CROSSNUMBER PUZZLES F O R S E C O N DA R Y M AT H E M AT I C S S T U D E N T S DAVID I CLARK
Published by
Australian Mathematics Trust University of Canberra Locked Bag 1 Canberra GPO ACT 2601 AUSTRALIA
Copyright ©2012 AMT Publishing Telephone: +61 2 6201 5137 www.amt.edu.au AMTT Limited ACN 083 950 341
Crossnumber Puzzles for Secondary Mathematics Students ISBN 978-1-876420-33-8
Contents
Introduction Crossnumber Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Author’s Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii iii iv iv
Puzzles Worked example . . . . . . . . . . . Puzzles with simple arithmetic . . . Puzzles with a difference . . . . . . Puzzles from Middle Earth . . . . . More challenging puzzles . . . . . . Puzzles with secondary mathematics Canberra Maths Day puzzles . . . .
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Guides
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Solutions
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Appendices Conventions . Hints . . . . . Glossary . . . A little Maths
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Introduction
Crossnumber Puzzles Crossnumber puzzles are similar to their more familiar cousins, crossword puzzles, in that they consist of interlocked grids of across and down answers, each of which is the answer to a specific clue. The answers are, of course, numbers rather than words. Unlike crossword puzzles the clues typically involve more than one answer, and answers often appear in more than one clue. It is rare that the answer to a clue can be determined in isolation. Another difference from crossword puzzles is that crossnumber puzzles are solved a digit at a time rather than a whole answer at a time. A digit is teased out of one clue, and this in turn helps in finding a digit in another answer. Crossnumber puzzles are something akin to detective stories. Clues are given, but the implications of the clue needs to be worked out before it is applied in furthering the solution of the puzzle. Crossnumber puzzles are brain-teasers, and you can anticipate many hours of pleasant occupation in solving the puzzles in this book. The more challenging puzzles may take an hour or more, but the puzzles themselves and the solvers’ experience are so varied that it is not easy to suggest a par time. The only puzzles constructed with a particular solution time in mind are the Canberra Maths Day puzzles (see below). A unique feature of this book is that each puzzle comes with a solution guide which gives one possible order for solving the puzzle. If you are stuck, the solution guide will tell you which cell or cells to tackle next, without telling you how to do it. You can then still have the fun and satisfaction of solving the puzzle. If you are really stuck, you can look up the solution. But again, the solution guide will tell you which cells to look up so that you can continue solving the rest of the puzzle. Many of the clues in the puzzles in this book have operations that go beyond simple arithmetic operations of addition, subtraction, multiplication, division and averaging, but they are still accessible to solvers who are unfamiliar with them. There is a glossary of mathematical terms used in the book. There is also a fully worked solution to a simple puzzle and some useful hints about solving crossnumber puzzles, particularly starting them. The puzzles are organized into six sections as described below. Section 1. These puzzles mainly use standard arithmetic operations — addition, subtraction, multiplication, division and averages. i
Section 2. These puzzles are a bit different. The contexts are different, and therefore solution techniques have to be adapted. There tends to be a bit more logic in them. Section 3. Each of these puzzles has a small story associated with it. You are asked to solve a problem akin to a logic problem, but using a crossnumber puzzle. Section 4. These problems are rather harder than the others. You will generally need to use more than one clue to make progress. Section 5. These problems have more mathematics in them. They incorporate most of the concepts in secondary maths, including algebra, trigonometry, differentiation and integration, summation and powers and logarithms. Students will need a thorough understanding of these topics to solve the puzzles. Section 6. These problems were devised for the Canberra Maths Day, where schools send a team of five final-year college students to compete against other teams from other schools in a fun and challenging day. The crossnumber contest is one of four events on the day. An interesting feature of the event is that each team is split into two halves, one half receiving the across clues, the other receiving the down clues. When one half of the team deduces the digit in a particular cell they tell the supervisor who tells them whether they were correct and lets the other half of the team know. Thus each half of the team relies on the other half to make progress. Typically just a few of the teams finish in the allotted 45 minutes, with most of the others fairly close.
Acknowledgments This book grew out of the puzzles I constructed for the Canberra Maths Day, and includes some of them. The maths day has run annually since the middle 1980s when it started life as the University of Canberra Maths Day. It was given generous administrative and financial support from the Australian Mathematics Trust. Thanks are due to John Matthews, Ian Lisle, Tracy Huang, Vance Brown and Malcolm Brooks for their checking of puzzles in the book, and to Malcolm Brooks for helping to work out how the crossnumber contest could be made to work as part of the maths day. This book was typeset in TEX, the mathematical typesetting system designed by Donald Knuth. Many contributors have given freely of their time and expertise to extend the capabilities of TEX. The TikZ / PGF package written by Till Tantau was used in the diagrams in this book. Finally, special thanks are due to Ian Lisle for expert advice on all things TEX.
