In addition to logical puzzles and mathematical paradoxes, this original collection features a variety of word problems and sports-related riddles. The puzzles, which vary in complexity from playful propositions to tough mathematical conundrums, are suitable for ages 12 and up. Detailed solutions appear at the end.
In addition to logical puzzles and mathematical paradoxes, this original collection features a variety of word problems and sports-related riddles. The puzzles, which vary in complexity from playful propositions to tough mathematical conundrums, are suitable for ages 12 and up. Detailed solutions appear at the end.
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Overview
In addition to logical puzzles and mathematical paradoxes, this original collection features a variety of word problems and sports-related riddles. The puzzles, which vary in complexity from playful propositions to tough mathematical conundrums, are suitable for ages 12 and up. Detailed solutions appear at the end.
Product Details
| ISBN-13: | 9780486315706 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 08/21/2013 |
| Series: | Dover Math Games & Puzzles |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 160 |
| File size: | 3 MB |
About the Author
Read an Excerpt
Golf on the Moon
Entertaining Mathematical Paradoxes and Puzzles
By Dick Hess
Dover Publications, Inc.
Copyright © 2014 Dick HessAll rights reserved.
ISBN: 978-0-486-31570-6
CHAPTER 1
PLAYFUL PUZZLES
Humanity has advanced, when it has advanced, not because it has been sober, responsible, and cautious, but because it has been playful, rebellious, and immature.
—Tom Robbins
1 RELATIONS PUZZLE
"Nieces and nephews have I none but that man's father is my father's son." What is the gender of the speaker? To whom is the speaker referring?
2 BIRTHDAY 1
Twin brothers, Brian and Ryan, born 5 minutes apart, turned 21 in 2008. Each celebrated his birthday on the actual day he was born, but Brian celebrated 3 days before Ryan. Explain.
3 SHOPPING PROBLEM
Penny and Nick wish to buy a computer game and each has a positive whole number of dollars. Penny finds she is $2 short; Nick finds he is $32 short. If they pool all their money they still are unable to buy the computer game. What is the price of the computer game if it costs a whole number of dollars?
4 FAST THINKING
Quickly state 30 English words that don't contain the letter a.
5 ISOSCELES TRIANGLE
Two sides of an isosceles triangle are 3 and 7. What is its perimeter?
(a) 16
(b) 13
(c) 17
(d) Can't tell for sure
6 LOGICAL QUESTION
The figure on the next page shows squares of areas 9, 16 and 25. Is area A larger or smaller than area B?
7 BIRTHDAY 2
Several years ago Nick stated: "Sometime last year I was still 21. In two days I'll be in my 25th year." What day of the year is Nick's birthday and on what day of the year is he speaking?
8 PET PROBLEM
Bill: "All of my pets but two are cats; all of my pets but two are dogs; all of my pets but two are birds."
Alice: "How many pets does Bill have?" Find and explain two different correct answers to Alice's question.
9 GOLF BALLS
Two manufacturers of golf balls arrange the circular dimples on the surface in a triangular pattern. This is accomplished by possibly having dimples of different sizes. Some dimples are surrounded by 6 others and other dimples are surrounded by 5 others. Manufacturer A places a total of 384 dimples on the ball and Manufacturer B places a total of 396. Which ball has more dimples surrounded by only 5 others?
10 HAIRY QUESTION
The average number of hairs on a person's head is about one hundred thousand (105). The number of Americans is about 300 million (3 × 108). Let S be the sum of the number of hairs on each head for all Americans. Let P be the product of those 300 million numbers. What is your best estimate of S and P?
11 FIVE LOGICIANS
Five logicians are seated at the restaurant and the server asks, "Do all of you want either tea or coffee?"
The first logician answers: "I don't know."
2nd logician: "I don't know."
3rd logician: "I don't know."
4th logician: "I don't know."
5th logician: "No."
At this point the server is needed by another table and the 5th logician uses the time to use the gent's room. The server reappears to ask the remaining four "Do all four of you want tea?"
