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 A Weighty Problem
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TchrWill
Advanced Member

USA
79 Posts

Posted - 07/22/2013 :  11:11:18  Show Profile  Reply with Quote
There is a king who has 5 bags of gold. One is a fake bag, which weighs either less or more than the rest. The king has a direct read scale to weigh the bags (not a balance scale). But he is only allowed to use the scale 3 time. How is it possible to find the fake bag no matter which bag it is? (more than one bag can be on the scale at the same time.)
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someguy
Advanced Member

Canada
143 Posts

Posted - 07/28/2013 :  13:51:56  Show Profile  Reply with Quote
An interesting diversion for a Sunday morning.

It can be done by first weighing bags 1, 2, 3 and 4.

Then weighing bags 1 and 2.

If the averages weight of a bag is the same in both cases, bag 5 is the fake.
You can weigh it to figure out if it is heavier or lighter.

If the average weight of the first group of bags is different then the average weight of the second group of bags, then weigh bags 1, 3 and 5.

I don't won't to spoil it for others, so I will leave it at this point for now. Just check the order of the average weights. In the few cases that lead to the same order, do some arithmetic.
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TchrWill
Advanced Member

USA
79 Posts

Posted - 08/16/2013 :  14:55:37  Show Profile  Reply with Quote
quote:
Originally posted by someguy

An interesting diversion for a Sunday morning.

It can be done by first weighing bags 1, 2, 3 and 4.

Then weighing bags 1 and 2.

If the averages weight of a bag is the same in both cases, bag 5 is the fake.
You can weigh it to figure out if it is heavier or lighter.

If the average weight of the first group of bags is different then the average weight of the second group of bags, then weigh bags 1, 3 and 5.

I don't won't to spoil it for others, so I will leave it at this point for now. Just check the order of the average weights. In the few cases that lead to the same order, do some arithmetic.



A clever takeoff of the more traditional weighing problems.

Let the 5 bags be labeled A, B, C, D and E.
Weigh the bags as follows remembering that any pair can be weighed.

Case 1
1--Let A+B = X lb.
2--Let C+D = X lb.
3--Then, by definition, E is the odd bag and weighing it with A, B, C or D will tell you whether it is heavier or lighter.

Case 2
1--Let A+B = X
2--Let C+D = X + Y, Y being the excess weight above X.
3--C or D is heavier than A or B.
4--Weigh A+C together.
5--If A+C = X, then D is the heavy bag.
6--If A+C = X + y, then C is the heavy bag.

Case 3
1--Let A+B = X
2--Let C+D = X - Y
3--C or D is lighter that A or B.
4--Weigh A+C together.
5--If A+C = X, then D is the lighter bag.
6--If A+C = X - Y, then C is the lighter bag.

Steps 1 and 2 can be reversed leading to the same results.


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