Identify the Lighter Packet in One Weighing

36. Identify the Lighter Packet in One Weighing

You have ten sealed packets of pens. Each packet is supposed to contain ten identical pens weighing 10 g each, but in one of the packets every pen is faulty and weighs only 9 g.

You may use an accurate digital scale only once. By choosing any number of pens from each packet and placing them all on the scale together, how can you determine which packet contains the lighter pens?

Added 15 March 2011

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Hint:

Take a different number of pens from each packet so that each packet contributes a unique amount to the total weight.

Solution:

Label the packets 1 through 10. From packet 1 take 1 pen, from packet 2 take 2 pens, from packet 3 take 3 pens, and so on until you take 10 pens from packet 10. In total you place 1 + 2 + … + 10 = 55 pens on the scale.

If every pen weighed 10 g, the total would be 55 × 10 = 550 g. Because one packet contains 9 g pens, the reading will be smaller. Let the actual reading be W grams. The number of grams missing from 550, namely 550 − W, equals the number of pens taken from the faulty packet—which is precisely the packet’s label. For example, if the scale reads 547 g (3 g less than 550), then packet 3 is the one with 9 g pens.

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