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        riddles >> hard >> 8 apples
        
(Message started by: Altamira_64 on Feb 21^st, 2016, 11:16am)

Title: 8 apples
Post by Altamira_64 on Feb 21^st, 2016, 11:16am

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.

Title: Re: 8 apples
Post by Hippo on Mar 9^th, 2016, 12:15am

[hide]
... My original bet was wrong ... :)

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.
[/hide]

Edit: I forget 8 could not stand before )

Title: Re: 8 apples
Post by Altamira_64 on Mar 9^th, 2016, 10:01am

Yes but how do we know which pair contains 98 (or 102)?

Title: Re: 8 apples
Post by Hippo on Mar 11^th, 2016, 9:25am

on 03/09/16 at 10:01:00, 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.

Title: Re: 8 apples
Post by Altamira_64 on Mar 11^th, 2016, 11:44am

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!!

Title: Re: 8 apples
Post by Hippo on Mar 11^th, 2016, 1:43pm

Write inequalities how mesures ended and I will tell you which case it was.
[hide]
<<< 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
[/hide]

Title: Re: 8 apples
Post by Altamira_64 on Mar 11^th, 2016, 2:09pm

What do you mean by (3,0), (1,2) etc??

Title: Re: 8 apples
Post by Hippo on Mar 11^th, 2016, 2:12pm

on 03/11/16 at 14:09:07, Altamira_64 wrote:

What do you mean by (3,0), (1,2) etc??

Read the original post.

Title: Re: 8 apples
Post by Altamira_64 on Mar 13^th, 2016, 1:26am

You are brilliant!!!

on 03/11/16 at 14:12:24, Hippo wrote:

Read the original post.