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Page 1: Mathematical Puzzles

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 Vessels

1. How can you measure out exactly 4 litres of water from a tap using a 3 litre and a 5 litre bucket? Ans

2. 3litre 5litre3. ----- ------4. 0 55. 3 26. 0 27. 2 08. 2 59. 3 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? Ans

11. 24 13 11 512. ----------13. 24 0 0 0 14. 11 13 0 0 15. 6 13 0 516. 6 2 11 517. 8 0 11 518. 8 5 11 019. 8 13 3 020. 8 8 3 521. 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

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23. Basket 1 Basket 224. ---------------------25. Stone up Prince down26. King down Prince up27. nothing up Stone down28. Queen down Stone and King up29. nothing up Stone down30. Prince down Stone up31. nothing up Stone down32. King down Prince up33. 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 a

37. for a=0:1038. 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. end43. end44. end

Ferries

1. 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? Ans

2. MAN AND HEN ->3. <- MAN 4. MAN AND FOX ->5. <- MAN AND HEN6. MAN AND CORN ->7. <- MAN 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

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missionaries will be eaten. How can everyone cross safely? Ans

10. CANNIBAL and MISSIONARY ->11. <- MISSIONARY12. CANNIBAL and CANNIBAL ->13. <- CANNIBAL14. MISSIONARY and MISSIONARY ->15. <- CANNIBAL and MISSIONARY16. MISSIONARY and MISSIONARY ->17. <- CANNIBAL18. CANNIBAL and CANNIBAL ->19. <- CANNIBAL20. CANNIBAL and CANNIBAL ->

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? Ans

22. BOY and BOY ->23. <- BOY24. MAN ->25. <- BOY26. BOY and BOY ->27. <- BOY28. MAN ->29. <- BOY30. BOY and BOY ->

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

Picking Captains

1. 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 script howmanyatstart=30;howmanyatend=15;

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first=1;leap=9;

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

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 digit 2. SEND3. +MORE4. -----5. MONEY

Ans: 9567+1085 = 10652

6. This sum uses all the digits 7. 28*8. +**49. ----10. ****

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 digits 17. EEO18. xOO19. -----20. EOEO21. EOO22. ----- 23. OOOOO

Ans: 285x39

24. P represents prime digits

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25. PPP26. xPP27. -----28. PPPP29. PPPP30. -----31. PPPPP

Ans: 775x33

32. Some more additions o THEo TENo MENo ----o MEET

o Ans: 490+407+107=1004 o SLOWo SLOWo OLDo ----o OWLS

o Ans: 4712 o SALo SEEo THEo SUEZo -----o CANAL

o Ans: 920+977+547+9876=12320 o FIVEo FIVEo NINEo ELEVENo ------o THIRTY

o Ans: 4027+4027+5057+797275

Letters

Agree 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.

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Numbers

1. 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?

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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 using 18. d = D+Y+(Y-1)/4 - (Y-1)/100 + (Y-1)/400 mod7

where d=1 would mean Sunday, etc. Can the first day of each century (e.g. 1st Jan 2001, 1st Jan 1901) be any day? Ans: No. Just Monday, Tuesday, Thursday or Saturday. See "Mathematical Bafflers"

19. Pick 3 digits (not zero) and make 6 2-digit numbers from them. Add up all these numbers, add up all the original digits and divide the first total by the second. Ans: 22

20. How many presents did the "true love" send during the 12 days of Christmas? Ans: 364

21.

At a fairground stall there are 3 piles of cans. You get 3 throws. You can only knock off the top can of a pile. The 2nd throw counts double, the 3rd triple. How do you get exactly 50? Ans: 7, 8, 9

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Enumerations

1. Holding it's hands out, palms upward a child starts counting on all its fingers and thumbs, going to and fro. If it counts up to 1982 which finger does counting end on? Ans: (middle finger is 3 or 7 (mod8)).

