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? 2.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? 3.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? 3.a)One out of 8 otherwise identical balls is overweight. How can it be identified after 2 weighings? 3.b)One out of 27 otherwise identical balls is overweight. How can it be identified after 3 weighings? 4)Ferries 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? 5.3 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? 6.2 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? 7.SMP and CSE 1974 extend this to cover the case of n men. Picking Captains 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? 8.12 black 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? 9.20 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? 10.30 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? Incomplete Sums (Worked Examples in J.A.H. Hunter's "Mathematical Brain Teasers"). 11.Each letter represents a different digit SEND +MORE ----- MONEY 12.This sum uses all the digits 28* +**4 ---- **** 13.This subtraction sum uses all the digits from 1 to 9. 9 * * - * 4 * ----- * * 1 14.O represents odd digits E represents even digits EEO xOO ----- EOEO EOO ----- OOOOO 15.P represents prime digits PPP xPP ----- PPPP PPPP ----- PPPPP 16.Some more additions THE TEN MEN ---- MEET 17.SLOW SLOW OLD ---- OWLS 18. SAL SEE THE SUEZ ----- CANAL 19. FIVE FIVE NINE ELEVEN ------ THIRTY