Scales And Vesells1. 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

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

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? 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 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. <- BOY

28. 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;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