2. You have 3 bricks, each measuring 18 x 9 x 6 cm. How many different heights can you build up with them? Ans: 14

3. How many right-angled triangles with integral sides have one side of 15? Ans

4. 9, 12, 15 5. 15, 20, 256. 15 112 1137. 8 15 178. 15 36 39

9. Minibuses seating 10, 12 and 15 passengers can be used to convey 120 passengers. There are 5 of each size of bus. How many different ways can the buses be used so that all the ones used are full? Which way uses the least buses? Ans

10. 10 12 1511. --------12. 3 5 213. 4 3 314. 5 4

15. Roosters cost 5 pounds, hens 3 pounds and 3 chicks cost 1 pound. Buying at least one of each type of bird how can you buy 100 birds for exactly 100 pounds? Ans

16. Rooster Hen Chick17. 4 18 7818. 8 11 8119. 12 4 84

20. A woman puts 120p on the counter. "some 4p stamps, 6 times as many 2p stamps and the rest in 5p stamps please". What does she need? Ans: 5 4p's.

21. A cube is painted white and cut into 27 small cubes. How many of these little cubes are painted on i) 1, ii) 2, iii) 3, iv) 0 sides? See SMP E for answers and developments

Geometry

1. 9 geraniums have to be planted so that there are 3 plants in each row. How can they be planted so that there are a) 8 b) 9 c) 10 lines.

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Ans:

2.o How many straight lines of unique lengths can you draw from dot to dot?

Ans: 5 o How many non congruent triangles can you draw?

Ans: 8

Miscellaneous

1. A man buys 5 cigarettes a day for 25 days. He keeps the stubs. From 5 stubs he made a new cigarette. How many cigarettes does he smoke during this period?

2. The bacteria in a test-tube double each minute. It is full in one hour. When was it half full? Ans: 59 mins.

3. In a knockout with 39 players, how many games are played? Ans: 39-1

4. In a knockout with 39 players, how many byes? Ans: 3 (find 2**n such that 2**n>39 then express 2n-39 in base 2 and count 1's in the binary number)

5. A necklace is made from 4 chains each of 3 links. It costs 1p to cut a link and 2p to resolder it. How can a necklace be made for 9p? Ans: Cut one chain into 3 links. Use these to connect up the rest.

6. If a crocus costs 8p, a tulip 7p and a daffodil 11p, how much does a hyacinth cost? Hint: count the vowels and consonants

7. Arrange a line of 5 coins as follows 8. Head Head Head Tail Tail Tail

In 3 moves (each move consisting of turning over a pair of adjacent coins), arrange the coins so that Heads and Tails alternate. Ans: Swop 3 and 4, 4 and 5, then 2 and 3

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9. Town A and town B are 99 km apart. There are 98 km posts between them. On each post is the distance to A and the distance to B. On how many posts are only 2 different digits used? (for example, the post that says 'A 33, B 66' only uses 2 digits). From SMP. 18 posts: 11, 18, 22, 27, 33, 36, 44, 45...

10. Clocks - o How many times is a stopped clock correct each week?o A clock goes at the right speed, but backwards. How many times each

week does it tell the right time?

Knights

1.

If you follow the jumps of a chess knight from word to word you can make a 20 word sentence about the ages of Sue and Sal. Sue is in her teens, so how old is Sal? Ans: Start at Sal in the bottom row. Sal is 8 (thanks to Jenny Leppington-Clark for discussing this puzzle, and thanks to Sweta Singh for pointing out the right answer).

The sentence is "SAL IS NOW HALF AS OLD AS SUE WAS WHEN SAL WAS A THIRD AS OLD AS SUE IS NOW". So let's create some variables - SueNow and SalNow are the ages now, and SueThen and SalThen are the ages X years ago. Then SalNow = SueThen/2and SalThen = SueNow/3

Also SueNow = SueThen + X SalNow = SalThen + X

Substituting, we get SalThen + X = SueThen/2and SalThen = (SueThen + X)/3

Subtracting, we get X = SueThen/2 - SueThen/3 - X/3

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so 4X/3 = SueThen/6and hence 8X = SueThen

So, assuming X is an integer and Sue is now a teenager, X=2 and SueThenwill be 16. That makes SueNow 18 and SalNow 8.

