Friday, February 24, 2023

Coin Weighing Puzzle

 Here's a favorite coin-weighing puzzle.

There are six bags of coins. Within each bag, all coins are identical.  Some bags have gold-plated coins that each weigh 5 grams.  The other bags have solid gold coins that each weigh 6 grams.  You have a scale that measures precisely and accurately to the nearest 1 gram.  You may remove coins from any or all bags and weigh them, but you are allowed only ONE weighing after which you must identify which bags have gold coins and which bags have plated coins. 

Try to come up with a solution before reading mine.

Solution:  Label the bags 1 through 6.  Remove 32 coins from bag No. 1, 16 coins from bag No. 2, 8 coins from bag 3, 4 coins from bag 4, 2 coins from bag 5 and 1 coin from bag 6.  Put all the coins together onto the scale and record the total weight, W, in grams.  Determine R=W-315.   Express R in binary notation. For example, if W=368 g. then R=368-315=53 grams.   In binary form, 53= 1 1 0 1 0 1.  The six binary digits correspond to the six bags in that order, i.e., Bag 1 = 1, Bag 2 =1, Bag 3 =0, Bag 4 =1, Bag 5 =0 and Bag 6 =1.  Bags with a "1" have solid gold coins and bags with a "0" have gold plated coins.

 

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