The Most Beautiful Mathematical Magic Games & Puzzles (01) 宽柔 Problem Solving Strategies Creative Thinking Skills SP 25 Jan 2007 The Art of Investigation By TengCH

16 of The Most Beautiful Mathematical Magic, Games & Puzzles (01) The Flash Mind Reader Crystal ball magic Sum of 10 numbers Fibonacci Magic 3-digit numbers, abc Magic Number 9 Five Tetrominoes $10K Puzzle 5 x 4 rectangle Magic Tables Binary Magic Secret of Dies Traffic Jam Leap frogs Best Team-building game Tower of Hanoi （河内之塔） Mathematical Recurrency Sum to 20 Game strategy 3 levels. Bai Qian Mai Bai Ji 百钱买百鸡） Problem of the 100 Fowls Han Xin Dian Bin （韩信点兵) Remainder Theorem 9 Flips Consecutive Sum The Singapore Polytechnic Lockers Winners & the Chocolates $5 & $2 notes Who keep the Fish? (谁家养鱼?) & More The Art of Investigation

MIND Reader The Flash Mind Reader Think of a two digit number Add both digits together Subtract the total from your original. Look up on the chart for your final number. Find the relevant symbol. Click on the crystal ball. The Flash Mind Reader

Fibonacci Magic Get two participants as Volunteers 2 Sum of 10 numbers Fibonacci Magic Get two participants as Volunteers Each of them suggests a number, any number between 1 to 20. The third number is the sum of the first two numbers, the forth number will be the sum of second & third number, so on and so forth, The subsequence number will be the sum of the previous two numbers, until you have all the 10 numbers Now, ask the volunteers to add up all the 10 numbers. ( Someone will be able to tell you the SUM well before they have completed the calculation. Why?) Fibonacci Magic

Cast out 9, Divisible by 9 Think of any three digit number ABC 3 Magic Number 9 Think of any three digit number ABC Rearrange the same three digits in any order to form another number, eg. BAC Work out the difference of the 2 numbers. You get xyz or xy Remove one of the digit (except 0)from your answer, and show me the remaining digits. I will be able to tell the digit that you had removed. Why? How? Three different digits http://trunks.secondfoundation.org/files/psychic.swf Cast out 9, Divisible by 9 Cast out 9, Divisible by 9

4. The Five Tetrominoes magic/puzzle The $10K Puzzle Using the 5 different shapes of tetrominoes. Can you fit them together to form a 4 x 5 rectangle as shown? Pieces may be turned over and placed with either side up. You will be rewarded with $10K if you form it within one hour Trace the five shapes shown in the Figure on a sheet of cardboard or stiff paper, and cut them out. Can you fit them together to make the 4 x 5 rectangle as shown in ? Pieces may be turned over and placed with either side up. 4 SQ 5 pieces

5. Magic Tables Table A Table B Table C 1 3 5 7 2 3 6 7 4 5 6 7 9 11 13 15 10 11 14 15 12 13 14 15 17 19 21 23 18 19 22 23 20 21 22 23 25 27 29 31 26 27 30 31 28 29 30 31 Table D Table E 8 9 10 11 16 17 18 19 12 13 14 15 20 21 22 23 24 25 26 27 24 25 26 27 28 29 30 31 28 29 30 31

The Secret of Dice

7. Traffic Jam - Fishing Boat Leap-Frog Ten Men are fishing from a boat, five in the front, five in the back, and there is one empty seat in the middle. The five in front are catching all the fish, so the five at the back want to change seats. To avoid capsizing the boat, they agree to do so using the following rules: A man may move from his seat to and empty seat next to him. A man may step over only one man to an empty seat. No other move are allowed. What is the minimum number of moves necessary for the men to switch places? If there are n men from each side, how many moves is needed for the swap?

Geometrical Series 8. Tower of Hanoi 河内之塔 http://www.mathsnet.net/puzzles/hanoi/ Geometrical Series

Geometrical Series How many moves are required? 8. The Tower of Hanoi 河内之塔 The French mathematician Edward Lucas (1842-1891) constructed a puzzle with three pegs and seven rings of different sizes that could slide onto the pegs. Starting with all the rings in one peg in order by size, the problem is to transfer the pile to another peg subject to two conditions: Rings are moved one by one, and no ring is ever placed on top of a smaller ring. Legend has it that an order of monks had a similar puzzle with 64 large golden disks. The monks supposedly believed that the world would crumble when the job was finished. How many moves are required? For n rings? http://www.mathsnet.net/puzzles/hanoi/ Geometrical Series

Sum to 20 Select all the cards with 1 to 5 9. Select all the cards with 1 to 5 You are now having a pool of cards with 4 sets of cards from 1 to 5, all cards are open, facing up. Play between 2 players (0r 2 teams of players) The players take turns to choose a card from the pool, and sum up the numbers of all the cards selected from both players Whoever gets the last card that the total sum reaches 20 win the game. Who will win? How?

