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Altamira_64
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8 apples  
« on: Feb 21st, 2016, 11:16am »
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We are given 8 apples of which 2 are rotten; one is lighter than the good apples and the second is heavier, while all the good apples weight the same.
Using a two-arm balance scale and in only 3 weightings, can you tell if the two rotten apples together weight more or equal or less than two good ones? Obviously there are no visible signs of the rotten apples.
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Hippo
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Re: 8 apples  
« Reply #1 on: Mar 9th, 2016, 12:15am »
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... My original bet was wrong ... Smiley
 
Suppose apple weights are 100,100,100,100,100,100 and either  
A) {98,101},
B) {99,101}, or
C) {99,102}.
Our goal is to detect which of A), B), C) cases  happened.
 
Let our scalings are as follows:
 
abcd?efgh
abef?cdgh
abgh?cdef
 
Let us discuss possible outcomes:
 
If either of ? is = that mean the weight on both sides is 400 and case B) happened.
 
Otherwise once the pair of distinct apples were on the same side of the scale and as there was no = it must be either A) or C) case.
 
Count for each apple pair ab, cd, ef, gh how many times it was on < side and how many times on > side.
In the A) case there is pair (containing 98 ) three times on < side.
In the C) case there is pair (containing 102) three times on > side.
 
The chosen weights were just for easier explanation.

 
Edit: I forget 8 could not stand before )
« Last Edit: Mar 11th, 2016, 9:26am by Hippo » IP Logged
Altamira_64
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Re: 8 apples  
« Reply #2 on: Mar 9th, 2016, 10:01am »
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Yes but how do we know which pair contains 98 (or 102)?
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Hippo
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Re: 8 apples  
« Reply #3 on: Mar 11th, 2016, 9:25am »
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on Mar 9th, 2016, 10:01am, Altamira_64 wrote:
Yes but how do we know which pair contains 98 (or 102)?

 
Luckilly 98 resp. 102 could be detected (the extreme pair contains the apple), but the task was just to find if its 199,200 or 201 case.
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Altamira_64
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Re: 8 apples  
« Reply #4 on: Mar 11th, 2016, 11:44am »
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From the above 3 cases:
 
abcd-efgh  
abef-cdgh  
abgh-cdef  
you can have  
1. Both rotten apples at the left scale
2. both at the right
3. one plus one
 
Only by identifying which side is > or <, how can you tell which of the above 3 cases we have? Also, by this, how can you tell if the two bad apples together are lighter or heavier or equal to 2 good ones?
Can you elaborate on this with a diagram?
 
Many thanks!!
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Hippo
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Re: 8 apples  
« Reply #5 on: Mar 11th, 2016, 1:43pm »
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Write inequalities how mesures ended and I will tell you which case it was.

<<< A (3,0) (1,2) (1,2) (1,2) 98\in ab
<<> C (2,1) (2,1) (2,1) (0,3) 102\in gh
<>< C (2,1) (2,1) (0,3) (2,1) 102\in ef
<>> A (1,2) (3,0) (1,2) (1,2) 98\in cd
><< C (2,1) (0,3) (2,1) (2,1) 102\in cd
><> A (1,2) (1,2) (3,0) (1,2) 98\in ef
>>< A (1,2) (1,2) (1,2) (3,0) 98\in gh
>>> C (0,3) (2,1) (2,1) (2,1) 102\in ab
« Last Edit: Mar 11th, 2016, 1:56pm by Hippo » IP Logged
Altamira_64
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Re: 8 apples  
« Reply #6 on: Mar 11th, 2016, 2:09pm »
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What do you mean by (3,0), (1,2) etc??
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Hippo
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Re: 8 apples  
« Reply #7 on: Mar 11th, 2016, 2:12pm »
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on Mar 11th, 2016, 2:09pm, Altamira_64 wrote:
What do you mean by (3,0), (1,2) etc??

Read the original post.
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Altamira_64
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Re: 8 apples  
« Reply #8 on: Mar 13th, 2016, 1:26am »
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You are brilliant!!!
 
on Mar 11th, 2016, 2:12pm, Hippo wrote:

Read the original post.

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