For Christmas this year, the king of Florin received 10 bags of identical gold coins from the king of the neighboring country of Guilder. Each bag contains 100 gold coins and the coins weigh 1 ounce each. However, he was tipped off that one of the bags was switched out in transit with a bag of 1.1-oz. counterfeit coins.
Florinese technology is very limited, so the king's digital scale, accurate to the nearest tenth of an ounce, can be used only once. How can a weigh-in be arranged that will identify the counterfeit bag?
Solution:
Number the bags 0 thru 9. Take 0 coins from bag 0, 1 coin from bag 1, 2 coins from bag 2, and so on up to bag 9 and put them on the scale. The tenths place of the scale holds your answer. If the scale displays a whole number, then there are no counterfeit coins on it and bag 0 is the counterfeit bag. If the scale displays X.1, then bag 1 is counterfeit. If the scale displays X.2, then bag 2 is counterfeit, and so on.
Out of curiousity, can we only use the digital scale? Or are we allowed to use other devices as well (such as a make-shift balancing scale)?
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