New Problems

Brain. Work at Friedey's restaurant sucks! Long hours, ugly clothes, and only one Brain to pass around. But the day is almost over, you've got one more easy task, and life is pretty good. If you can just get your hands on the Brain. The owner, who's rather partial to you, decides to help you out. He picks a natural number from one to sixteen and promises you the Brain if you guess his number; you may ask him seven yes/no-questions before you make your guess.

You are familiar with his motto "one lie a day keeps the doctor away" and you've been informed that he has yet to lie today. As he is aware of your strong mathematical background, he is confident this will be an easy one for you. Taking in consideration that he MAY lie to you AT MOST ONCE, will the brain be yours tonight?

Boys. http://www.thewizardofodds.com/math/ A woman is chosen at random among all women that have two children. She is asked do you have at least one boy, and she answers 'yes.' What is the probability her other child is a boy? Assume every pregnancy has a 50/50 chance to be a boy or a girl.

Candy Sharing .What happens when a group of people, each with a pile of candy, sits in a circle and passes pieces to the left according to a prescribed formula. For example,

 Player A B C D E Fraction passed 1/3 2/5 1 1/2 3/4

Do the piles stabilize, oscillate, or not show any pattern?

Card Flips. On a table in a lightless room, you have a pile of 11 cards, 10 of which are face up and one face down. Can you manipulate the cards into two piles so that the same number of cards are face up in each pile? Flips are allowed. What if there are 12 cards, 10 face up? 13? Larger numbers?

Chicken fingers. Machuskies sells chicken fingers in three different order sizes: 6,9,20. What is the largest number of chicken fingers that you CANNOT obtain by combining order sizes? (For example, an order of 7 chicken fingers is impossible.)

`Chocolate. 1. First of all, pick the number of times a week that you would like to have chocolate. (more than once but less than 10)   `
`2. Multiply this number by 2 (just to be bold) `
`3. Add 5. (for Sunday) `
`4. Multiply by 50. I'll wait while you get the calculator................ `
`5. If you have already had your birthday this year add 1755.... If you haven't, add 1754 ..... `
`6. Now subtract the four digit year that you were born. You should have a three digit number.`
` `
`               The first digit of this was your original number (i.e., how many times you want to have chocolate each week). `
`               The next two numbers are .....YOUR AGE! (Oh YES, it is!!!!!) `
` The year 2005 is the only year in which this trick will work. `

Cryptarithm.

Digit Design.  Arrange the digits 1-9 so that left to right pairs are products of single digit

numbers. For example, 126345789 won’t work because 26 isn’t a product of single digit numbers.

Folding Fun.

Place the two squares back to back and tape the two top edges together. Cut slits along the cell boundaries indicated by double lines. The task is now to fold the paper, only along the boundary lines of the cells, so that you end up with a four by four square showing respectively four 1’s, four 2’s, and so on up to four 8’s.

Fuses. (The Actuarial Review, February 2000)  You have two fuses, each 12’’ long. Each fuse burns in exactly one hour, but does not necessarily burn at a uniform rate. Also, the two fuses do not necessarily burn at the same rate over corresponding segments. But a given segment on a given fuse burns in the same amount of time in either direction. How do you use these two fuses to time 15 minutes?  Extra credit - how do you time 15 minutes using only one fuse?

Easter Bunnies. A candy maker manufactures 7 Chocolate Easter bunnies of increasing sizes. His plan was to use 10 grams of chocolate for the first bunny he made and 10 additional grams for each of the successive ones. So, at the end, the bunnies should weigh 10, 20, 30, 40, 50, 60, and 70 grams respectively. However, after making the first k bunnies, he takes a lunch break and during that time a mischievous helper messes up his scale so at the end, the remaining 7-k bunnies wind up weighing 10 grams more than they should. Suppose that you are to determine when he went out to lunch by means of a [balance] scale with 3 outputs; less than, equal, and greater than.

What is the minimum number of weighings you need to determine k?

Construct a scheme of weighings that achieves this minimum.

What if there are originally eight bunnies (10 through 80)?

Clarifications: There are only seven bunnies, made in order (10, 20, etc.). 'k' can be any number from 0 to 7.