The first logician answers: "I don't know."
2nd logician: "I don't know."
3rd logician: "I don't know."
4th logician: "No."
The 5th logician now returns to the table. Assuming no customer wants both tea and coffee, what should the server serve to each of the five logicians and why?
12 BRIDGE ENDING
South's on lead with spades trump. How will he guarantee taking all seven tricks?
13 SLAM CHANCE
You are playing south in a [??]6 slam and get the [??]J lead. How can you guarantee the contract?
14 SMALLEST PRIME DIVISOR
Consider the sums S1 = 11, S2 = 12 + 21, S3 = 13 + 22 + 31, S4 = 14 + 23 + 32 + 41, .... Find the smallest prime divisor of
(a) S6
(b) S10
(c) S22
(d) S100
15 PULLEY QUESTION
The ideal pulley system below has weightless and frictionless elements except for the weights shown. In the first instant after release from rest will the 99-weight go up or down?
CHAPTER 2DIGITAL PUZZLES
Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. —John von Neumann
16 PUMPKINS
Having sorted them by weight a farmer takes his crop of pumpkins to the wholesale market. He sells the 42 lightest pumpkins to customer A and notes that they account for 25% of the total weight. He sells the 50 heaviest pumpkins to customer B and notes that they account for 30% of the total weight. How many pumpkins were originally in the crop?
17 YEAR PROBLEM
Find all integer pairs, (m, n), with m< n such that m + n + mn = 2012.
18 EXTREME SINES
Josh and Andy are computing f = sin(nm), where nm is given in degrees and both n and m are positive integers under 50. (a) What are the extreme values they can achieve for f? (b) Suppose m and n are allowed to be positive integers under 100? (c) Suppose nm is taken in radians; what now are the answers to (a) and (b)?
19 PANDIGITAL POWERS
Find a number such that if its square and its cube are written down, each digit from 0 to 9 appears exactly once.
20 POWERFUL NUMBERS
Suppose n = abcd in decimal notation and define f (n) = aa + bb + cc + dd. For example, f (1125) = 11 + 11 + 22 + 55 = 3131. (a) Find a four-digit number, n, such that f (n) = n. (b) If we define 00 = 0 then find a larger number, N, such that f (N) = N.
21 MAKE IT PYTHAGOREAN
The example triangle below can be made Pythagorean by inserting the single digit 2 in some manner to each side length as shown. In each of the subsequent triangles, insert the same single digit of your choice to each side to make it a Pythagorean triangle.
22 SQUARE DIGITS
Take the number abcd in decimal notation and split it into two pieces such as a and bcd; abc and d; or ab and cd. We can ask whether it is possible thatabcd = a2 + (bcd)2, abcd = (abc)2 + d2 or abcd = (ab)2 + (cd)2. Clearly, numbers like 10,000 and 10,001 have this property, so such "easy" solutions don't count. (a) Find two four-digit numbers with this property. (b) Find numbers with 6, 7, and 8 digits having this property. It's OK for any digit other than a to be 0.
23 CUBIC DIGITS
Take the number abcd in decimal notation and split it into two pieces such as a and bcd; abc and d; or ab and cd. We can ask whether it is possible that abcd = a3 + (bcd)3, abcd = (abc)3 + d3 or abcd = (ab)3 + (cd)3. Clearly, numbers like 1000 and 1001 have this property, so such "easy" solutions don't count. Find an infinite set of numbers with this property. It's OK for any digit other than a to be 0.
24 NEGATIVE BASE
Find as many palindromes as possible that are written the same in base 10 and base -11. In base -11 the number 10310 is written as 124-11 = (-11)2 + 2 × (-11) + 4.
25 MEDIAN PROBLEM
A triangle has integer sides that are consecutive even numbers. One of its medians, also an integer, is longer than the shortest side. What is the area of the triangle?
