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We are given a bag of sugar, a two-pan balance, and a weight of 1 gram. How do we obtain 1 kilogram of sugar in the smallest possible number of weighings?

Mathematics
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can this be done in less than a thousand moves? start with the gram weight on one scale, then fill the sugar up on the other scale until it balances. both sides now have 1 gram. then move the gram over to the side with sugar so its combined weight is 2 grams, then fill the other side with sugar until it balances. now both sides have 2 grams. do this 1000 times you can greatly increase this if you are allowed to transfer the sugar from one balance to the other on each move. this will reduce the # of moves to less than 10
if you can transfer sugar on each trial then maybe.. use the gram weight to balance one gram of sugar remove the gram weight, then balance the other side with sugar. now both sides have 1 gram of sugar. then combine the sugar on one side, then balance the empty side. both sides now have 2 grams of sugar. repeat. now both sides have 4 grams of sugar. use the gram weight to sequentially add 1 gram until you have 7 grams of sugar. then use the previous technique to double this to get 14 grams, then 28 grams, then 56 grams, then 112 grams, then 224 grams, then 448 grams. use the gram weight to sequentially add 2 more grams to get 500 grams then use the previous technique to double this to get 1000 i think this is about 15 weighing moves total
1--------1-1 ---1 gram of sugar 2--------2-2 ---3 gram of sugar 3--------4-4----7gram of sugar 4-------8-8----15 gram of sugar 5------16-16 ----31gram of sugar 6------32-32 ----63 gram of sugar Now, 7-----63-63 ------126 gram of sugar Now, we take out 1 gram of sugar from 126 gram of sugar 8 -----1-1 ------125 gram of sugar 9 --------125-125 ---250gram of sugar 10--------250-250 -----500 gram of sugar 11-------500-500-----1000 gram of sugar

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Other answers:

So, I guess 11

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