# Challenging Math Problems

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ISBN-13: 9780486795539 Dover Publications 10/21/2015 Dover Books on Mathematics First Edition, First 112 816,387 5.90(w) x 8.90(h) x 0.60(d)

Terry Stickels is the author of more than 25 puzzle books as well as two nationally syndicated columns: FRAMEGAMES, appearing in USA Weekend, and STICKELERS, a daily feature that appears in several papers, including The Washington Post, The Chicago Sun-Times, and The Seattle Post-Intelligencer.

#### Challenging Math Problems

By Terry Stickels

#### Dover Publications, Inc.

ISBN: 978-0-486-80857-4

CHAPTER 1

PUZZLES

1

If a teacher can place his students eight to a bench, he will have only three students on the final bench. If he decides to place nine students on a bench, he'll have only four students on the final bench. What is the smallest number of students this class could have?

2

Here's a probability puzzle that may test even some of the best mathematicians.

Imagine you have two opaque boxes. One box has one white marble and the other box has one white marble and one black marble. Of course, you can't see into either box. Simultaneously, you reach into both boxes, grab one marble from each, and quickly switch the marbles without looking at them. You then pick one of the marbles from one of the boxes. What is the probability the marble you pick is white?

3

Molly places 200 kilograms of watermelons in cases in front of her shop. At that moment, the watermelons are 99% water. In the afternoon, it turns out that it is the hottest day of the year, and as a result, the watermelons start to dry out. At the end of the day, Molly is surprised because she wasn't able to sell a single watermelon, and the melons are now only 98% water. How many kilograms of watermelons does Molly have left at the end of the day?

4

One of the most interesting studies in mathematics is the way numbers grow as they approach infinity. So, for this puzzle consider if you were to take the limit, as "n" approaches infinity, put the following in order, going from lowest to highest in value. Note: "c" is any positive integer you choose greater than 1. All values of "n" are positive and greater than one.

nc 1n n n! n cn nn

5

What number comes before the 3 to start this sequence?

? 3 11 31 75 155 283 471

6

A positive whole number is divisible by 3 and also by 5. When the number is divided by 7, the remainder is 5. What is the smallest number that could work?

7

What are the missing digits in the division below? All the 3s are given. The placeholder Xs may be any digit from 0 – 9, excluding 3.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

8

In the figure below, the total area of rectangle BCDE plus triangle ABE is 192 sq. units. What is the length of AB?

[ILLUSTRATION OMITTED]

9

The mean of three numbers is 12 more than the least of the numbers and 18 less than the greatest. The median of the three numbers is 9. What is the sum of the three numbers?

10

Alison and Amelia live 14 miles apart. Alison started to drive her car toward Amelia's house. At the same time, Amelia started to drive toward Alison's house. When they met, Alison had been driving for three times as long as Amelia and at 3/5 Amelia's rate. How many miles had Amelia driven when they met?

11

On a normal die, the sum of the dots on opposite faces is seven.

[ILLUSTRATION OMITTED]

On the four dice above, the opposite faces total seven but one of the pictures is incorrect because of the orientation of the dots. Which die is incorrect? There is enough information from the four views to make this determination.

12

Below are 24 matchsticks arranged in a 3 × 3 grid. The circles represent the matchsticks that would have to be removed so that there are no squares of any size remaining in the grid. The minimum number, as you can see, is six matchsticks that need to be removed. There are several ways to accomplish this.

[ILLUSTRATION OMITTED]

Now, what is the minimum number of matchsticks that need to be removed in a 4 × 4 grid of 40 matchsticks? Remember, there can be no squares, of any size, remaining.

[ILLUSTRATION OMITTED]

13

For what value K is the following system consistent?

1)P + Q = 6 2)KP + Q = 18 3)P + KQ = 30

14

Molly's yard has twice the area as Mickey's yard and 6 times as much area as Maggie's lawn. Maggie's mower cuts 1/3 as fast as Mickey's mower and 1/5 as fast as Molly's mower. If they all start to mow their respective lawns at the same time, who will finish last?

15

A factory makes gears for heavy machinery. Because of the intense heat and friction, a solid form of resin sticks is manually inserted into one of the machines at regular intervals to aid the grinding process.

Each resin stick has enough left over residue so that four pieces of it can be reformed to make another whole new piece of resin. At the beginning of the second shift, a worker notices 24 individual left over residue pieces. How many new resin sticks will these 24 make?

16

SEND +MORE/MONEY
D = 7, E = 5

The well-known puzzle above first appeared in 1924 and is known as a cryptarithm or alphametic. Each letter has a positive integer value and for most alphametics, the range of values is between 0 and 9. No word can begin with a zero but that's about the only restriction. Often times, some of the values of letters are given to the solvers as hints. This also helps in keeping the answer to a single solution. Many alphametics have multiple solutions unless values are stipulated.

