Mathematical Puzzles

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Mathematical PuzzlesThese puzzles have been adapted from various sources to be used with pupils who finish classwork early. Most of the questions were chosen with enthusiastic, bright 11 year olds in mind. Some of the puzzles are also appropriate for class work - an initial worked example on the board will help a lot. There are a few trick questions. Some questions can be quickly answered if you chance upon the right approach, but the 'long' solution isn't too arduous. Remove Answers Show Answers

Scales and Vessels1. How can you measure out exactly 4 litres of water from a tap using a 3 litre and a 5 litre bucket? Ans2. 3. 4. 5. 6. 7. 8. 9. 3litre ----0 3 0 2 2 3 5litre -----5 2 2 0 5 4

10. A 24 litre bucket is full of lemonade. 3 men want to have equal amounts of it to take home, but they only have a 13 litre, a 5 litre and an 11 litre bucket. How do they do it? Ans11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 24 13 11 5 ---------24 0 0 0 11 13 0 0 6 13 0 5 6 2 11 5 8 0 11 5 8 5 11 0 8 13 3 0 8 8 3 5 8 8 8 0

22. A Queen (78kg), the Prince (36kg) and the King (42kg) are stuck at the top of a tower. A pulley is fixed to the top of the tower. Over the pulley is a rope with a basket on each end. One basket has a 30kg stone in it. The baskets are enough for 2 people or 1 person and the stone. For safety's sake there can't be more than a 6kg difference between the weights of the baskets if someone's inside. How do the people all escape? Ans

23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

Basket 1 Basket 2 --------------------Stone up Prince down King down Prince up nothing up Stone down Queen down Stone and King up nothing up Stone down Prince down Stone up nothing up Stone down King down Prince up Stone up Prince down

34. One out of 9 otherwise identical balls is overweight. How can it be identified after 2 weighings? Ans: Weigh 3 against 3, then you'll know which group of 3 contains the heavy ball. Pick 2 balls from that group and weigh one against the other. 35. One out of 27 otherwise identical balls is overweight. How can it be identified after 3 weighings? Ans: Weigh 9 against 9, then 3 against 3. 36. How many ways can you put 10 sweets into 3 bags so that each bag contains an odd number of sweets? Ans 15 solutions. The first trick is to realise that if you put one bag inside another, then sweets in the inner bag are also in the outer bag. The only workable configuration is to put one bag inside another and leave the third alone. The answers can be obtained using the following octave script, where bag b is inside bag a37. for a=0:10 38. for b=0:(10-a) 39. c=10-a-b; 40. if (rem((a+b),2)==1 && rem(b,2)==1 && rem(c,2)==1) 41. fprintf('a=%d b=%d c=%d\n',a,b,c) 42. end 43. end 44. end

Ferries1. A man has to take a hen, a fox, and some corn across a river. He can only take one thing across at a time. Unless the man is present the fox will eat the hen and the hen eat the corn. How is it done? Ans2. 3. 4. 5. 6. 7. 8. MAN AND HEN ->

9. 3 missionaries and 3 obediant but hungry cannibals have to cross a river using a 2-man rowing boat. If on either bank cannibals outnumber missionaries the

missionaries will be eaten. How can everyone cross safely? Ans10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. CANNIBAL and MISSIONARY ->

21. 2 men and 2 boys need to cross a river in a boat big enough for 1 man or 2 boys. How do they do it? Ans22. 23. 24. 25. 26. 27. 28. 29. 30. BOY and BOY ->

SMP and CSE 1974 extend this to cover the case of n men.

Picking Captains1. 6 boys pick a captain by forming a circle then eliminating every n'th boy. The 2nd boy in the counting order can choose n. If he wants to be captain what's the smallest n he should pick? Ans: 10 2. 12 black mice and 1 white mouse are in a ring. Where should a cat start so that if he eats every 13th mouse the white mouse will be last? Ans: If the white mouse is 1st in the counting order, the cat should start at the 7th mouse : hint - start anywhere, see how far out you are, then make the necessary correction 3. 20 passengers are in a sinking ship. 10 are mathematicians. They all stand in a ring. Every 7th climbs into the lifeboat which can only hold 10 people. Where should the mathematicians stand in the ring? Ans: 1 4 5 7 8 9 14 15 16 17 4. 30 passengers are in a sinking ship. They all stand in a circle. Every 9th passenger goes overboard. The lifeboat holds 15. Where are the 15 lucky positions in the circle? Ans: 1 2 3 4 10 11 13 14 15 17 20 21 25 28 29 The last of these questions can be answered using the following octave scripthowmanyatstart=30; howmanyatend=15;

first=1; leap=9; x=first-1; a=1:howmanyatstart; while (length(a)>howmanyatend) x=rem(x-1+leap,length(a)); a(x+1)=[]; end a

By changing the initial values in this script you can solve questions 2 and 3.

