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1. weigh 1: A and B balance; fake coin is in C (10 coins) weigh 2: CONFIRM WHETHER FAKE COIN IS LIGHTER OR HEAVIER (comparing with A or B) weigh 3: fake coin is in D (5 coins) weigh 4: fake coin is one among 2 or 1 which is left out. If the fake one is one among the 2 we have to go for weigh 5! Great! The arguments given go well. You have left discussing the case 9a and 9b balance, 9a and 9c do not balance!
2. bed bugs, head lice, mosquitoes
3. 0 (zero)
4. 140 mph
5. This part is fine!
6. I am sorry, you have misunderstood the problem. You cannot assume that the fake one is lighter or heavier and then proceed. It could be lighter or heavier which we don't know. We have to find out the fake one and also find out whether it is lighter or heavier.
7. Thank you, sir.
8. The fake one could be heavier also!
9. Suppose you have 30 coins identical in every way except one is either slightly lighter or slightly heavier than the other 29 coins. You can not find this difference by mere perception. You have a balance beam scale to weigh coins against each other. Using it only 4 times, can you figure out which is the counterfeit coin and whether it is lighter or heaver than all the other coins?
10. Dr.BASKARAN R THOOTHUKUDI INDIA aarbaskaran@gmail.com keep 4 separately. put 4 and 4 in each pan. weigh (i). 1) If they are not equal, the 4 coins kept aloof are originals. take 3 coins which appear to be with less weight and keep them marked. have the remaining 1 supposed with less weight and 2 supposed with more weight on one pan. have 1 original and the remaining 2 supposed with more weight on the other pan. weigh (ii). a) EQUAL: one of the 3 coins which appear to be with less weight and kept marked is fake. put one in each pan and weigh (iii). if equal, third one is the fake, otherwise, one with less wight is the fake one. Let O,L and M denote original, less weight and more weight coins. b) if L1 + M2 weighs less then L1 or one of M2 in the other pan is the fake coin. put 1 M in each pan and weigh (iii). if equal, L1 is the fake coin; otherwise one with more weight is the fake coin. c) if L1 + M2 weighs more then one of M2 with L1 is the fake coin. put 1 M in each pan and weigh (iii). one with more weight is the fake coin. 2) if they are equal, one of the 4 coins kept aloof is fake; the other 8 are originals. keep 1 aloof from these 4. put 2 in one pan. put the remaining 1 with 1 O in the other pan. weigh (ii). a) if equal, one kept aloof is fake, weigh (iii) with that in one pan and 1 O in the other to decide whether it is of less weight or more weight. b) if not equal, suppose the 2 unknown coins weigh more; (similar argument will hold for weighing less) 1 of these M2 is fake or 1 unknown in the other pan is fake with less weight. put 1 M in each pan and weigh (iii). if equal, the other L1 is fake. otherwise 1 with more weight is fake.
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