This particular puzzle has one solution only ... and if you'd like to try to solve it, I'll give you two clues:

D = 7 and E = 5. There are eight different letters used, which means only eight numbers somewhere between 0 and 9 will be used.

The lead-in with this puzzle is usually stated as something to the effect that this was a note from a college student to his parents asking for a specific amount of money to cover tuition ... but that was a long time ago, so it's time for an update. Here's the new revised version:

SEND MORE LOTS +MORE/GREEN
Let D = 7, E =1, and ) = 6

Now, see how long it takes you to come up with both solutions ... share them with family and friends. These puzzles can be great fun.

17

You have two pieces of string of varying thickness that each burn for 40 seconds. You want to measure 30 seconds. You can't cut either string in half because the fuses have different thicknesses and you can't be sure how long each will burn.

How can you accomplish this without bending, folding, or cutting the strings?

18

Let's say you wanted to calculate the number of license plates that could be issued by using three letters and three numeric characters (e.g., AAA123), using all letters of the alphabet and numbers 000 thru 999.

1)Assume that any of the letters or numbers can be in any of the six positions, that is: AAA000 or 000AAA or A0A0A0 or A1B2C3.

2)You cannot create plates that have all numbers (123456) or all letters (ABCDEF). Each plate must have three numbers and three letters.

How many possibilities for plates are there this way?

19

You are in a dark room sitting at a table. On a speaker, a voice tells you there are 26 nickels on the table – 10 are heads and 16 are tails. You are then told you will win \$1,000 if you can separate the coins into two groups with the same number of coins heads up in each group, and you have one minute to accomplish this in total darkness. Could you do it?

20

If the stack of cubes below was originally 4 × 4 × 4, is A, B, C, D, or E the missing piece from the broken cube? Note1 : All rows and columns run to completion unless you actually see them end. Note 2: The missing piece is to be inserted upside down to complete the cube.

[ILLUSTRATION OMITTED]

21

When a certain positive whole number (N) is divided by x, the remainder is 7. If the original number (N) is multiplied by 5 and divided by x, the remainder is 10. What is the least possible value of N?

22

I once wrote a puzzle where I asked the solver to use four different weights of counterbalance to weigh objects on a scale. I told them the weights have to be able to balance any whole number weight from 1 to 40 ounces. While I stated they could use four weights only, it is readily apparent that they will need more than one each of some of the weights. I asked what that minimum number was and how many of each of the weights would meet the requirements. It turns out that using two weights each of 1, 3, and 9 ounces plus one weight of 27 ounces will work (for a total of seven weights).

Here's the new puzzle. Is it possible there is an alternative solution(s) using four different weights? Again, some of the weights will have to be used more than once but seven weights total can't be exceeded.

23

Bella is doing a lab experiment and just realized she has put a certain solution in the wrong cylindrical jar. She needs another cylindrical jar with 30% larger diameter but the same volume as the jar she's currently using. If the diameter of the new jar is increased by 30% without the volume changing, by what percent must the height be decreased? Remember, the volume of a cylinder is V = πR2h.

24

The zeros that end a given number are called "terminal zeros." For example, 830,000 has four terminal zeros and 803,000 has three terminal zeros. How many terminal zeros are in the number represented by 15!?

25

One of the figures below doesn't belong with the others. The other four figures can be drawn without lifting a pencil, retracing or crossing any lines. Which is the odd one out?

[ILLUSTRATION OMITTED]

26

Can you create the following figure without lifting your pencil, retracting, or crossing another line? You can create the lines by going from vertex to vertex or line segments.

[ILLUSTRATION OMITTED]

27

How many squares can you create in this figure by connecting any four dots (the corners of a square must lie upon a grid dot).

[ILLUSTRATION OMITTED]

28

Two friends wish to meet for breakfast twice a month throughout the year but never on consecutive days. In how many ways can they choose those two days of the month so they never meet on consecutive days? (Consider a month to have 30 days.)

29

Triangle ABC is a right triangle with an altitude drawn from the vertex of the right angle to its hypotenuse.

[ILLUSTRATION OMITTED]

What is the length of AB? AD = 18, DC = 8

30

A certain city has done a thorough investigation and found that 2% of its citizens use drugs. That figure is solid and not in dispute. Suppose a drug test is employed that is 98% accurate. This means if a citizen is a user, the test will be positive 98% of the time. If the individual is a non-user, the test will be negative 98% of the time.

A person is selected at random, given the test, and the results are positive. What is the chance the person is a drug user?