2. On a 5 by 5 chessboard, a knight on the centre square would have 8 possible moves. Draw a 5 by 5 grid and put "8" in the middle square. Fill in the other squares with the number of moves that a knight would have from that square.

3. Draw a 5 by 5 chessboard. Put a "0" in the top left corner square. Imagine a knight there. Put a "1" in all the squares that it can reach in 1 go, a "2" in all the squares that it can reach in a minimum of 2 goes, etc, until all the squares are filled in.

Digital Displays

_ _ _ _ _ _ _ _| | | _| _| |_| |_ |_ | |_| |_||_| | |_ _| | _| |_| | |_| _|

Make a table showing how many lines are used for each digit. While changing from 0 to 1, four bars are turned off. No new ones are turned on.

Make a table of what happens as the digits change ON OFF 0->1 0 4 1->2 ... 9->0

Add up the "OFF" and the "ON" columns. What do you notice?

Palindromes

1. A car milometer shows 15951. After 2 hours it shows another palindromic number. How fast was the car going on average?

2. Pick a 2 digit number. Reverse it then add to the original number. Have you got a palindrome? If not then repeat the process. Try all the 2 digit numbers (as a class activity) (from Mathematical Bafflers). Ans

3. Sum of digits 1-9 10 11 12 13 14 15 16 17 18 4. Num of additions 1 2 1 2 2 3 3 6 24 6

References

1. Mathematical puzzles and pastimes, A. Bakst, Princeton,N.J, 1965.

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2. More mathematical activities: a resource book for teachers, Brian Bolt, Cambridge University Press, 1985.

3. The Corgi Book of Problems, Jameson Erroll, Transworld Publishers Ltd., 1964.4. Mathematical Bafflers, Angela Dunn, Dover Publications, 1980.5. Mathematical puzzles & diversions, M. Gardner, London, 1961.6. More mathematical puzzles & diversions, M. Gardner, London, 1963.7. Mathematical puzzles for the connoisseur, P.M.H. Kendall and G.M. Thomas,

London, 1962.8. Mathematical games and puzzles, T. Rice, London, 1973.9. The Canterbury Tales, Henry Dudeney, Thomas Nelson and sons, 1949

See also Mathematical games

Updated May, 2009 [email protected]

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Grade Five Problems

1)

How many addition signs should be put between digits of the number 987654321 and where should we put them to get a total of 99?

2)

Divide the face of the clock into three parts with two lines so that the sum of the numbers in the three parts are equal.

3)

According to experts the first 4 moves in a chess game can be played in 197299 totally different ways. If it takes 30 seconds to make one move, how long would it take one player to try every possible set of 4 moves?

4)

Batman has been imprisoned by the Riddler. To escape he must find the quickest way to move the tower of plutonium disks from one post to another so that the disks have the same arrangement as on the original post. He may move only one disk at a time. What is the minimum number of moves he must make in order to move the ten disk tower and have it appear the same?

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5)

A man has to be at work by 9:00 a.m. and it takes him 15 minutes to get dressed, 20 minutes to eat and 35 minutes to walk to work. What time should he get up?

6)

Replace each blank with the correct digit.

4 3 _ 2 4 5 _ _ 1 2 7 --------- 8 8 9 3

7)

In the first year of production a play sells 1572 tickets, in its second year it sells 1753 tickets, in its third year it sells 152 less than in its second year. How many tickets are sold in 3 years?

