Problem of the week : mathbyjoe.com
Problem 1: Which set has more numbers: {1,2,3,...} or {3,4,5,...}? Note: The three dots ... means the sets go on forever in the same pattern. Justify your answer.
Problem 2: Why are manhole covers round as opposed to some other shape? Hint: Think of what could possibly happen if these covers were not in this shape.
Problem 3: What is the largest number you can write using three 2's?
Problem 4: On a cold windy morning, a man shoots his wife and then drowns her. Later that same evening, the man and this very same wife were enjoying dinner together. Explain how this is possible.
Problem 5: A palindromic number is one which is the same read backwards and forwards. For example, 121 and 3443 are numeric palindromes. What is the next palindromic number after 89100198 and how much more is it?
Problem 6: Irrational numbers are numbers that cannot be expressed in the form a/b, where a and b are both integers. When written as decimals, irrational numbers have decimal representations that do not repeat and do not terminate. Give two examples of an irrational number by illustrating a decimal representation which neither repeats nor terminates.
Problem 7: You are offered two choices by a friendly genie who wants to give you money. The genie says that for the next thirty days you can have either $1 on the first day, then $2 on the second day, then $3 on the third day, and so on until the end of the thirty days; or you can have 1 penny on the first day, then two pennies on the second day, then four pennies on the third day, and so on until the end of the thirty days. Which option would you choose and why?
Problem 8: Your school is hosting a student-exchange program and you have been invited to participate. There will be students from 10 different countries and, as part of the greeting ceremonies, each one will have to shake hands with everyone else. How many total handshakes will be exchanged among all 10 different students?
Problem 9: The Water Jug Problem. This famous problem has been around for centuries and a variant of it was used in the movie Die Hard with a Vengeance, starring Bruce Willis and Samuel Jackson. In this movie, the two characters had to defuse a bomb by measuring exactly 4 gallons of water from a 5-gallon jug and a 3-gallon jug. Can you figure out how to do this?
Problem 10: The Goldbach Conjecture is one of the oldest unsolved problems in mathematics. This conjecture states that every even number greater than 2 can be written as the sum of two prime numbers. For example, 6 = 3 + 3 and 12 = 5 + 7. Given the number 48, can you write all the different ways that this number can be expressed as the sum of two primes? Hint: There are 5 distinct ways.
Problem 11: The decimal represented by 0.9999... in which the sequence of 9's goes on forever can be shown to be exactly equal to 1. At first blush this seems odd because there seems to be missing a very small part to get to 1. Yet strange things happen when you deal with infinities, in this case the infinite sequence of 9's. Can you think of a way to show that this decimal is exactly equal to 1?
Problem 12: In mathematics, the harmonic series is the infinite sequence of terms that goes 1 + 1/2 + 1/3 + 1/4 + ... The reason this is called the harmonic series is because the terms are related to the harmonic overtones of strings in music. In considering what the sum of this infinite series might be, you might be surprised to learn that there is no sum and that the sum of this series is bigger than any number you can imagine---no matter how large. Can you think of a way of showing how this could be so?
Problem 13: A number of children are standing in a circle, evenly spaced about its circumference. The fourth child is standing directly opposite the seventeenth child. How many children are standing around the circle?
Problem 14: If the difference of two numbers is 8 and their product is 12, what is the sum of their squares?
Problem 15: Each child in a family has at least five brothers and three sisters. What is the smallest number of children the family might have?
Problem 16: If a and b are positive integers and a2 - b2 = 7, then what is
a + b?
a + b?
Problem 17: If one over x over x equals x, then what is x?
Problem 18: Which letter comes next in the sequence: A C F J O _____ ?
Problem 19: General Mills' Lucky Charms Cereal was invented in 1962. For General Mills, this particular brand was not only the fastest cereal introduction to the market---taking only six months from the day the challenge to invent a new cereal was undertaken, to market-launch---but also was the first cereal to ever contain marshmallows. The original shapes consisted of moons, stars, hearts, clovers, and the company mascot Leprechaun. If the manufacturing assembly line turned out these shapes in this order, and you randomly selected shape number 2,093 on line, which one would it be?
Problem 20: Try this brain teaser: I have two U.S. coins that total 55 cents. One is not a nickel. What are the two coins?
Problem 21: An old riddle goes as follows: One brick is one kilogram and half a brick heavy. What is the weight of one brick?
