Classic Puzzles Series 2 Spud bashing

Transcription

1 Spud bashing A horticulture research team has 100 kg of potatoes. When tested the potatoes are found to be 99% water. They are left lying around for several months and they shrivel and dry out. The potatoes are tested again and they are found to be only 98% water. How much do the potatoes now weigh?

2 The Truel Mr. Black, Mr. Grey and Mr. White are going to fight a truel. This is like a duel but with three protagonists. They will take it in turns to fire a shot at one of the others. Mr White will go first because he is the worst shot, hitting his target only 1/3 of the time. Mr Grey, if he is not hit, will go second. He hits his target 2/3 of the time. Mr. Black will go third, if he is still standing. He always hits his target. All three participants know the abilities of the others and all three act in their own best interests. Where should Mr. White aim to maximise his chances?

3 Fly mow Adam can mow the lawn in 20 minutes, but his dad takes 30 minutes to do the same job. Using two mowers they cut the lawn together. How long does it take them?

4 Waiter, waiter Three friends go for a meal in a posh restaurant. They each write a cheque for 100. The waiter takes the 300. The bill is actually only 250, so there should be 50 change. But the waiter can t divide 50 by 3 so he pockets 20 and gives the three friends 10 each. So the friends have paid 90 each and the waiter has 20 in his pocket. That comes to 290. The waiter took 300. What happened to the other 10?

5 Imbalance The pivot of the scales is not quite in the middle. The dealer places the gold in the side with the longer arm, so the other side has to have a heavier weight to make it balance. He places some gold on the pan and balances it with 100 grams on the other side. He charges the King for 100 grams of gold. The wily King suspects the dealer of trickery. He insists that the gold and weights are put in the reverse positions. The scales no longer balance. In fact the gold now only needs 81 grams to make the scales balance. What is the true weight of the gold?

6 Oranges and lemons Alice, Ben and Catrin have oranges and lemons in their bags. Each one has 7 pieces of fruit in his or her bag, a mixture of oranges and lemons. Catrin has twice as many lemons as Alice but she has fewer oranges than Ben. Ben has half as many oranges as Alice. What does each one have in his or her bag?

7 Jail break A jail has 100 cells and 100 guards, who all want to be in charge. At night the cells are all locked. In the morning guard number one comes along and turns the key in every lock, thus opening all the cells. Right behind him comes guard number two, who proceeds to turn the key in every second lock: cells 2, 4, 6, 8... thus re-locking half the cells. Just behind him comes guard number three who turns the key in every third lock: cells 3, 6, 9, Then comes guard number four who turns the key in every fourth lock: cells 4, 8, 12, and so on until guard number one hundred turns the key only in the one hundredth lock: cell 100. Which cell doors are now open?

8 Nelson s column The column is 60m high and is cylindrical. The circumference of the column is 9m. The yellow streamer spirals around the column for five complete turns. How long is the streamer?

9 Tipping the scales You have 12 golf balls. One is faulty and does not weigh quite the same as the others. You do not know whether it is lighter or heavier than the others. You may use the balance only three times. How do you find the faulty ball? And is it heavier or lighter than the others?

10 Orbit Two planets are circling their sun. They both move in a clockwise direction. The red planet will take 3 Earth years to complete a full orbit and the green planet will take 8 Earth years. They are now in line with each other and their sun. When will they next form a straight line with each other and their sun?

11 128 = 129 The area of each trapezium A = 3+5 x 5 = 20 4 x 20 = 80 2 The area of each triangle B = 8 x 3 = 12 4 x 12 = 48 2 Total area = = 128 A B B A A B A B But there are 129 squares. How can this be so?

12 As easy as ABC In this multiplication each letter stands for a different numeral. ABCDE x 4 EDCBA Find the values of A,B,C,D and E.

13 Classic Puzzles Series II Answers Page 1 Tipping the Scales This is the most difficult Classic Puzzle yet! Let the balls be called A,B,C,D,E,F,G,H,I,J,K and L Green indicates that a ball is established as a good ball. Red is bad. 1. Weigh ABCD v. EFGH If ABCD=EFGH then: 2. Weigh I J v. KH If IJ=KH then: 3. Weigh L v. H If L>H then L is heavy If L<H then L is light If IJ>KH then: 3. Weigh I v. J If I=J then K is light If I>J then I is heavy If I<J then J is heavy If ABCD>EFGH 2. Weigh ABE v. CFJ If AEB=CFJ then 3. Weigh G v. H If G =H then D is heavy If G>H then H is light If G<H then G is light NOTE: Since ABCD are all good, the bad ball is light. If AEB>CFJ then: 3. Weigh A v. B NOTE: Since sides have not If A=B then F is light changed, either A or B is heavy If A>B then A is heavy or F is light If A<B then B is heavy If AEB<CFJ then: NOTE: Bad ball has changed 3. Weigh E v. J sides so either C is heavy If E=J then C is heavy or E is light If E<J then E is light As Easy as ABC 1 A is less than 3 Reason: Otherwise answer would be a 6 digit number. 2 A is not 0 Reason: Zeros not at the beginning of numbers. 3 A is not 1 Reason: In column 5, there is no value for E which gives A as 1. 4 Therefore A is 2 5 E is 3 or 8 Reason: In column 5, 4xE=12 or 32. These are the only results which make A=2 in that column. 6 Therefore E is 8 Reason: In column 1, 4x2=8 7 B is less than 3 Reason: In column 2, 4x3=12, which involves carrying 1 to next column. We have already established that A is 2 and E is 8. 8 B is not 0 Reason: In column 4, 4xD + 3 (the number carried), cannot end in 0. 9 Therefore B is 1 10 D is 2 or 7 Reason: In column 4, 4xD + 3 = 11 or 31. These are the only results which make B=1 in that column. 11 D is not 2 Reason: A is 2 12 Therefore D is 7 13 Therefore C is 9 Reason: In column 3 4x9 + 3 = 39. This is the only result in which C is the same in both instances.