8)

The Riddler has left a clue for Batman to follow at the scene of each crime. These are the clues that Batman has found:

(1) There is a 1 in the thousands place. (2) The digit in the tens place is 9 times the digit in the thousands place. (3) Multiply the digit in the thousands place by 2. (4) The digit in the ones place is a hand without a thumb. (5) The digit in the hundreds is 2 less than the number in the tens.

Solve the riddle to find the number and help Batman stop the Riddler.

9)

Scientists have just discovered people on Neptune. There are 3 villages, 2 cities and 1 super city on Neptune. These are the populations in 1994 and 1996.

1994 1996

Eilosa 129 204 Vertu 308 292

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Pridi 90 50 Dedrun 500 600 Antran 700 693 Maran 1200 1500

List the places in order of increasing size in 1994 and 1996.

In which year was the population of the planet greater?

10)

Rachel, Kim and Shawn went to Cuba for a vacation. On the way to Cuba, the plane made the trip in 315 minutes. On the return trip, the flight took 216 minutes. They stayed in Cuba for 3629 minutes. Estimate how long the trip took to the nearest 10.

How long did the trip really take?

11)

The mass of the Great Pyramid is 557t greater than that of the Leaning Tower of Pisa. Stone Henge has a mass of 2695t which is 95t less than the Leaning Tower of Pisa. There once was a Greater Pyramid which had a mass twice that of the Great Pyramid, what was the mass of the Greater Pyramid.

12)

Find the number of faces, edges and vertices on the figure shown.

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What would this figure be called? What would the two figures contained in it be called?

13)

Given a cube, draw a second figure with a square base, having 8 edges, 5 vertices and 5 faces.

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Answer

14)

Given a cone having one edge and one vertex and two faces, make two intersecting cuts through the flatside and find the number of vertices, edges and faces each piece will have.

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Answer

15)

How many different cubes can you make if each face of a given cube has a line connecting the center points of two opposite edges?

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Answer

16)

If every vertex of a regular pentagon is connected to every other vertex, how many triangles are formed?

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Answer

17)

The weather during Pith Possum's vacation was strange. It rained on 15 different days, but it never rained for a whole day.

Rainy mornings were followed by clear afternoons. Rainy afternoons were preceeded by clear mornings. There were 12 clear mornings and 13 clear afternoons in all. How long was the vacation?

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Answer

18)

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Festival Antigonish keeps a set of five counter weights to help stagehands move heavy props. The weights can balance exactly any load that is a multiple of 10kg, from 10kg up to a total of the five weights. Give the masses of weights which meet these conditions and allow the stagehands to counterbalance the max possible load.

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Answer

19)

If you begin with a one digit integer, multiply by 3, add 8, divide by 2 and subtract 6, you will get the integer back.

Find the number.

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Answer

20)

This stairway is made of cubes. How many cubes would be needed to make the steps 9 steps high?

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Answer

21)

Look at the chart and fill in the numbers.

The key is to find a pattern.

4 5 9 16 72 2 4 5 7 12 3 5 6 8 44 6 20 42 25 63 2

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Answer

22) This is a turtle graphics problem.

A bug is walking across the doorstep of a house. When he starts out he is facing the house at the opposite side of the doorstep. If he goes 3 units left, 1

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unit right, 2 units left, 8 units left, what will the bug have to do to get back where he started as quickly as possible without retracing his steps?

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Answer

23)

If Jane is older than Kim, Kim is older than Shawn. Shawn is younger than Jane and Rachel is older than Jane

List the people from oldest to youngest.

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Answer

24)

There are 12 people in a room. 6 people are wearing socks and 4 people are wearing shoes, 3 people are wearing both. How many people are in bare feet?

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Answer

25)

One morning grasshopper fell down a hole 2 metres deep. He would climb 1/4 of a metre every day but at night he slid down 1/8 pf a metre. At this rate, how many days until the grasshopper gets out?

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Answer

26)

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Kim, Raelene, Jim and Rick finished first, second, third and fourth in a motor bike race. If their numbers were 5, 17, 1 and 7, use the following clues to find out who had what number and what order they finished in.

