A king demands a tax of 1,000 gold sovereigns from each of 10 regions of his nat
ID: 3110912 • Letter: A
Question
A king demands a tax of 1,000 gold sovereigns from each of 10 regions of his nation. The tax collectors for each region bring him the requested bag of gold coins at year end. An informant tells the king that one tax collector is cheating and giving coins that are all 10% lighter than they should be, but does not know which collector is cheating. The king knows that each coin should weigh exactly one ounce. (a) The King has a balance scale that can compare two quantities. What is the fewest number of times he can use this device to detect which region is cheating (and how does he do so)? (b) The King has an enchanted digital scale that can give the exact weight of a quantity. How can he use this device once to find which region is cheating?Explanation / Answer
Answer:
a).
Step 1: Select one gold coin from each of the 10 regions. Divide the gold coins into 4 groups, three groups containing 3 coins each (name them A, B, C) and last group containing one coin(X).
Step 2: Weigh two groups A and B of 3 coins on weigh scale.
Case 1: Weights are equal (A = B): If the weights are equal, defective coin is in C group or the lonely one(X).
Step 3: Label the coins in group C as C1, C2 and C3.
Weigh C1 and C2
Case 1: Weights are equal (C1 = C2): If the weights are equal, defective coin is either C3 or X.
Again weigh C3 and X to find the coin which is defective.
Case 2: Weights are not equal (C1 != C2): If the weights are unequal, defective coin is the one which weighed less.
Case 2: Weights are not equal (A != B): If the weights are unequal, defective coin is in groups A or B. Choose the coins which weighed less.
Assume B group weighed less.
Step 3: Label the coins in group B as B1, B2 and B3.
Weigh B1 and B2
Case 1: Weights are equal (C1 = C2): If the weights are equal, defective coin is B3.
Case 2: Weights are not equal (C1 != C2): If the weights are unequal, defective coin is the one which weighed less.
Thus, minimum number of trials required to find the defective coin is 3.
b).
Collect one coin from first region, two coins from second region, three coins from third region and so on which results in ten coins from tenth region and thus resulting in total of 55 coins. Place this collection on the weighing device and look for the difference from the value it actually should be.
E.g. If the actual weight is .30 ounces less then bag three is light and collector three is the cheat.
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