26 SHORTCUT
The figure below shows two paths I can walk from my home, H, to work, W. The longer path is HABCW and the shorter path is HACW. It so happens that all adjacent points are a whole number of meters apart and the corner at B is a right angle. I also note that HA and CW are the same length, each amounting to 40% of the path length HACW. If HACW is less than 300 meters and I must walk more than an extra 8% distance if I take HABCW, how far do I walk to and from work along HACW?
27 DIGIT GAME
Andy and Bob play a digit game in which Andy picks a digit (0 to 9) for his first move and Bob then puts a digit of his choice to the left or to the right, making sure that the two-digit result is divisible by 2, but he's not allowed to leave 00 after this turn. Andy then does the same, making sure the result is divisible by 3; Bob must make his next result divisible by 4, and so on, until the last one to play is the winner. Their game could develop as 7, 74, 174, 1740, 01740, and so on. (a) To make it simpler they first decide to play in base 2. Who wins the game and by what strategy? (b) They next play in base 3. Who wins this game and by what strategy? They also play a cooperative game in base 3 to find the largest number of turns possible before the game ends. How might that proceed? (c) They finally play the cooperative game in base 4 to find the largest number of turns possible before the game ends. How might that proceed?
28 MATH CONVENTION
(a) At last year's math convention, each participant was assigned a unique positive whole number less than 105. I noted my number, calling it a, and noticed it was a perfect square. I met two other participants with numbers b1 and c1 and noted with interest that a + b1, a + c1, b1 + c1and a + b1 + c1 were all square numbers. During the convention I was amazed to find a total of 5 pairs of people with numbers having the same property. That is, a + bi, a + ci, bi + ci and a + bi + ci were all squares for i = 1 to 5. What was my number?
(b) This year we were assigned unique, positive whole numbers less than 106 and again mine was a perfect square that I called A. This time I found 11 pairs of people such that A + bi, A + ci, bi + ci and A + bi + ciwere all squares for i = 1 to 11. What was my number?
29 NON-ZERO DIGITS
Define N = 22013. Show how to find a multiple of N that does not contain the digit 0.
30 NEAR MISS
Using up to four integers adding to less than a million, can you beat this approximation for π? ln (6403203 + 744)/v163 = π + ε, where ε = 2.2373 ... × 10-31 and ln is the natural logarithm. You may use up to five operations including ln, exp, roots, +, -, ×, ÷, powers, factorials, and as many parentheses as you like.
CHAPTER 3GEOMETRICAL PUZZLES
There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry. —Martin Gardner
31 NON-OVERLAPPING TRIANGLES
The first figure on the next page shows how to draw a straight line through a pentagram to form 7 non-overlapping triangles as shown by the dots. (a) Draw two straight lines through the pentagram to produce 10 non-overlapping triangles. (b) Draw two straight lines through the augmented pentagram to produce 11 non-overlapping triangles. (c) Draw three straight lines through the H-shape to produce 7 non-overlapping triangles.
32 PUNCTURED 9 × 9 CHESSBOARD
Cover the white section of the 9 × 9 punctured chessboard shown below using rectangular tiles. Use as few 1 × 1 tiles as possible.
33 RECTANGLE WITHIN A TRIANGLE
What is the maximum area of a rectangle contained entirely within a triangle with sides of 9, 10, and 17?
34 TETROMINOES
Take three different tetrominoes from the set of 5 shown below and fit them together to make a form that has 180° rotational symmetry.
35 PENTIAMONDS
The four pentiamonds are shown below. Separate them into two pairs that form two identical shapes. No overlapping is allowed, but pieces can be turned over.
36 NO TRIG, PLEASE
The triangle in the figure below has angles at C and D equal to 45° and 60°,respectively. Also AD = 2 and CD = 1. Without using trigonometry, determine the angle ? at vertex A.
37 39 DEGREES
Given a regular pentagon how could you construct a 39° angle with a compass and a straightedge?
38 4 EASY PIECES
Cut the cake shown below into 4 pieces of the same shape and size using straight lines to connect grid points. Can you find 12 or more solutions?