Author’s Comments This book has been a labour of love. I have enjoyed constructing the puzzles, and coming back and solving them some months or even years later. I hope that it will be a fun and challenging for those who enjoy mathematics. I also hope that using mathematics in a different context will enhance understanding and occasionally lead to further explorations – I am a firm believer in indirect learning. David Clark University of Canberra February, 2012.
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Puzzle 1.4 1
2
3
5
11
12
7
6
8
9
10
13
4
14
16
15
17
Across
Down
One fifth of 13d see 16a, 2d One less than a square A multiple of 4d see 3d, 14d see 14d A multiple of 7 12d − 17a see also 17a, 11d 16 Three times 3a see also 2d 17 15a + 5 see also 15a 1 3 6 7 8 10 13 15
1 2 3 4 5 9 11 12 13 14
1
5d + 13d 3a + 16a A permutation of 8a see 7a see 1d The square of 13d 15a + 12d see 15a, 11d see 1a, 1d, 9d A divisor (10a − 8a)
Puzzle 2.9 In this puzzle each of the letters stands for a different digit. 1
2
3
4 A
7
6
5 B
8
C 9
10
11
12 D
13
14 E
F
16
15 G
H 17 J
I
Clues Across 1 3 6 8 9 11 13 14 16 17
Down
12d − 14a see 4d Half of 11a A multiple of 13a 11a − 7d A square see 6a, 9a see 8a see 1a, 10d see 10d A multiple of 13d
1 2 4 5 7 10 11 12 13 15
2
A power of 2 13d = 11d mod 2d 3a + 5d see 4d see 9a 16a − 14a see 2d see 1a see 17a, 2d, 15d 13d × 5
Puzzle 3.1 The Middle Earth Treasure Hunt The Middle Earth treasure hunt was contested by a team of hobbits and a team of elves. The hobbits were Bilbo, Frodo, Merry, Pippin and Sam, while the elves team consisted of Celeborn, Galadriel, Arwen, Legolas and Gimli, the elf friend. A maximum of 100 points was possible, with a prize of a mithril ring for a score of 90 or more; a gold ring for a score of 80 or more; a silver ring for a score of 70 or more; a bronze ring for a score of 60 or more and an iron ring for a score of 50 or more. Every member of each team won a ring, although only one member of each team won a mithril ring and no one won a gold ring. At least one member of each team only won an iron ring. The most successful elf was Arwen while Sam was the best of the hobbits. Bilbo scored lowest for the hobbits and Galadriel lowest for the elves. As to be expected, the scores of Gimli and Legolas were very close. Solve the puzzle below to determine how many points each contestant got. 1
2
5
3
4
6
7
8
10
14
11
9
12
13
15
Across
Down
1 3 5 6 7 8 10
1 Gimli’s score 2 Total points scored by hobbits 3 From the point of view of winning rings, scores over 90, 80, 70, 60, 50 were wasted. 3d is the total wasted score. 4 Bilbo’s score 9 Total points scored by the elves 10 Legolas’ score 11 Those competitors who got iron rings were a little disappointed. 11d is the number of extra points they would have needed to score between them so that they all won bronze rings. 13 Arwen’s score
The average elves’ score Galadriel’s score Pippin’s score Sam’s score Frodo’s score Merry’s score Total points scored by hobbits and elves 12 Average hobbit’s score 14 Celeborn’s score 15 The range of hobbits’ scores
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Puzzle 5.10 1
2
5
7
3
6
8
9
10
11
13
12
16
18
4
14
15
17
19
Clues Across
Down 1 see 1a, 5a 2 see 1a 3 3d is a square. see also 10a 4 see 3a, 7a 6 see 14a, 16d, 17d 8 see 19a 9 9d is a multiple of 12a 12 tan(10a + 12d) = 1 see also 16a 13 see 14a, 16a 15 15d = 9a + 10a 16 11a = log2 6d + 16d
1 7a, 2d, 1a and 1d form an increasing AP. 3 log3 3a = 1 + log3 4d 5 75a = 71d × 718a 7 4d = 9 × 7a see also 1a 9 9a mod 5 = 2 9a mod 3 = 0 see also 15d 10 10a = −1 + 2 − 3 + · · · + 3d see also 19a, 12d, 15d 11 see 16d, 17d 12 see 9d 14 214a = 6d × 213d 16 12d = 16a + 13d 18 see 5a 19 8d = 10a + 19a
Z
Z
6d
1 dx =
17 0
4
17d
1 dx 11a
Appendices
Conventions The puzzles are presented in a fashion similar to crossword puzzles. And as in crossword puzzles, a means “across”, d means “down”. Because answers can appear in more than one clue, cross references may be given to other clues. These are in the form of “see ...”. No answer has first digit zero (this can be useful in solving the puzzles). Angles are in degrees rather than radians.