31

On the planet Xenon there is only one airport and it is located on its North Pole. There are only three airplanes on the planet. Each plane's fuel tank holds just enough fuel to allow each plane to make it one way to the South Pole. There is an unlimited supply of fuel at the airport and the airplanes can transfer their fuel to one another. Your mission is to fly around the globe/planet with at least one airplane covering the circular journey and flying over the South Pole on its return to the North Pole. The planes may stop at any time along the way. (Note: the journey must be accomplished by flying in a "great circle." All great circles of a sphere have the same circumference and the same center as the sphere. A great circle is the largest circle that can be drawn on a sphere.) How can this be done?

32

Imagine you have a square piece of ordinary paper. Can you tell me a way in which you can divide this square into thirds, using a pencil and straightedge only? No compasses, protractors, or any measuring devices. You may fold the square once.

33

You are standing outside a 30-story building holding two identical glass spheres. You are told that either sphere, if dropped from the roof, would shatter upon hitting the earth, but that it would not necessarily break if dropped from the first story. Your task is to identify the lowest possible floor from which you can drop a ball and break it.

In the general case, what is the smallest number of drops required to guarantee that you have identified the lowest story? NOTES: 1) Both balls have the same minimum breakage story; 2) You only have two balls to use. If one breaks, it cannot be used for the rest of the experiment.

34

A mathematician needs to get through a train tunnel by foot. She starts through the tunnel and when she gets 1/3 the way through the tunnel, she hears a train whistle behind her. She begins to wonder how much faster the train is going than she is. At this point, we don't know how far away the train is, or how fast it is going (or how fast she is going). Here's what is known:

1)If she turns around and runs back the way she came, she will just barely make it out of the tunnel alive before the train hits her.

2)If she keeps running through the tunnel, she will also just barely make it out of the tunnel alive before the train hits her.

Assume she runs the same speed regardless of which path she chooses and reaches top speed instantaneously. Assume the train misses her by an infinitesimal amount (so that there are no extenuating circumstances that would make the puzzle unsolvable or a non-puzzle. In other words, don't make a case that she shouldn't be in the tunnel in the first place or that she still might get squished or anything that would cancel a solution.)

How fast is the train going compared to the mathematician?

35

If the stack of cubes below was originally 5 × 5 × 5, is A, B, C, D, or E the missing piece from the broken cube? Note 1: All rows and columns run to completion unless you actually see them end. Note 2: The missing piece is to be inserted upside down to complete the cube.

[ILLUSTRATION OMITTED]

36

There is a fun old puzzle that asks the following: At one point, a remote island's population of chameleons was divided as follows:

13 red chameleons
15 green chameleons
17 blue chameleons

Each time two different colored chameleons would meet, they would change their color to the third one (i.e. If green meets red, they both change their color to blue.) Is it ever possible for all chameleons to become the same color? Why or why not?

The answer is "no" – it is not possible. One of the most interesting things about this puzzle is that most of the explanations found online are either wrong or are not clear ... and none I have found offer an example of when the answer is "yes." So, the new puzzle is this: give me two different examples (with different sets of numbers/colors) that can give chameleons all with the same color.

37

There are four cards that have a letter on one side and a number on the other side. Someone has laid them out and the cards appear as 2, 7, G, A. There is a rule that a card with an odd number on one side must have a vowel on the other. What is the minimum number of cards you should turn over to prove the rule is true?

38

You die and the devil says he'll let you go to heaven if you beat him in a game. He sits you down at a round table. He divides a large pile of quarters in two so you both have the same amount of quarters in your own pile. He says, "Ok, we'll take turns putting quarters down, no overlapping allowed, and the quarters must rest on the table surface. The first guy who can't put a quarter down loses. You cannot shift or try to squeeze any quarter into a space that moves another quarter." The devil says he wants to go first.

You realize that if the devil goes first, he probably has a strategy to win. You haven't had time to think this through yet, so you ask for some time to consider your options. He grants you 15 minutes. At the end of that time you know how to beat him ... but you also know you must go first for your strategy to win. You convince him to let you go first, saying you really didn't have enough time to consider a strategy so you should at least go first. What is your winning strategy?

39

There's an old puzzle that asks, "If a hen and a half lays an egg and a half in a day and a half, how many hens will it take to lay 6 eggs in 6 days? The answer is 1.5 hens. Here's the new puzzle: at this rate, how many eggs will one hen lay in one day?

(Continues...)

Excerpted from Challenging Math Problems by Terry Stickels. Copyright © 2015 Terry Stickels. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

#### Contents

Introduction,
Acknowledgements,
Puzzles,
Solutions,
Afterword,

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