17 said she would have placed higher if her bike had not stalled at the start. Rick finished before 7 but after Raelene. 1's father, said he was very proud of his daughter's finish. Kim finished after 5. Jim wasn't third.

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Answer

27)

An ice cream stand has nine different flavours. A group of children come to the stand and each buys a double scoop cone with two flavours of ice cream. If none of the children choose the same combination of flavours, and every different combination of flavors is chosen, how many children are there? FLAVOURS

1 Vanilla2 Maple3 Chocolate4 Tiger5 Raspberry6 Strawberry7 Coffee8 Moon Mist9 Cherry Vanilla

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Answer

28)

Place the digits 9,4, 7, 6, 5, 1, in the boxes in order to get the largest result. [ ][ ] x [ ][ ] + [ ] x [ ] = ?

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Answer

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29)

Four boys work together painting houses for the summer. For each house they paint they get $256.00. If the boys work for 4 months of summer and their expenses are $152.00 per month, how many houses must they paint for each of them to have one thousand dollars at the end of the summer?

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Answer

30)

In her first year a dog breeder's dogs produce 2 puppies. In her second year her dogs produce three-times as many puppies. In her third year her dogs produce 5 times as many puppies as the first year. How many puppies will the breeder have produced in her first 3 years?

If she sells the puppies for $200.00 each, how much will she have made?

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Answer

31)

Two frogs are hopping along a path one lily pad wide, going in the same direction. There are 18 lily pads and one frog jumps 3 lily pads at a time, the other jumps two lily pads at a time. If the frogs leave the shore one after the other, how many lily pads will be jumped on by both frogs?

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Answer

32)

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A grocery store has a sale on bananas. If you buy six bananas you get the sale price. If the grocer has 489 bananas how many bunches of six can he sell at his sale price? In this case how many can be sold at the regular price?

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Answer

33)

Some students hold a bake sale to raise money for their school library. They sold fudge and squares and cookies. One of each type of treat was put into 95 paper bags and the students were allowed to keep the left overs. If they started out with 110 cookies, 130 pieces of fudge and 116 squares, the students put the same number of each kind of treat in each bag(eg. 2cookies, 1 fudge, 1 square. How many of each treat did they keep?

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Answer

34)

Describe the shaded area of the following figure in terms of a fraction.

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35)

Sixteen players enter a tennis tournament. If there can be no draws how many games must be played if each player can be eliminated by one loss.

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Answer

36)

If a man can walk 4km in 1 hour and his wife can walk 5km in 1 hour, how many minutes will it take them to meet if they begin walking at the same time at opposite ends of a 12km road, and have agreed to meet at the half way point?

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Answer

37)

In how many different ways can the cars of a three car train be arranged?

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Answer

38)

In 1989 the price of a soccer ball was $8.00. A financial analyst predicted that the price would rise $0.25/year for the next 10 years. In what year will the price be $9.75?

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Answer

39)

A math student interviewed 50 fifth graders. 41 said they like peanut butter sandwiches, 35 liked jam sandwiches and 30 liked both on their sandwiches.

How many students liked neither?

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Answer

40)

A unit fraction is a fraction where the numerator is 1 and so it looks like 1/n.

Can you express the fraction 19/72 as the sum of two unit fractions?

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Answer

41)

How many unit cubes are needed to build this solid figure?

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Answer

42)

See how many triangles you can draw using any three of the five points on the circle as vertices.

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Answer

43)

Nobody knows how old Aunt Helen is but she gave a few hints. She had passed 1/20 of her life before she started school. She spent 3/20 of her life in school; She worked for 1/10 of her life before she got married. She was married for 2/5 of her life. Her husband died after 7/10 of her life.

From reading Uncle Harry's gravestone you find out that she has been a widow for 24 years.

How old is Aunt Helen?

