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Topic: Weighings - A sack of rice (Read 10768 times) |
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Altamira_64
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Weighings - A sack of rice
« on: Jan 29th, 2012, 6:07am » |
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A grocer has a sack of rice of 10 kgs, 2 weights of 200 and 300 grams and a 2-discs balance. Unfortunately, he does not have any other scale in his store, so he is limited to selling the rice at multiples of 100 grams, and only if he can measure the desired quantity in only 3 weighings. What is the probability that he can serve the first customer that enters the shop, who may desire to buy any quantity, starting from 100 grams and up to 10 kgs? Suppose that each weighing is completed once the 2 discs balance and also that the bags have negligible weight.
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rmsgrey
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Re: Weighings - A sack of rice
« Reply #1 on: Jan 30th, 2012, 9:18am » |
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Some preliminary thoughts: 1) Why is he limited to multiples of 100g? With three weighings, I could produce 25g, 50g, 75g, 100g, 125g, 150g, 175g, 200g, ... Some achievable quantities aren't even whole numbers of grams... 2) The (sensible) operations you can do with the scales all consist of partitioning the set of two weights into three (left pan, right pan, unused), possibly adding a bag of rice to either or both pans, then taking one bag of rice and dividing it into two, either so they differ by the difference between left and right pan weights, or so one of the output bags is the difference between the pan weights. After each weighing, you can merge bags of rice, so, after 3 weighings, you can have up to 4 bags of rice of known weight. 3) With (up to) one weighing, you can measure: 0g 10000g 5000g 100g 9900g 4950g 5050g 200g 9800g 4900g 5100g 300g 9700g 4850g 5150g 500g 9500g 4750g 5250g Of which, 6 are not multiples of 100g, and we can probably eliminate 0 too, leaving 12 "valid" quantities. From the second weighing, you can also use existing bags of rice as weights, so the branching number may increase (some operations would require you to borrow rice from somewhere, and then you'd have negative rice to keep track of too - avoiding this would limit the branching) For one weighing, I make the answer 12%. Someone else can work out the rest
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Altamira_64
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Re: Weighings - A sack of rice
« Reply #2 on: Feb 1st, 2012, 11:37am » |
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Very nice approach! However, I guess, to get some of the weights you suggest, you need 2 weighings, isn't it? For example, to get the 4900, you first need to split the 10kg into two parts (by using the scale once), then place 5kgs and 200 grams on one pan and the desired quantity plus 300 grams on the other pan. Then if they balance, we have the 4900 weight. This is 2 weighings. Same goes for 5100.
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towr
wu::riddles Moderator Uberpuzzler
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Re: Weighings - A sack of rice
« Reply #3 on: Feb 1st, 2012, 11:59am » |
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No, to get 5100 and 4900 you put 200 grams on in one pan of the scale, then split the 10kg over the pans to balance out. So it's one weighing.
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Wikipedia, Google, Mathworld, Integer sequence DB
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Altamira_64
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Re: Weighings - A sack of rice
« Reply #4 on: Feb 1st, 2012, 12:51pm » |
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OK. By the 2nd weighing, we can have the following: 100 200 300 400 500 600 700 800 900 1000 1100 2500 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100 8400 8500 8600 8700 8800 8900 9000 9100 9200 9300 9400 9500 9600 9700 9800 9900 10000 (Including the ones from the first weighing). These are 57, so we have 57% so far. Now deep breaths so as to move on to the 3rd weighing!! I will try to focus only to the weights that are missing!
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« Last Edit: Feb 1st, 2012, 1:13pm by Altamira_64 » |
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Altamira_64
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Posts: 116
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Re: Weighings - A sack of rice
« Reply #5 on: Feb 1st, 2012, 1:28pm » |
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Well, if I haven't made any obvious mistake, the 3rd weighing gives us all the rest: 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2600 2700 2800 2900 3000 3100 3200 3300 6200 6300 6400 6500 6600 6700 6800 6900 7000 7100 7200 7300 7400 7500 7600 7700 7800 7900 8000 8100 8200 8300 So we now have a 100% probability to cover every request.
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Altamira_64
Junior Member
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Re: Weighings - A sack of rice
« Reply #6 on: Feb 1st, 2012, 1:43pm » |
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Ahh yes you are right! on Feb 1st, 2012, 11:59am, towr wrote:No, to get 5100 and 4900 you put 200 grams on in one pan of the scale, then split the 10kg over the pans to balance out. So it's one weighing. |
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Altamira_64
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Re: Weighings - A sack of rice
« Reply #7 on: Feb 1st, 2012, 2:00pm » |
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Hmmm... it seems I was not thinking straight... Some of these weights or combinations of weights cannot be used together! I must go through all of them again... on Feb 1st, 2012, 1:28pm, Altamira_64 wrote:Well, if I haven't made any obvious mistake, the 3rd weighing gives us all the rest: 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2600 2700 2800 2900 3000 3100 3200 3300 6200 6300 6400 6500 6600 6700 6800 6900 7000 7100 7200 7300 7400 7500 7600 7700 7800 7900 8000 8100 8200 8300 So we now have a 100% probability to cover every request. |
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antkor
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Re: Weighings - A sack of rice
« Reply #8 on: Feb 25th, 2014, 3:58am » |
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It is 100%!
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