A mysterious box, contents unknown, might weigh anywhere from one to forty pounds, though it definitely weighs a whole number of pounds (you don't have to worry about ounces). You must determine its weight. Instead of a scale, you are given a pan balance. Furthermore, this party is byow, bring your own weights. If you bring a 5 pound weight you can easily determine whether the box weighs 5 pounds. If you also bring a 7 pound weight you can determine whether the box weighs 12 pounds, and so on. Now the catch - you can only bring four weights. Which weights should you bring?
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Question

A mysterious box, contents unknown, might weigh anywhere from one to forty pounds, though it definitely weighs a whole number of pounds (you don't have to worry about ounces).  You must determine its weight.  Instead of a scale, you are given a pan balance.  Furthermore, this party is byow, bring your own weights.  If you bring a 5 pound weight you can easily determine whether the box weighs 5 pounds.  If you also bring a 7 pound weight you can determine whether the box weighs 12 pounds, and so on.  Now the catch - you can only bring four weights.  Which weights should you bring?
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anonymous · Answer

extend this result for 1 to 81 pounds

anonymous · Answer

So you should bring weights of 1, 3, 9, and 27 pounds.
You can weigh 2 pounds by putting 3 on one side, and 1 on the other side.
4 = 3 + 1
5 = 9 - 3 - 1
etc
For 1, 3, 9, 27, 81 pounds, you can weigh up to 81+27+9+3+1 = 121 pounds.