Eggs. http://www.thewizardofodds.com/math/  You are a cook in a remote area with no clocks or other way of keeping time other than a 4-minute hourglass and a 7-minute hourglass. You do have a stove however with water in a pot already boiling. Somebody asks you for a 9-minute egg, and you know this person is a perfectionist. What is the least amount of time it will take to prepare the egg?

Friendship.  In a group of 5 people we have the situation that any two persons have precisely one common friend. Is there always a person who is everybody’s friend?

Coins. http://www.thewizardofodds.com/math/ A box contains two coins. One coin is heads on both sides and the other is heads on one side and tails on the other. One coin is selected from the box at random and the face of one side is observed. If the face is heads what is the probability that the other side is heads?

Four Weights. Using a balance scale and four weights you must be able to balance any integer load from 1 to 40. How much should each of the four weights weigh?

Heads or Tails. You are blindfolded before a table. On the table are a very large number of pennies. You are told 128 of the pennies are heads up and the rest are tails up. How can you create two subgroups of pennies, each with the same number of heads facing up?

Light Bulbs.  Light bulbs and switches have been the subject of a variety of popular and perplexing mathematical puzzles. Here's one that Martin Gardner mentions in an article on the binary Gray code.

Imagine a light bulb connected to n switches in such a way that it lights only when all the switches are closed. A push button opens and closes each switch, but you have no way of knowing which push opens and which closes. What is the smallest number of pushes required to be certain that you will turn on the light regardless of how many switches are set at the outset?

One way of solving the problem involves using a binary Gray code. In essence, a Gray code is a way of encoding numbers so that adjacent numbers have a single digit differing by 1 (see http://mathworld.wolfram.com/GrayCode.html ). For example, 193 and 183 would be adjacent counting numbers in a decimal Gray code (the middle digits differ by 1), but 193 and 173 would not be.

This switch effect is perplexing enough that it forms the basis for an amusing magic trick. Available in magic stores (sometimes under the name Electronic Monte), the device has three push-button switches and one light (http://www.thetrickery.com/?nd=full&key=5576 ). "The magician demonstrates how a single button seems to control the light," Gardner notes, "but the control mysteriously changes from one push button to another, like the pea in a three-shell game."

`Guess My Number.  http://digicc.com/fido`

Surgeons. How can three surgeons with only two surgical gloves, with open wounds on their hands, operate on a patient so that nobody is directly exposed to anybody else's blood?

Bicycle Crossing. Three people (A, B, and C) need to cross a bridge. A can cross the bridge in 10 minutes, B can cross in 5 minutes, and C can cross in 2 minutes. There is also a bicycle available and any person can cross the bridge in 1 minute with the bicycle. What is the shortest time for all to get across the bridge? Each person travels at his own constant rate.

Camel Cargo. It is your task to deliver as much grain as possible from city A to city B. The cities are 1,000 miles apart. You initially have 3,000 pounds of grain. Your camel may carry up to 1,000 pounds and eats 1 pound of grain per mile traveled. You may leave grain along the way and return to it later. How much grain can you deliver to city B?

Plane Facts. Two identical planes take off from their home airport on the equator to circumnavigate the globe. They each fly part of the way and land. Plane 1 gives some of her fuel to Plane 2 then returns home, with just enough fuel to make it. Plane 2 continues on and completes the circumnavigation with just enough fuel to make it. What fraction of her fuel did Plane 1 give to Plane 2? What fraction of the way did they fly before exchanging fuel?

Thirteen. Why are 6-digit numbers like 245,245 and 176,176 always divisible by 13?

Twenty Unfaithful Husbands (from A Mathematician Plays the Stock Market, by J. Paulos).

Long ago and far away there was a village with unusual social practices. First, if a husband was unfaithful to his wife, every wife in the village knew about it except his own. Second, if a wife learned for certain that her husband had been unfaithful, she would kill him before midnight.

One spring there were 20 unfaithful husbands living happily in the village until the arrival of a traveling wise man. The sage gathered the wives of the village in the square and announced: “At least one of you has an unfaithful husband.” Nothing happened for 19 days, but on the 20th day, each of the 20 unfaithful husbands was killed by his wife. Explain.