Incomplete Sums(Worked Examples in J.A.H. Hunter's "Mathematical Brain Teasers"). 1. Each letter represents a different digit2. 3. 4. 5. SEND +MORE ----MONEY

Ans: 9567+1085 = 10652 6. This sum uses all the digits7. 8. 9. 10. 28* +**4 ---****

Ans: 289+764 = 1053 11. This subtraction sum uses all the digits from 1 to 9.12. 9 * * 13. - * 4 * 14. ----15. * * 1

Ans:927 - 346 = 581 16. O represents odd digits E represents even digits17. EEO 18. xOO 19. ----20. EOEO 21. EOO 22. ----23. OOOOO

Ans: 285x39

24. P represents prime digits25. 26. 27. 28. 29. 30. 31. PPP xPP ----PPPP PPPP ----PPPPP

Ans: 775x33 32. Some more additionso o o o o o o o o o o o o o o o o o o o o o o o o o THE TEN MEN ---MEET SLOW SLOW OLD ---OWLS

Ans: 490+407+107=1004

Ans: 4712SAL SEE THE SUEZ ----CANAL

Ans: 920+977+547+9876=12320FIVE FIVE NINE ELEVEN -----THIRTY

Ans: 4027+4027+5057+797275

LettersAgree on a font of capital letters.

Put the letters into sets according to line symmetry Put the letters into sets according to rotational symmetry Put the letters into sets according to topology How many letters only use straight lines? There is only one number whose English name uses as many straight lines to write as the number itself. Think of a number. Write it out in words. Write in words the number of letters you've used (E.g. SIXTEEN-SEVEN-FIVE-FOUR). Continue do so and see what

happens. Try 3 other numbers. > You always end at FOUR. In german you end up at VIER.

Numbers1. Alan, Bill and Chris dug up 9 nuggets. Their weights were 154, 16, 19, 101, 10, 17, 13, 46 and 22 kgs. They took 3 each. Alan's weighed twice as much as Bill's. How heavy were Chris's nuggets? Ans: 272 2. The product of 3 brothers' ages is 175. Two are twins. How old is the other one? Ans: 7 3. A man has 2 bankcards, each with a 4 digit number. The 1st number is 4 times the 2nd. The 1st number is the reverse of the 2nd. What is the first number? Ans: 8712 4. Tom has 7 sandwiches, Jan has 5, Simon has none. They share them out equally. Simon leaves, paying for his sandwiches by leaving 12 biscuits. What's the fairest way for Tom and Jan to share out the biscuits? Ans: 3 to Jan 5. A cyclist buys a cycle for 15 pounds paying with a 25 pound cheque. The seller changes the cheque next door and gives the cyclist 10 pounds change. The cheque bounces so the seller paid his neighbour back. The cycle cost the seller 11 pounds. How much did the seller lose? Ans: 21 pounds? 6. Using four "4"s and common symbols (including the square root, factorial and recurring decimal symbols), make sums whose answers are 0, 1, 2....100 (See Mathematical Bafflers) 7. Make fractions (each using all the digits from 1 to 9) with these values 1/2, 1/3....1/9 Ans:8. 6729/13458, 5823/17469, 3942/15768, 2697/13485, 2943/17658, 9. 2394/16758, 3187/25496, 6381/57429

10. A greengrocer was selling apples at a penny each, bananas at 2 for a penny and pears at 3 for a penny. A father spent 7p and got the same amount of each type of fruit for each of his 3 children. What did each child get? Ans: 1 apple, 2 bananas and 1 pear. 11. A woman bought something costing 34c. She only had 3 coins: $1, 2c and 3c. The shopkeeper had only 2 coins: 25c and 50c. Fortunately another customer had 2 10c coins, a 5c coin, 2 2c coin and a 1c coin. How did they sort things out? Ans: They pool the money. The woman takes 74, the shopkeeper takes 109 and the customer 28. 12. Mr and Mrs A are 120 km apart. A bee is on Mr A's nose. The couple cycle towards each other, Mr A at 25km/h and Mrs A and 15km/h. The bee dashes from Mr A's nose to Mrs A's nose and back again and so on at 60km/h. How far does the bee travel before the cyclists crash? Ans: The cyclists crash after 3 hours so the bee flies 180km.

13. Pick a number. If it's even, divide by 2. If it's odd multiply by 3 and add 1. Continue this until you reach "1". Eg 3-10-5-16-8-4-2-1. Which integer less than 100 produces the longest chain? Ans: 97 leads to a 119 link chain. The lowest number that causes a long chain is 27 (112 links). 14. Pick a number. Multiply the digits together. Continue until you get a single digit. What is the only 2 digit number which would require more than 3 multiplication? Ans: 77 15. Starting with 1, place each integer in one of 2 groups so that neither contains a 3 term Arithmetic Progression. How far can you go? Ans: Up to 8. 16. The following 2 questions use the following result - "Given integers a and b the biggest number that can't be expressed in the form ia + jb is ab - a - b." o Apples are packed in boxes of 8 and 15. What is the biggest number of apples that would require loose apples? Ans: 97 o A country only has 5p and 7p coins. Make a list of prices that you could give exact money for. What is the highest prices that you couldn't give exact money for? Ans: 23 17. If D = the day (1-366) in year Y, then the day of the week can be calculated using18. d = D+Y+(Y-1)/4 - (