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Answer

44)

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If a ball is dropped from a height of 100m, each time it hits the ground it bounces 3/5 of the height it fell. How far will the ball have travelled in the 5th bounce?

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Answer

45)

Two girls went on a vacation together and they agreed to split all expenses. They trip was prepaid so Mary wrote a cheque for food and one for lodging. Clare wrote the cheques for gas and entertainment. How much does Clare owe Mary? FOOD LODGING Mary: $56.97 $83.29

GAS ENTERTAINMENT

Clare: $41.96 $37.47

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46)

The record for the greatest number of consecutive jumping jacks is 14500 in 5 hours and 35 minutes. If Joe does 55 jumping jacks per minute, how long will it take Joe to tie the record?

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47)

A line of length 4 is divided into nine equal segments. Find a fraction to describe the length AB.

Page 28: Mathematical Puzzles

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48)

Rachel opened her math book and found that the sum of the facing pages was 243. What pages did she open to?

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Answers

Grade Five Answers

1)

Answer:

     1)  9+8+7+65+4+3+2+1 = 99   -> 7 addition signs.

     2)  9+8+7+6+5+43+21 = 99    -> 6 addition signs.

2)

Answer:

Page 29: Mathematical Puzzles

26+26+26 = 78

3)

Answer:

     30 seconds * 4 * 197299 = 23675880 seconds.

4)

Answer:

     The quickest way is to move each disk onto the first post     except for the last disk -> which is 9 moves.

     Then move the last disk to the second post which is one     move.  

     Then move each of the disks on the first post to the second     post which is an additional 9 moves.

               9 + 9 + 1 = 19 moves

5)

Answer:

     9 hours - (15 + 20 + 35)     9 hours - 70 minutes     7 hours + 120 minutes - 70 minutes = 7 hours and 50 minutes = 7:50 AM

6)

Answer:

Page 30: Mathematical Puzzles

          4 3 1 2            4 5 4          4 1 2 7         ---------          8 8 9 3

7)

Answer:

     +1572          1753     +1753          -152     +1601         ------     ------         1601      4926

8)

Answer:

     2794

9)

Answer:

                    1994      1996

     Eilosa         129       204     Vertu          308       292     Pridi           90        50     Dedrun         500       600     Antran         700       693     Maran         1200      1500                  -----------------                   2927      3339

    3339 - 2927 = 412 therefore the population in '96 is greater

10)

Answer:

             ESTIMATE       ACTUAL               320            315               220            216              3630           3629

Page 31: Mathematical Puzzles

               ------         ------                4170           4160

11)

Answer:

Mass of Stone Henge + 95t = Mass of Leaning TowerMass of Leaning Tower + 557t = Mass of Great Pyramid

Therefore the mass of the Great Pyramid  = 2(2695t + 95t + 557t) = 6694t

12)

Answer:

     17 edges       9 faces        10 vertices

     Pentagonal Prism contains a  rectangular prism and a right     trianglular prism.

13)

Answer:  

        4 edges, 4 vertices and one         face come from square base.

          12E                8E           8V                5V           6F                5F

14)

Answer: There are many solutions. Here we have one solution.

 

              4V              6E              4F

Page 32: Mathematical Puzzles

15)

Answer:

        23 = 8

16)

Answer:

    1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10  = 10 triangles    1,2 / 2,3 / 6,4 / 4,1 / 10,7 / 7,6 / 9,8 / 8,10 / 9,5 / 5,3 = 10 triangles    7,0,5 / 4,0,8 / 2,0,7 / 5,0,4 / 8,0,2 = 5 triangles    1,2,3 / 1,4,6 / 6,7,10 / 10,8,9 / 9,5,3 = 5 triangles    2,0,7,8,10 / 4,0,8,5,9 / 7,0,5,2,3, / 8,0,2,4,1 / 5,0,4,7,6 = 5 triangles

    Answer:   35 triangles

17)

Answer:

     5 totally clear days and 15 half and half.          In the 15 half and half days:     7 <= 12 of the clear mornings were followed by rain,

     8 <= 13 of the clear afternoons were preceeded by rain.