Simple math required. Create the number 24 using only a 1, 3, 4, and 6. You may only use +, -, /, and *. Parentheses are allowed. For example if I asked for 23 an answer would be ((6-1)*4)+3. This is not a trick question, for example the answer does not involve a number system other than base 10 and does not allow for decimal points.

Ferry Boats. (Contingencies, July/August 2001) Two ferry boats leave opposite sides of a river at the same time, both traveling at right angles to the shore. Each boat travels at a (different) constant speed. They pass each other 500 yards from the closest shore. Each boat takes 15 minutes to unload and reload passengers. On the return trip they pass each other 300 yards from the shore. How wide is the river?

Coconuts. On a deserted island live five people and a monkey. One day everybody gathers coconuts and puts them together in a community pile, to be divided the next day. During the night one person decides to take his share himself. He divides the coconuts into five equal piles, with one coconut left over. He gives the extra coconut to the monkey, hides his pile, and puts the other four piles back into a single pile. The other four islanders then do the same thing, one at a time, each giving one coconut to the monkey to make the piles divide equally. What is the smallest possible number of coconuts in the original pile?

Ping Pong.  Three players – Al, Bea, and Cat – are playing ping pong. Two players play a game and the loser drops out. When they finally stop, Al has won 10 games and Bea has won 21.  How may times did Al and Bea play against each other? If Cat won 17 games, who lost the last game?  SIMPLIFY.

Missing Square. A single square is removed from an 8 by 8 chessboard. Can you cover the remaining 63 squares with L-shaped Trominoes?  Simplify. Try 2 by 2 and 4 by 4.  Pause and Reflect. Does it work for larger chessboards?

Missing Missionaries. Long ago there were 26 Indian villages way out west in California that needed to be converted. For planning purposes, the villages were labelled A,B,C,…,Z. At various times during the 18th and 19th centuries, 26 missionaries - conveniently named A,B,C,…,Z - were sent to convert the villages. Each missionary began at the village labeled by his name. If the village was not converted, then the missionary converted the village and moved on to the next letter in the alphabet (from Z he went to A). On the other hand, if the village was already converted, the villagers viewed the arrival of the missionary as “just too much” and killed him. How many villages remained converted after all the missionaries were sent?

Egg Drop. You stand in front of a 36-story building holding 2 red eggs (this is an Easter problem!). What is the minimum number of tries that is required to figure out the highest floor from which you can throw an egg without breaking? Throwing an egg from the nth floor and seeing what happens is considered as one try. Of course, once an egg is broken it can't be used again.

Lost in the Woods.  After a Rip-van-Winkle-like sleep, you wake up in the forest. At your feet is a signpost that indicates that it is one mile to the road, and you know that this road is the only one in the area and that it runs straight. Unfortunately, the signpost has fallen down, so you know only the distance to the road, not the direction.

What search path should you follow so that the worst-case distance you go before finding the road is as small as possible?

Tsoureki Cuts. What's the maximum number of pieces into which one can cut a large tsoureki using ten straight cuts? (Tsoureki is a shiny glazed sweet Greek Easter bread. Three long dough ropes are traditionally braided, and sometimes the loaf is sprinkled with sesame seeds, almond slivers, and decorated with dough designs or red eggs). For the purpose of this problem, you can assume tsoureki is a convex, large, 3-dimensional figure.

Moussaka Matters. In a Greek family of 89 adults, each one of them has a different amount of money in his/her bank account. I happen to know all of them (the adults that is, not their bank accounts) from way back when, and decided to invite them for dinner. After they are all seated around my big round table (I am still rushing around finishing the moussaka), they challenge me to determine one of them who has more money in his/her bank account than each of his/her two neighbors (on the left and on the right) by checking the bank accounts of no more than 10 of them.  Needless to say I did live up to the challenge and the moussaka was delicious as usual.

More Moussaka. Muge’s famous moussaka has ten ingredients. Alas, everyone who is coming over for moussaka dislikes at least one of the ingredients. Muge knows that while different people's dislikes may overlap, no one person's dislikes completely include someone else's. The number of people coming over for moussaka is the largest possible for this to happen. How many forks will the Muge need?

