What equipment is at hand? does that include a 40 lb hay bale? Cut the bale in quarters so you have 10 lbs = a 1 and 9 lb weight? then 3/4 so 30 lbs = 27 + 3?

At first you are tempted to run up the powers of 2. Weights of 1, 2, 4, and 8 pounds, in various combinations, can weigh everything from 1 to 15 pounds, but that’s nowhere near 40. The trick is, some of the weights can sit on the left pan, with the box. Each weight can be on the left, the right, or not used at all. This suggests powers of 3.

Using 1 and 3 pound weights, you see immediately how to measure 1, 3, and 4 pounds. If the box weighs 2 pounds, put the 1 pound weight on the left pan with the box, and the 3 pound weight on the right, and everything is in balance. Next, bring in a 9 pound weight and put it on the right. We already showed we can add a net weight of 1 through 4 to the box, so if the box weighs anywhere from 5 to 8 pounds, we’ll be able to balance it against the 9 pound weight. Of course there is no trouble if the box weighs 9 pounds, and if it weighs anywhere from 10 to 13 pounds, we achieve balance by adding (net) 1 to 4 pounds to the right. Now we’ve covered everything from 1 to 13 pounds, and the last weight to bring in is 27 pounds. This lets us weigh everything from 27-13 pounds up to 27+13 pounds, or 40 pounds. An inductive proof shows successive weights should always be powers of 3.

To weigh a box from 1 to 81 pounds, double the weights. Thus the weights are 2, 6, 18, and 54 pounds. If the weight of the box is even, you can determine it using the procedure outlined above. If the weight is odd, the balance will always tip one way or the other. Determine that the box is heavier than n and lighter than n+2. For instance, the box might be more than 30 pounds and less than 32 pounds, whence it is 31 pounds.

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