Hints 1. A 3-digit number which is the sum of two 2-digit numbers must start with 1. 2. If a 2-digit number is the average of a 2-digit and a 3-digit number, the 3-digit number must start with 1. 3. If a 3-digit number is n times a 2-digit number, with n < 2, the 3-digit number must start with 1 and the 2-digit number must be > 50. 4. A multiple of 5 must end in 0 or 5. (This can be used in conjunction with numbers not starting with 0.) 5. If a 2-digit number is 5 (or more) times another 2-digit number, the latter must start with 1. 6. Squares must end in 0, 1, 4, 5, 6 or 9. 7. Pythagorean triples are often of the form (a + 1, a, b) with b an odd number. In which case, b2 = 2 × a + 1. For example (5,4,3), (13,12,5), (25,24,7). They may also be multiples of (a + 1, a, b). For example (50, 40, 30), (39,36,15). (The first 5 hints are often used in starting a puzzle.)
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Glossary Arithmetic progression (AP) Three or more numbers a0 , a1 , a2 , . . . form an AP if the difference between successive numbers is constant. That is, ai+1 − ai = ai − ai−1 . For example 3, 7, 11, 15 form an AP, with a common difference of 2. Binary numbers Binary numbers are numbers with base 2. Normal (decimal) numbers have base 10, so that 100010 = 1 × 103 + 0 × 102 + 0 × 101 + 1, whilst 10012 = 1 × 23 + 0 × 22 + 0 × 21 + 1 = 910 . Composite A composite number is not prime. For example, 6 (= 2 × 3) is composite. Congruent Two numbers are congruent modulo n if their remainders after being divided by n are equal. Or, equivalently, if n is a divisor of their difference. For example, 64 and 28 are congruent modulo 12. This can be expressed as 64 ≡ 28 (mod 12). Coprime Two numbers are coprime if their only common factor is 1. For example 16 and 21 are coprime, and 16, 21 and 25 are mutually coprime. div Integer divide. Any remainder is discarded. For example, 14 div 5 is 2. (cf mod) Divisor A divisor of a number is any number which divides it without leaving a remainder. A (positive) divisor of n that is different from n is a proper divisor of n. For example, the positive divisors of 10 are 2, 5 and 10. The proper divisors of 10 are 2 and 5. Factor A synonym of divisor. Fibonacci numbers The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, ... Each Fibonacci number after the second is the sum of the preceding two Fibonacci numbers. Geometric mean √ The geometric mean of two numbers a and b is a × b. For example, the geometric mean of 8 and 18 is 12. Geometric progression (GP) Three or more numbers a0 , a1 , a2 , . . . form a GP if if the ratio between successive numbers is constant. That is, ai+1 /ai = ai /ai−1 . For example 2, 6, 18, 54 form a GP with common ratio 3, and 20, 30, 45 form a GP with common ratio 1.5. 6
Greatest common divisor (GCD) The greatest common divisor of two numbers is the largest number which is a factor of both numbers. For example the greatest common divisor of 12 and 20 is 4. Lowest common multiple (LCM) The lowest common multiple of two numbers is the smallest number which is a multiple of both numbers. For example the lowest common multiple of 12 and 20 is 60. Magic square In a magic square, the sums of the rows, columns and diagonals are equal. Normally, duplicates are not allowed, but in some of the puzzles in this book there may be duplicates. Examples: No duplicates Duplicates 5 1 6 4 9 2 5 4 3 3 5 7 2 7 3 8 1 6 mod (modulo) a mod n is the remainder after a is divided by n. For example, 12 mod 5 is 2. (cf div) Number of ways of choosing na n of choosing a elements from n elements without replacement. a is the number of ways n n = n × (n − 1)/2 and 2 3 = n × (n − 1) × (n − 2)/6. For example, the number of ways of choosing 2 elements from 10 elements is 45. Octal numbers Octal numbers are numbers with base 8. Normal (decimal) numbers have base 10, so that 43210 = 4 × 102 + 3 × 101 + 2, whilst 4328 = 4 × 82 + 3 × 81 + 2 = 28210 . Palindrome A number (or word) which is the same when reversed. For example, 1221 is a palindrome. Prime The only factors of a prime number are itself and 1. For example, 11 is prime. Prime factor A factor which is a prime. For example, 2 is a prime factor of 20, but 4 is not. Pythagorean triple Three numbers a, b and c form a Pythagorean triple if a2 = b2 + c2 . Examples of Pythagorean triples are (3, 4, 5) and (30, 40, 50). Triangular numbers The triangular numbers are 1, 3, 6, 10, 15, 21, ... The nth triangular number is n × (n + 1)/2.
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