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Transcript of Mathematical Puzzles - FBE - · PDF fileHochschule Bremen Department of Computer Science...

  • Hochschule BremenDepartment of Computer Science

    Mathematical Puzzles

    Prof. Dr. Th. Risse

    An amusing,brisk and cool,

    enriching and entertaining,informative and oriented towards practical applications,

    playful,relevant and rewarding,


    little contribution to the (general) mathematical education!

    c 20022014 risse(at)hs-bremen.deLast Revision Date: August 8, 2014 Version 0.5


  • Table of Contents

    0. Introduction1. Riddles [7]

    Measuring with Two Jugs Races Census and its Boycott Zig-Zag between Trains Outward and Return JourneyMagic Squares Conspicuous Text No Talk about Money Corrupt Postal System Equal Opportunities Two andMore Eyes

    2. More Riddles [11] Matches Decanting Analytical Riddles I AnalyticalRiddles II Analytical Riddles III Analytical Riddles IV Analytical Riddles V Crossing a Bridge Synthetic Rid-dles I Synthetic Riddles II Synthetic Riddles III Syn-thetic Riddles IV Dialectic Riddle Riddles, 588 Riddles,622 Labyrinth, 652 Riddles, 680 Riddles, 708 Riddles,734 Riddles, 750 Riddles, 772

    3. Prime Numbers

  • Table of Contents (cont.) 3

    Fermat-Numbers Euler-Numbers Mersenne-Numbers4. Computations with Remainders

    Crucial is What is Left Over Computing With Remain-ders Adroit Computing With Remainders Euclid & littleFermat Fermat, Euler and More Chinese Stuff GaloisFields GF(p) Galois Fields GF(pn)

    5. Cryptography Caesar and Cohorts Caesar in General Vigenere andAccomplices Permutations DES Public Keys? RSA AES Elliptic Curves over R Elliptic Curves over GF(p) Elliptic Curves over GF(2m) Elliptic Curve Cryptogra-phy, ECC

    6. Compression Exploiting Relative Frequencies Using Dictionaries

    7. Probability & Intuition Cards & Goats Algorithms to Generate Chance? Whatis Randomness?

    8. Sources and LinksSolutions to Problems

  • 4

    0. Introduction

    To begin with Youll find some mathematical riddles. But there ismore serious stuff. Several algorithms to be tried are provided by thisdocument to explore procedures of cryptography, coding, probability,etc.There are other in this sense interactive documents, (German)

    The functionality of pdf-documents provides

    convenient selection of problem areas of interest or of single prob-lems and, uniquely, execution of algorithms

    easy navigation between problem and solution and vice versa,

    simple visit of the numerous links to informations on our webDAVserver or in the WWW.

  • 5

    1. Riddles [7]

    Measuring with Two Jugs

    Problem 1. There are two jugs at hand with a capacity of p ` andq ` liters and any amount of water.What quantities m of water can be measured out?

    (a) p = 5, q = 3, m = 4(b) p = 5, q = 3, m = 1(c) p = 4, q = 9, m = 1, 2, . . . , 13(d) p = 6, q = 3, m = 4


    Problem 2.

    (a) Climbing a 3000m mountain top Sisyphos makes 300m a day onlyto loose 200m each night again.Wenn does Sisyphos reach the top?

  • Section 1: Riddles [7] 6

    (b) At a 100m race the first runner A beats the second B by 10m,and the second B beats the third C by 10m.How many meters is the first runner A ahead of the third C whencrossing the finishing line?

    Census and its Boycott

    Problem 3.

    (a) At a census there is the following dialog:Field helper: number of children?Citizen: three!Field helper: age of Your children in whole numbers?Citizen: The product of the years is 36.Field helper: This not a sufficent answer!Citizen: The sum of the ages equals the

    number of the house of our next neighbour.(Field helper acquires the number.)

    Field helper: That is still not a sufficient answer!

  • Section 1: Riddles [7] 7

    Citizen: Our eldest child plays the piano.How old are the three children?

    Zig-Zag between Trains

    Problem 4.

    (a) Two trains start on the same line 100km apart to drive at 50km/htowards each other. A fly flies from one to the other at 75km/h.How many kilometres has the fly travelled up to its unavoidablefate?

    Outward and Return Journey

    Problem 5.

    (a) In A somebody gets up at sunrise and walks with many rests toB where he arrives at sunset.The next day he walks back on the same route, again pausing abit here and there.

  • Section 1: Riddles [7] 8

    There is a point of the route the roamer at the same time of dayhits both on the outward as on the return journey.