14 Classic Puzzles Series II Answers Page 2 Nelson s Column Imagine the column as a paper tube 60cm tall. Cut it into five pieces, one for each circuit of the streamer. Now you have five shorter tubes, each 12cm tall. Cut one of these straight down its length. Flatten the paper and you have a rectangle. The width of the rectangle is 9cm, the circumference of the column. The diagram shows the yellow streamer as well. By Pythagoras the streamer line s is 15cm. Therefore the streamer is 5x15 = 75m long s 9m 12m Waiter, waiter The waiter took 300 and this is made up of 250 for the food, three lots of 10 returned to the friends and 20 in the waiter s pocket. The explanation given in the question is nonsense. It should read: The friends have paid 90 each which is 270 i.e. 250 for the food and 20 for the waiter. Orbit All distances are measured in terms of rotation i.e. in degrees. o Red s speed, Sr = 120 per year o Green s speed, Sg = 45 per year After 3 years: Red is back at its starting place o = 0 Green is at 3 x 45 = 135o So Red will pass Green during its next circuit and, at the moment that Red passes Green, all 3 bodies will be aligned again. Time = Distance So at any moment T = Dr = Dg Speed Sr Sg So Dr = Dg But during the next circuit, Dr = Dg Substituting for Dr in the equation above: Dg = Dg o So Dg = 81 o Altogether Dg = 216 In time, Green will have travelled 216 = 4.8 years 45 Spud bashing Potatoes are made of water and some non water material; let s call it starch. To begin with, the starch is 1% of the weight and, therefore, weighs 1kg. This wieght is not going to change. After the second test, the water represents 98% of the total weight and so the starch now represents 2% of the weight. But that 2% still weighs 1kg. If 2% is 1kg then 100% will be 50kg. The potatoes now weigh 50kg.

15 Classic Puzzles Series II Answers Page 3 Jail Break A cell only changes its state from locked to open or vice versa when it is passed by a guard whose number is a factor of the cell number. Consider cell number 60. This is opened and locked succesively by guards: 1,2,3,4,5,6,10,12,15,20,30 and 60. It is not touched, for example, by guard number 7, who only changes cells: 7,14,21,28,35,42,49,56,63 and so on. Therefore, if a cell number has an even number of factors, that cell will finish locked. If a cell number has an odd number of factors, that cell will finish open. Only square numbers have an odd number of factors, so only cells with square numbers will finish open: 1,4,9,16,25,36,49,64,81,100. The Truel If White aims at Black, he might hit him, in which case Grey will then aim at White before White gets his second turn. If White aims at Grey he might hit him, in which case Black will then shoot White. So neither of these options ensures White gets another turn. There is another option. White could aim into the air, ensuring he hits nobody. In this case Grey will aim at Black, perhaps followed by Black aiming at Grey. Either Black or Grey will be hit and White is ensured a second turn, this time with only one rival to face. White must therefore aim into the air. 128 =129 By using thick lines to outline the shapes the small gap between the two top triangles has been hidden. The drawing below shows the long thin gap appearing and this gap has the area of the extra square. A B B A Oranges and Lemons From the information given: From 1st clue: Catrin can have only 2, 4 or 6 lemons. From 1st clue: Alice can have only 1, 2 or 3 lemons. From 1st clue: Ben can have only 2, 3, 4 or 5 oranges. From 2nd clue: Alice can have only 2 or 4 oranges. The only way Alice can have 7 pieces of fruit is 3 lemons and 4 oranges. So Ben must have 2 oranges and hence 5 lemons. Catrin must have only 1 orange and hence 6 lemons. This is best developed in a table of possibilities.

16 Classic Puzzles Series II Answers Page 4 Imbalance First weighing Second weighing 100 G G 81 a b a b Using moments of force: 100a = Gb 100a = G b a b... = 81b a a = 9 b 10 Ga = 81b G = 81b a So if a = 9 and b = x 9 = G x 10 Therefore G = 90 The gold weighed 90 grams Fly mow Ratio of time spent, Adam:Dad = 2:3 But mowing speed is in inverse proportion to time spent. So Ratio of distance mowed, Adam:Dad = 3:2 Therefore Adam cuts 3/5 of lawn while Dad cuts 2/5 of lawn. So Adam spends 3/5 of 20 min cutting lawn = 12 minutes. Or Dad spends 2/5 of 30 min cutting lawn = 12 minutes.