     20 days of the vacation.

18)

Answer:

               counter       Possible                  weights       loads               10kg           10               20kg           20               40kg           30               80kg           40              160kg           50

Page 33: Mathematical Puzzles

                              60                              70                              80                              90                             100                             110                             120                             130                             140                             150                             160                             170                              :                              :                             310

19)

Answer:

        Trial and error

        [(2 * 3 + 8)/2] - 6 = 1  <- NO        [(6 * 3 + 8)/2] - 6 = 7  <- NO        [(8 * 3 + 8)/2] - 6 = 10 <- NO        [(4 * 3 + 8)/2] - 6 = 4  <- YES!!  The number is 4

Algebraically: [(3*x + 8)/2] -6 = 4

20)

Answer:

     Find a pattern.

     2 steps        3 cubes             2 + 1     3 steps        6 cubes             3 + (2 + 1)     4 steps       10 cubes             4 + (3 + 2 + 1)     5 steps

     9 steps       45 cubes             9 + (8 + 7 + 6 + 5 + 4 + 3 + 2 + 1)

     You will need 45 cubes.

21)

Answer:

          4 = 2 + 2              2 * 2 = 4

Page 34: Mathematical Puzzles

          5 = 2 + 3              2 * 3 = 6

        4       5       9       13      10      15      16      3       7        2       2       4       7       5       7       7       1       3        2       3       5       6       5       8       9       2       4        4       6       20      42      25      56      63      2       12

22)

Answer:

     1 unit left and 3 units left.

23)

Answer:

24)

Answer: 

Page 35: Mathematical Puzzles

          12 - 7 = 5 people in bare feet!

25)

Answer:

     Each day the grasshopper goes 1/8 m until the day when the      grasshopper is at 1.75m in morning, he gets out of the hole that      day.  1.75/(1/8)=14  and 14 + 1 = 15 days.

26)

Answer:

        Race Number             Position

K       5  (17)  (1) 7          (1) 2 3 4 RA      5  (17)  (1) 7          1 (2) 3(4)J       5  17     1  7          1  2  3(4)

Ri      5  17     1  7          1  2 (3)4

     1 and 17 are female -- from clue 1 and 3     5 and 7 are male -- from above     J did not come 3rd -- from clue 5     17 is not first -- from clue 1     Kim is not first.     Rick finished before 7, therefore he must be 5     after Raelene who was first,     therefore,      Raelene is #1 and first     Rick is #5 and second     Kim is #17 and third     Jim is #7.

27)

Answer:

     The number of children = number of different combinations of     two different flavours.

        1,2     2,3     3,4     4,5     5,6     6,7     7,8     8,9        1,3     2,4     3,5     4,6     5,7     6,8     7,9        1,4     2,5     3,6     4,7     5,8     6,9        1,5     2,6     3,7     4,8     5,9        1,6     2,7     3,8     4,9        1,7     2,8     3,9

Page 36: Mathematical Puzzles

        1,8     2,9        1,9

               36 = 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

               There are 36 children.

28)

Answer:

     [9][5] * [7][6] + [4] * [1] = 7224

29)

Answer:

     Must earn 4 x $1,000.00 + 4 x $152.00= 4608

     Number of houses = 4608/$256/house                       = 18 houses.

30)

Answer:

          2 + 3(2) + 5(2) = 18

          18($200.00) = $3600.00

31)

Answer:

     How many numbers between 1 and 18 are multiples of 2 and 3?                6, 12, 18

     [ ][o][x][o][ ][x/o][ ][o][x][o][ ][x/o][ ][o][x][o][ ][x/o]

                       1                   2                   3

     There will only be 3 lily pads that both frogs jump on     - Number 6, 12 and 18.