    Magic Squares

    Problem 6.

    (a) Magic squares are natural numbers arranged in a square grid, i.e.a quadratic matrix, such that the sum of all numbers in each row,in each column, and in each diagonal are all the same!

    a b c

    d e f

    g h i

    mita + b + c = s . . .a + d + g = s . . .a + e + i = s . . .

    Taking symmetry into consideration, there is exactly one magicsquare consisting of the natural numbers 1, 2, . . . , 9 arranged ina 3 3-matrix.

  • Section 1: Riddles [7] 9

    Conspicuous Text

    Problem 7.

    (a) Study this paragraph and all things in it. What is virtually wrongwith it? Actually, nothing in it is wrong, but you must admitthat it is most unusual. Dont zip through it quickly, but studyit scrupulously. With luck you should spot what is so particularabout it. Can you say what it is? Tax your brains and try again.Dont miss a word or a symbol. It isnt all that difficult.

    No Talk about Money

    Problem 8.

    (a) The boss in an office wants to acquire the average salary of hisemployees without getting to know individual salaries und thusbreaking privacy. How does he proceed?

  • Section 1: Riddles [7] 10

    Corrupt Postal System

    Problem 9.

    (a) In a corrupt postal system each letter is opened and the contentstolen independently of its value. Only securely closed strongboxes are delivered reliably (because it takes too much hassle toopen them).How can Bob send a valuable item to Alice in some strong boxwhich can be locked with several locks when they both can com-municate about the transfer?

    Equal Opportunities

    Problem 10.

    (a) Alice and Bob live in different cities and decide to go to see eachother in turns. They want to find out who starts to drive to theother by tossing a coin.How do they find out if they live in different cities?

  • Section 1: Riddles [7] 11

    Two and More Eyes

    Problem 11. It is called the Two Eyes Principle if two persons eachwith a separate key are necessary to open a treasure box, or if twopasswords are necessary to open a file.Each person opens her/his lock of the treasure box by her/his ownkey or adds her/his part of the password to complete the password.

    (a) Alice, Bob and Claire own a treasure box with several locks. Theywant to make sure that only at least two persons together can getat the content of the treasure box.How many locks and how many keys to each lock do they need?

    (b) Now, Alice, Bob, Claire and Denis want to be sure that only atleast two persons together can open the treasure box.Minimally how many locks and minimally how many keys to eachlock do they need?

    (c) Only at least m persons together out of a total of n persons aremeant to be able to open the treasure box.How many locks, how many keys do they need?

  • 12

    2. More Riddles [11]


    Problem 12. Move a given number of matches in order to generatea given number of equally sized squares.

    (a)Move four matches in order to generate threeequally sized squares.

    (b)Move two matches in order to generate fourequally sized squares.

    (c)Move three matches in order to generate threeequally sized squares.

    (d)Move three matches in order to generate fiveequally sized squares.

  • Section 2: More Riddles [11] 13


    Problem 13.

    (a) How can one get 6 litres water from a river if there are only a fourlitre and a nine litre bucket available?

    (b) How can one get exactly 1 litre from a container if there are onlya 3-litre and a 5-litre container available?

    (c) A 8 litre canister is filled with wine. How to decant 4 litre if thereare only a 3-litre and a 5-litre jug available?

    (d) A barrel contains 18 litres wine. There is a 2-litre can, a 5-litrejug and a 8-litre bucket. How to distribute the wine such that thebarrel contains half of it, the bucket a third, the jug a sixth?

    Analytical Riddles I

    Problem 14.

    (a) Let the sum of the ages of a family of four, father, mother, andtwo children, be 124. The parents together are three times as

  • Section 2: More Riddles [11] 14

    old as the children. The mother is more than twice as old as theoldest child. Age of father minus age of mother is nine times thedifference age of the oldest minus age of the youngest child. Howold is each member of the family?

    (b) Emil is 24 years old. Hence, he is twice as old as Anton has beenwhen Emil was as old as Anton is now. How old is Anton?

    (c) In a supermarket one gets a deduction of 20%, but has to pay 15%turnover tax. What is best, first to deduct the discount or first topay the tax?






    The L-shaped area is to be divided into four con-gruent subareas.

    (e) If Fritz was 5 years younger then he was twice as old as Paul waswhen he was 6 years younger. Wenn Fritz was 9 years older thenhe was trice as old as Paul when Paul was 4 years younger. Howold are Fritz and Paul?

  • Section 2: More Riddles [11] 15