Problem of the Hundred Fowls 10 Bai Qian Mai Bai Ji 古代中国算经 Ancient Chinese Mathematical Problem 百钱买百鸡 A man paid exactly 100 dollars for 100 chicken A rooster cost $5 each, a hen cost $3 each, and a dollar for 3 chicks How many roosters, hens and chicks did the man buy? Problem of the Hundred Fowls The 100 monks and 100 Buns Problem

（韩信点兵） 11 Han Xin Dian Bing 1/2 韩信 Han Xin, an Han dynasty general, devised a method to count the exact number of his soldiers. He arranged them in rows of 5, 6, 7 and 11, from the remainders, he will be able to know the exact number of his soldiers. How did he do that? With the respective remainders of 1,5, 4,10, What is the exact number of 韩信’s soldiers? Art of Simplification 2111

Han Xin Dian Bing Solution Multiplication of remainder 韩信点兵 Simplification Methods No Divisor D Remainder X Multiplication of remainder value N 3 2 5x7=35 1 35 5 3x7=21 63 7 3x5=15 30 Sum 128 LCM 3x5x7 105 Final Answer N= 23 Two Remainder Theorems: 余数定理 Number X multiply by M, remainder also multiply by M Addition of Multiple of divisor, X + D x M, Remainder unchanged Art of Simplification

Han Xin Dian Bing Solution Beautiful Remainder Theorem Two Remainder Theorems: 余数定理 Number X multiply by M, remainder also multiply by M Addition of Multiple of divisor, X + D x M, Remainder unchanged 韩信点兵 Han Xin Dian Bing; the real question Number Divisor remainder X Multiplication of remainder Final value N 5 1 6 7 4 11 10 Sum LCM N= Beautiful Remainder Theorem

X 9 lkjihgfedcba Simplify abcdefghijkl 12. 9 Flips abcdefghijkl X 9 lkjihgfedcba What is the 12 digit number abcdefghijkl ? Suppose that N is a positive number written base 10, and that 9xN has the same digits as N but in a reversed order. Then we shall say for short that N is a 9-Flip Find all 9-flips with 12 digits Is it possible to say exactly how many 9-flips there are with precisely n digits? Simplify

Exactly which numbers have this property? For example, observe that; 13. Consecutive Sums Some numbers can be expressed as the sum of a string of consecutive positive numbers, Exactly which numbers have this property? For example, observe that; 5=2+3 9=2+3+4 =4+5 11=5+6 18=3+4+5+6 =5+6+7 What are the consecutive numbers that sum to 30? 30= ? How about 105? 315? 2310 = ??

Exactly which numbers have this property? 13. Consecutive Sums Some numbers can be expressed as the sum of a string of consecutive positive numbers, Exactly which numbers have this property? What are the numbers have no consecutive sum? Old or even integers? average Exactly How many solutions will it be? If there are more than one solution. How to determine the number of solutions? The Methodology? Fn= ? 1=, 2= 3=, 4=, 5=, 6=, 7=, 8=, 9=,10=,… The single solution problem. For example, observe that; 5=2+3 9=2+3+4 =4+5 11=5+6 18=3+4+5+6 =5+6+7 What are the consecutive numbers that sum to 30? 30= ? How about 105? 315? 2310 = ??

14. The Singapore Polytechnic Lockers At Singapore Polytechnic, there were 1,000 students and 1,000 lockers (numbered 1-1000). At the beginning of our story, all the lockers were closed. The first student come by and opens every locker. Following the first students, the second student goes along and closes every second locker. The third student changes the state, ( if the locker is open, he closes it; if the locker is closed, he opens it) of every third locker. The fourth student changes the state of every fourth locker, and so forth. Finally, the thousandth student changes the state of the thousandth locker. When the last student changes the state of the last locker, Which lockers are open?