Page 37: Mathematical Puzzles

32)

Answer:

     489 divided by 6 equals 81 with a remainder of 3

     He casn sell 81 bunches, which wuld leave him 3 to sell at regular     price.

33)

Answer:

     110 divide by 95 equals 1 with a remainder of 15  cookies     130 divide by 95 equals 1 with a remainder of 35  fudge     116 divide by 95 equals 1 with a remainder of 21  squares      110 - 95 = 15 cookies     130 - 95 = 35 fudge     116 - 95 = 21 squares

     The students kept 15 cookies, 35 pieces of fudge and 21     squares.

34)

Answer:

     A rectangle = (5-1)(4-1) = (4)(3) = 12     - each shaded area.

     (1)        (3-1)(2-1) = (2)(1) = 2     (2)        (4-2)(3-2) = (2)(1) = 2     (3)        (5-4)(4-3) = (1)(1) = 1     (4)        (3-2)(4-3) = (1)(1) = 1

                                      6

     Area shaded / Area of rectangle = 6/12 = 1/2

35)

Answer:

     ROUND          PLAYERS REMAINING        GAMES

       1                     16                8       2                   16 - 8              4

Page 38: Mathematical Puzzles

       3                    8 - 4              2       4                    4 - 2              1                                               ----                                               15

36)

Answer:

     Time for Man                  Time for Wife

     6km / 4km/hr                  6km / 5km/hr      =                             =     3/2 * 60min                   6/5 * 60min     =                             =     90 minutes                    72 minutes

     As the woman will get there first, she will have to wait for      the man.  The time taken to meet will be 90 minutes.

37)

Answer:

     (1)  [1] - [2] - [3]     (2)  [2] - [3] - [1]     (3)  [3] - [2] - [1]     (4)  [2] - [1] - [3]     (5)  [3] - [1] - [2]     (6)  [1] - [3] - [2]     There are three different ways.

38)

Answer:

     $9.75 - $8.00 = $1.75

     $1.75/$0.25/yr = 7 years

     Therefore, 1989 + 7 = 1996

39)

Answer:

Page 39: Mathematical Puzzles

             Only Jam        Only Peanut Butter      Jam/Peanut Butter        35 - 30              41 - 30                    30           5                    11

     Number that liked neither = 50 - (5+11+30) = 4

40)

Answer:

     72/18 = 4

     18/72 = 1/4

     18/72 + 1/72 = 1/4 + 1/72 = 19/72

41)

Answer:

     In a full cube 3 x 3 x 3 = 27

     There are 5 missing cubes, therefore there are 22 cubes needed to      build this figure.

42)

Answer:

             123  234   345             124  235             125  245             134             145

     There are 10 possible triangles.     Each will be used 6 times.

43)

Page 40: Mathematical Puzzles

Answer:

     (3/10)*? = 24  and 24/(3/10) = 80.    

     Aunt Helen is 80 years old.

44)

Answer:

          3/5(3/5)(100)   3/5(3/5)(3/5)(100)

          (1)       (2)         (3)     100 + 2(3/5)(100) + 2(3/5)(3/5)(100) + 2(3/5)(3/5)(3/5)(100)     + 2(3/5)(3/5)(3/5)(100) = 

     100 + 600/5 + 1800/25 + 5400/125 + 16200/625

               = 361.12m

45)

Answer:

               $56.97                83.29                41.96              + 37.47             ---------              $219.69

          1/2(219.69) = 109.85

     $109.85 - (Clare's payment) = $30.42 = amount Clare owes Mary.

46)

Answer:

     1min/55 * 14500 = 263.64 min                     = 4h 23.64 min.

47)

Answer:

Page 41: Mathematical Puzzles

4 * 3/9 = 12/9 = 4/3 = AB

48)

Answer:

243/2 = 121.5

121.5 pages are whole numbers therefore the pages are 121 and 122