Mathmistakes.com

Arithmetic Geometry Algebra Logic

Arithmetic Problems:

1. Write equations that equal the numbers 0 to 10 using exactly four 4's. You can use operations +, -, ×, ÷, or ( ). No other numbers are allowed. For example, ( 4 + 4 ) - ( 4 + 4 ) = 0 or 4/4 - 4/4 = 0 or 44 - 44 = 0

2. You need exactly four gallons of water, yet you only have two buckets: one five gallons and the other seven. How do you measure out four? Can you measure every number of gallons from one to twelve? How?

3. Three business executives eat lunch in a restaurant. They estimate that the bill should come to $30. They split the bill 3 ways and pay 10 dollars each. When the actual bill comes, it is $25. Since this is not divisible by three, they each take a dollar back and leave the extra two dollars tip for the waiter. Since each paid nine dollars and nine times three is 27, plus two for the waiter is 29. Where did the other dollar go?

4. A fruit stand manager has two piles of oranges from two different suppliers. One supplier wants $10 for thirty oranges, or three for a dollar. The other wants $15 for 30 oranges for, or two for a dollar. The manager decides to sell all sixty at five for two dollars. After selling twelve batches of five the manager only has $24, but he needed $25 dollars for the suppliers. Where is the other dollar?

5. On a long horseback trip through the Arizona desert, three horsemen start out with 7 one gallon canteens of water each. Half way through the trip they inventory their supply. They find seven full canteens, seven empty canteens, and seven half full. How do you divide the canteens so that each rider has the same amount of canteens and water for each? (You cannot transfer water for fear of spilling).

6. A Race track is one mile long, if a driver goes around once at 30 miles per hour, how fast must he drive in the second lap to average sixty miles per hour over both laps?

7. A man buys 9 gallons of paint, he can only carry 2 at a time to the car. How many trips from the store to the car does he have to make?

8. A train leaves Phoenix bound for Tucson at 60 miles an hour. Another train leaves Tucson bound for Phoenix 10 minutes later traveling at 50 miles per hour. Given that it is 110 miles in between, how far apart will the two trains be when they meet?

9. A bookworm lies on page 1 volume 1 of a two volume dictionary. He eats his way to the last page of volume 2. The pages of each book measure 2 inches thick, and the binding of each volume is 1/8th of an inch thick. How far did the bookworm go?

Geometry

1. A box that is 6 inches in length, 5 inches in width and 6 inches high. There is a spider in the top right front corner. There is a fly in the opposite bottom left rear corner. What is the shortest path necessary for the spider to walk, and how far will it walk?

2. Another spider walked straight for six inches, made a 90 degree turn to the right walked six more inches, made another 90 degree turn to the right, walked another six inches and ended up exactly where he started. How can this be?

3. Figure 1 is divided into 3 equal and identical shapes. Figure 2 is divided into four equal and identical shapes. Can you divide figure 3 into five equal and identical shapes?

Answer

4. Line AB||CD
Ð ABE = 40°
Ð BED = 70°
Ð CDE = ?

Answer

5. Given: Ð BAC = 20°
Ð ABC = Ð ACB
Ð ABE = 20°
Ð DCA = 30°
Ð BED = ?
And do it without Trigonometry!
Answer

6. Submitted by Bastiaan Scheppers:

7. A cattle farmer has 1000 meters of fencing and wants to fence in the largest pasture he can. He gives the problem to his two children, Bobby and Cindy. Bobby's solution is to use the rectangular formula A = lw, and find the largest A where 2l + 2w = 1000. a.) What are the dimensions of Bobby's pasture? b.) Cindy's more brilliant solution results in a pasture that is over 25% larger, what did she do?

8. How many squares can you draw connecting the dots below:

Algebra

1. Here is a challenging systems of equations problem to waste your time. Find a positive or negative number for each letter such that O + N + E = 1, T + W + O = 2, T + H + R + E + E = 3, etc. up to T + E + N = 10. Note: it is possible to go from ZERO to TWELVE.

2. You are on an edge of a thousand mile long desert with a camel and three thousand cobs of corn. The camel can carry one thousand cobs, but each mile you travel, the camel needs to eat one cob of corn. What is the highest number of cobs you can get to the other side and how?

3. Solve for x:

4.

5.

Logic

1. Three boxes are presented to you. One contains gold, the other two are empty. Each box has imprinted a clue as to its contents again only one message is telling the truth the other two are lying. Which box has the gold.

The gold is not here

The gold is not here

The gold is in the second box

2. Raymond Smullyan has made a career of writing logic problems. His most famous have to do with a mythical island of Knights and Knaves. Knights always tell the truth, and Knaves always lie.

As you approach the island, you spot three inhabitants on the shore. You call out to them, "Are you Knights or Knaves?" The first says something but you do not hear what he says, so you ask, "What did you say?" The second inhabitant says, "He says he is a Knight, he is and so am I." The third responds, "He is a Knave, but I am a Knight." What are the three inhabitants really?

3. In his book, What is the Name of This Book?, Smullyan relates the following story:

"Once when I visited the Island of Knights and Knaves, I came across two of the inhabitants resting under a tree, I asked one of them, 'Is either of you a Knight?' He responded and I knew the answer to my question."

What are the two inhabitants, really?

4. Victor, Virgil, Vincent, Vito, and Vance are brothers performing in a five man traveling circus known as the little big top. They are in no particular order, a clown, a juggler, an acrobat, a magician, and a strong man. Whenever they perform, their acts appear in the same order. The clown comes after Victor and Vito, but before the magician. The acrobat comes on third. Neither the strong man nor Vincent is the first or the last to perform. Virgil, Victor, and Vito perform in that order.

Which brother does what act, and in what order do they perform?

5. Five houses of different colors are in a row. Each is owned by a man with a different nationality, hobby, pet, and favorite drink. The Englishman lives in a red house, the Spaniard owns dogs, coffee is drunk in the green house, the Ukrainian drinks tea, the green house is directly to the right of the white one, the stamp collector owns snails, the antique collector lives in the yellow house, the man in the middle house drinks milk, the Norwegian lives in the first house, the man who sings lives next to the man with the fox, the man who gardens drinks juice, the antique collector lives next to the man with the horse, the Japanese man's hobby is cooking, and the Norwegian lives next to the blue house.

Who drinks water, and who owns a zebra?

6. Three siblings, Alice, Bob and Carol, truthfully report their grades to their parents as follows: Alice says, " If I passed Math, then so did Bob. I passed English if and only if Carol did. Each of us passed at least one subject."
Bob says, "If I passed Math than so did Alice. Alice did not pass History. Each subject was passed by at least one of us three."
Carol says, "Either Alice passed History or I did not pass it. If Bob did not pass English then neither did Alice. I did not pass as many subjects as Bob or Alice."

Which subjects did they each pass?

7. A chief of detectives had three likely candidates for an opening in his department. To test their powers of reasoning, he pulled out a red and black marker and told them, "I am going to either make a red or black mark on each of your foreheads. At least one will be black. Without any help other than your own reasoning skills, I want you to determine the color of the mark on your own forehead. The first to do this and give me a satisfactory explanation at how you arrived at a conclusion will get the job."
He then blindfolded the candidates and proceeded to put a black mark on each of their foreheads. After he removed the blindfolds, the three stared at each other for a few seconds, each seeing that the other two marks were black, then one of the candidates said, "I have a black mark."

How did the candidate arrive at this conclusion?