39 5 EASY PIECES
A 5 × 7 rectangular cake with a very thin, even layer of frosting on its top and sides is to be divided fairly among 5 people. Show how to cut the cake vertically into 5 pieces that are equal in cake and frosting.
40 3 EASY PIECES
Two rectangular cakes, each with a very thin, even layer of frosting on its top and sides, are each to be divided fairly among 3 people such that each piece is a quadrilateral and each person receives the same amount of cake and frosting. Vertical cuts can be done only.
(a) Team A reports success with Cake A and that they managed to each get part of a small dot of different colored frosting placed equidistant from two parallel edges of the cake. What were the relative dimensions of the top of Cake A and how did they divide it?
(b) Team B has Cake B with different dimensions from Cake A and also reports success. They note that a small dot of different colored frosting was equidistant from two parallel edges of Cake B, but only two of their team members got part of it and it was on a vertex of each of the two pieces involved. What were the relative dimensions of the top of Cake B and how did they divide it?
41 PUNCTURED 8 × 8 CHESSBOARD
The 8 × 8 square shown has one square removed. Please divide the remaining area into two pieces of the same shape (but of possibly different sizes).
42 TILING THE 1 × 2 RECTANGLE
How many ways are there to tile a 1 × 2 rectangle with three tiles of the same shape, but possibly of different sizes? Allowable tile shapes are rectangle, triangle, or L shape.
43 RESTORE THE SQUARES
A square has had a 45° right triangle cut from it and pasted to the right edge creating figure (a). Cut the figure into two pieces in a different way that assemble to make the original square. Turning over pieces is permitted.
Similar problems are presented in figures (b) and (c) where the cutand-pasted piece has been modified.
44 EQUAL AREA AND PERIMETER
The right triangle and isosceles triangle shown below have integer sides. The perimeter of R equals the perimeter of I, and the area of R equals the area of I. What is the difference between the smallest angles of the two triangles? That is, what is the value of α - β?
45 TILING THE CIRCLE
The figure below shows a circle tiled by 5 congruent pieces. Tile a circle with n congruent tiles (reflected pieces allowed) where at least one of the tiles does not touch the center of the circle. You may pick n.
CHAPTER 4LOGICAL PUZZLES
The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it. —Bertrand Russell
46 CIRCLE AND CUBE
V is the volume in cubic meters of a cube with side π meters. A is the area in square meters of a circle with a radius of p meters. Is V larger or smaller than A?
47 MEDICAL HEADACHE
You have a rare disease that requires that you take pills of four types each morning. Your dosage is one pill of type A, two pills of type B, four pills of type C and eight pills of type D. One morning you take the required pills from each bottle and put them into your hand to take when you realize you have interchanged the pills of types A and D. Unfortunately they all look alike and have been mixed up so all you know is you have eight of type A and one of type D along with the correct amounts of types B and C. How can you guarantee your proper dose today and succeeding days without having to discard any pills?
48 MOLE PROBLEM
Your garden is occupied by a mole whose only connection to the surface is a pattern of mole holes as shown below. Each morning at sunrise, he comes near the surface in one of the 9 holes and is vulnerable to being shot only at that time. You may guess or figure out where he will come near the surface each day and will kill him if you shoot into the hole he occupies. It is known he must move each day to a hole orthogonally adjacent to the hole he occupied the prior day. Also, you are limited to one shot per day.
(Continues...)
Excerpted from Golf on the Moon by Dick Hess. Copyright © 2014 Dick Hess. 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.
Table of Contents
Divide a rectangular cake into five even pieces, calculate the cost of running a successful casino, figure out the maximum number of losses on the road to winning the U.S. Open Tennis Championship, and solve other amusing and challenging puzzles—including a mathematical brainteaser involving a couple of holes of golf on the surface of the moon.
In addition to logical puzzles and mathematical paradoxes, this original collection features a variety of word problems and sports-related riddles. The puzzles, which vary in complexity from playful propositions to tough mathematical conundrums, are suitable for ages 12 and up. Detailed solutions appear at the end.
Dover (2014) original publication.
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www.doverpublications.com