15. Winners & the Chocolates (2/3) Geometrical Series 15. Winners & the Chocolates (2/3) After a mathematics quiz, Mrs Lai YM gave the three prize winners a box of chocolate Bars to share. The first winner received 2/3 of the chocolate Bars plus 1/3 of a bar. The second winner received 2/3 of the remainder plus 1/3 of a bar, The Third winner received 2/3 of the New remainder plus 1/3 of a bar. And there will no chocolate Bars left after this. How many chocolate Bars were there in all? How about if there was One bar Left? How about if there were 5 winners? Base 3 Draw a diagram

LCM units draw a diagram 16. $5 & $ 2 notes The number of $5 notes to $2 notes is in the ratio 3 : 2 . When $50 worth of $2 notes are converted to $5 notes, the new ration is 8 : 5. How many $5 notes are there? PSLE question LCM units draw a diagram

17 Who keep the Fish? 谁家养鱼? Albert Einstein once posed a brain teaser that he predicted only 2% of the world population would get. FACTS 1. There are 5 houses in 5 different colours 2. In each house lives a man with a different nationality 3. These 5 owners drink a certain beverage, smoke a certain brand of cigarette and keep a certain pet 4. No owners have the same pet, brand of cigarette or drink

CLUES 1. The Brit lives in a red house 2. The Swede keeps a dog 17 Who keep the Fish? 谁家养鱼? CLUES 1. The Brit lives in a red house 2. The Swede keeps a dog 3. The Dane drinks tea 4. The green house is on the left of the white house 5. The green house owner drinks coffee 6. The person who smokes Pall Mall keep birds 7. The owner of the yellow house smokes Dunhill 8. The man living in the house right in the center drinks milk

9. The Norwegian live in the first house 17. Who keep the Fish? 谁家养鱼? 9. The Norwegian live in the first house 10. The man who smokes Blend lives next to the one who keeps cats 11. The man who keeps horses lives next to the man who smoke Dunhill 12. The owner who smokes Blue Master drinks beer 13. The German smokes Prince 14. The Norwegian lives next to the blue ouse 15. The man who smokes Blend has a neighbour who drinks water The question is, who keeps the fish? This is not a trick question- it is a genuine logic puzzle.....

How? Rearrange the numbers, such that 18. 1 -9 Magic Number Rearrange the numbers, such that Sum of the 8 numbers in the larger circle Could be divisible by The product of the 3 numbers in the smaller circle How? What is your approach?

36 Hand-Shakes During the Foon Yew Maths Society gathering @ the auditorium All members will shake hands with each and everyone. If, there were all together 36 hand- shakes, How many members are there? Simplify

20. The Handshaking party （握手言欢） On one Saturday, At the Foon Yew High Alumni gathering @ City Square, only five married couples turn out (never happened, fictitious) No person shakes hands with his or her spouse. Of the nine people other than the host, Tan CH, no two shake hands with the same number of people. With how many people does Mrs. Tan, the hostess shake hands? Generalise n couples ?

21.Ages of my Three Children Two friends, Chia How and Chong Heng, met at a Foon Yew High gathering @ CITY Sqaure on Sat, after not having seen each other for many years. As they talk,Chia How asked, ”How many children do you have and what are their ages?” “I have three children, the product of their ages is 36, and the sum of their ages is your house number.” answered Chong Heng. Chia How thought for a moment and then said, “I need more information to solve the problem.” “Oh yes,” replied Chong Heng.”My oldest child is a girl.” With this additional information, Peter immediately found the answer. How did Chia How figure out the ages of the children, and what were their ages?

Permutation & Combination 22. Mathematicians 3 Mathematics Tutors are seated one behind another. Another person showed them 3 grey hats and 2 white hats, blindfolded them, put one hat on each head, and threw the rest away. When the blindfolds were off, they all looked in front of them. Each was asked in turn what colour hat she or he had. No one could answer. After a long thoughtful silence, the person in front who could see no one’s hat was able to respond correctly about his own hat. How was this done ? Permutation & Combination

23. Crossing the Desert A man has to deliver a message across a desert. Crossing the desert takes 9 days. One man can only carry enough food to last him twelve days. No food is available where the message must be delivered, but food can be buried on the way out and used on the way back. There are two men ready to set out together. Can the message be delivered and both men return to where they started without going short of food? Draw a model

-- f g h i j 3 3 3 3 3 Only One Solution? a b c d e 24. 0 -9 Magic Number a b c d e -- f g h i j 3 3 3 3 3 ‘abcde’ is a five digit number & ‘fghij’ is either a 5 digit or a 4 digit number The Alphabets a,b,c,d,e,f,g,h,i j represent distinct positive integers from 0 - 9 Find the two numbers What are a,b,c,d,e,f,g,h,I,j represent?? 4 Only One Solution? 4870215369

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