Problem 22: Solve the following algebraic equations by replacing each letter by a number from 1 to 9, so that the equations makes sense numerically. Each number can only be used once:
Problem 22: Solve the following algebraic equations by replacing each letter by a number from 1 to 9, so that the equations makes sense numerically. Each number can only be used once:
RE + MI = FA
DO + SI = MI
LA + SI = SOL
DO + SI = MI
LA + SI = SOL
Problem 23: The following is a very famous algebra problem and has an interesting history that will be discussed in the solution. Flying Between Two Trains: Two trains 150 miles apart travel toward each other along the same track, the first train at 60 mph, the second at 90 mph. A fly buzzes back and forth between the two trains until they collide. If the fly's speed is 120mph, how far will it travel?
Problem 24: This problem comes from a loyal reader B. Bishop, who actually has this situation in his family. He wrote to me and asked for my help in solving this problem. See how you fare with this challenging probability problem. Brian wrote: My wife, daughter, and brother all have birthdays that fall on the 27th day of different months. Ignoring leap years, what is the probability of this happening?
Problem 24: This problem comes from a loyal reader B. Bishop, who actually has this situation in his family. He wrote to me and asked for my help in solving this problem. See how you fare with this challenging probability problem. Brian wrote: My wife, daughter, and brother all have birthdays that fall on the 27th day of different months. Ignoring leap years, what is the probability of this happening?
Problem 25: A dartboard consists of a square incribed in a circle and another circle inscribed in that square. How many times more likely are you to hit within the larger circle than you are to hit within the smaller circle, assuming your dart lands on the board?
Problem 26: The Infinite Hotel Problem. This problem shows the "weirdness" of infinite sets. There is a beautiful lodging resort in south California known as the Infinite Hotel. You and your family arrive to stay but the concierge tells you and your family that although there are infinitely many rooms, there is no vacancy currently. Every room is occupied by a guest and none are expected to leave before next week. Your wife and kids are very disappointed as they were looking so forward to staying in this fabulous resort. Also, you have come all the way from the east coast and you don't want to return disappointed. After speaking with the concierge, the husband informs his wife and kids that they are going to stay after all. Explain what the husband suggested to the concierge to free up a room.
Problem 27: You are on your way to visit your Grandma, who lives at the end of the valley. It's her birthday, and you want to give her the cakes you've made.
Between your house and her house, you have to cross 7 bridges, and as it goes in the land of make believe, there is a troll under every bridge! Each troll, quite rightly, insists that you pay a troll toll. Before you can cross their bridge, you have to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake.
How many cakes do you have to leave home with to make sure that you arrive at Grandma's with exactly 2 cakes?
Problem 26: The Infinite Hotel Problem. This problem shows the "weirdness" of infinite sets. There is a beautiful lodging resort in south California known as the Infinite Hotel. You and your family arrive to stay but the concierge tells you and your family that although there are infinitely many rooms, there is no vacancy currently. Every room is occupied by a guest and none are expected to leave before next week. Your wife and kids are very disappointed as they were looking so forward to staying in this fabulous resort. Also, you have come all the way from the east coast and you don't want to return disappointed. After speaking with the concierge, the husband informs his wife and kids that they are going to stay after all. Explain what the husband suggested to the concierge to free up a room.
Problem 27: You are on your way to visit your Grandma, who lives at the end of the valley. It's her birthday, and you want to give her the cakes you've made.
Between your house and her house, you have to cross 7 bridges, and as it goes in the land of make believe, there is a troll under every bridge! Each troll, quite rightly, insists that you pay a troll toll. Before you can cross their bridge, you have to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake.
How many cakes do you have to leave home with to make sure that you arrive at Grandma's with exactly 2 cakes?
Problem 28: A jewel thief with long spindly fingers has a burlap bag containing 5 sets of emeralds, 4 sets of diamonds, and 3 sets of rubies. A "set" consists of a large, a medium, and a small version of each of these gems. The electricity is out, and it is dark. How many gems must the thief withdraw from his bag to ensure that he has a complete set of one of these gems? How many gems must he remove to ensure that he has removed all of the large gems?
Problem 29: Supply the missing number in the following sequence: 2, 71, 828, ?
Problem 30: This one is cute but really try and think about it for a second because it really makes sense. Why should you never mention the number 288 in front of anyone?
Problem 31: How can you add eight 8's to get the number 1,000? (Use only addition)
Problem 32: Two father's and two son's sat down to eat eggs for breakfast. They ate exactly three eggs, each person having eaten one egg. Explain how this is possible.
Problem 31: How can you add eight 8's to get the number 1,000? (Use only addition)
Problem 32: Two father's and two son's sat down to eat eggs for breakfast. They ate exactly three eggs, each person having eaten one egg. Explain how this is possible.
Problem 33: This is the famous "locker problem." I solved it after recognizing a pattern. Let's see how smart you really are. Here goes: A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:
There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?
There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?
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