Twenty-four. Create the number 24 using only these numbers once each: 3, 3, 7, 7. You may use only the operations: +, -, *, / and this is not a trick question. For example, the answer does not involve a number system other than base 10 and does not allow for decimal points.

Full Coverage (From Creative Puzzle Thinking, Nob Yoshigahara). Can you cover the 7 ´ 7 square with 11 of the small L-shaped pieces and the one large one? You are not allowed to turn over any of the pieces, but you may rotate them in the plane. (Stir)

Swimmers. Two swimmers dive into the pool at opposite ends and swim two lengths, back and forth. The first time they pass a distance a feet from one end and the second time a distance b feet from the other. How long is the pool?

Pizzas and Pythagoras. Two pizzas have radii a and b. A third pizza has radius c. When is there more to eat in the third pizza?

Circumference. A string is wrapped around the circumference of a basketball. Then the length of the string is increased by one foot and reformed into a circle (with the same center). Now imagine a string wrapped around the circumference of the earth. The length of this string is also increased by one foot and reformed into a circle centered at the center of the earth. Which string is farther away from the sphere it goes aroundt?

Tickets. The national touring company of Sesame Street Live visited Seattle, Washington, last year. The producers held a special promotion ticket sale for one hour.  During this time adult tickets sold for \$5, junior tickets (kids age 8 - 16) sold for \$2, and children’s tickets (kids age 0-8) sold for a ridiculously low price of 10 cents. During this sale, 120 tickets sold for exactly \$120. How many of each kind of ticket were sold during the sale?

Product and Sum. Erich Friedman -- P is given the product and S the sum of two non-zero digits (1 to 9).

1. P says "I don't know the numbers".
S says "I don't know the numbers".
2. P says "I don't know the numbers".
S says "I don't know the numbers".
3. P says "I don't know the numbers".
S says "I don't know the numbers".

4. P says "I don't know the numbers".

S says "I don't know the numbers".
5. P says "I know the numbers".

What are the two digits?

http://www.1800gotjunk.com/genie (guess my number)

Quickie Quiz.   1. You are participating in a race. You overtake the second person. What position are you in?

2. If you overtake the last person, then you are...?

4. Mary's father has five daughters: 1. Nana, 2. Nene, 3. Nini, 4. Nono. What is the name of the fifth daughter?

Traveling Salesman. A salesman’s office is located on a straight road. His 5 customers are all located along this road to the east of the office, with the office of the first customer one mile away, the second two miles away and so on up to the fifth customer who is five miles away. The salesman must drive to visit each customer once and then return to his office. Because he makes a profit on his mileage allowance, he would like to make the trip as long as possible. What is the maximum distance he can drive and how many different trips produce this distance? He must drive directly from one customer’s house to the next, but he can do this in any order.

Sandglasses. You have two sandglasses, one that measures 9 minutes and the other 13. You want to boil a stew for exactly 30 minutes. Can you do this if you must turn over the glass(es) for the first time just as the stew starts to boil?

Party people. If six people are chosen from a large party, must there either be 3 of them who know each other or 3 who do not know each other?

Pop Quiz.

`1. A murderer is condemned to death. He has to choose between three`
`rooms. The first is full of raging fires, the second is full of`
`assassins with loaded guns, and the third is full of lions that `
`haven't eaten in 3 years. Which room is safest for him?`
` `
`2. A woman shoots her husband. Then she holds him under water for over `
`5 minutes. Finally, she hangs him. But 5 minutes later they both go out`
`together and enjoy a wonderful dinner together. How can this be?`
` `
`3. What is black when you buy it, red when you use it, and gray when `
`You throw it away?`
` `
`4. Can you name three consecutive days without using the words Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, or Sunday?`
` `
`5. This is an unusual paragraph. I'm curious how quickly you can find out what is so unusual about it. It looks so plain you would think`
`nothing was wrong with it. In fact, nothing is wrong with it! It is`
`unusual though. Study it, and think about it, but you still may not `
`find anything odd. But if you work at it a bit, you might find out. Try to do so without any coaching!`