Problem of the Week

Scientific research done at Stanford University has shown that regular mental exercise, such as that performed in doing brain teasers, riddles, and the like, improves cognitive functioning and brain health.  Come work out your brain with these fascinating problems and brain teasers.  For the solution, join my free newsletter by clicking here Newsletter Tab.   Then email me to let me know.  Once I receive confirmation that you have joined my newsletter, I will send you the  link to the solutions page. 

Each week, the first 2 people who submit the correct solution to my email joe@mathbyjoe.com, will receive 50% off of any ebook of choice.  Use "Solution to Problem of the Week" as the subject line.  Once you email me the solution, I will give you the information on how to obtain your 50% discount and get your ebook.

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?

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:

RE + MI = FA
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 25: Find the area of the flipped back portion of the circle surface, with the flipped backed tip touching the center of the circle. The dimensions are given in the diagram.