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Take the right one (Posted on 2013-10-18) Difficulty: 3 of 5
Among 100 coins exactly 4 are fake. All genuine coins weigh the same; all fake coins, too. A fake coin is lighter than a genuine coin.

How would we find at least one genuine coin using two weighings on a balance scale?

Source: 2010 Euler math Olympiad in Russia- authored by A.Shapovalov

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

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Think I've got it | Comment 15 of 18 |
Thanks to the hint and an insight by Mrs. xdog. 

Weigh 33 against 33.

Say they don't balance.  Then the heavier side has either 0 or 1 light coins.  Pick two coins from the heavier side and weigh them.  If they balance they're both normal, otherwise the heavier is normal.

If the 33-33 weighing balances, each side can contain 0,1, or 2 light coins.  Here I complained to Mrs. xdog that I couldn't match 34 against 33.  She replied "Well, 33 and 1 equals 34".  Yes, yes it does.

Label the groups of 33 X and Y, the group of 34 Z.  Take a coin from X and add it to Y.  

Distribution of light coins 
          XYZ        XYZ
Case   before     after 
 1        004        004
 2        112        112
 3        112        022
 4        220        220
 5        220        130

Case 1, Y>Z, transferred coin normal
Case 2, Y>Z, transferred coin normal
Case 3, Y=Z, remaining X normal
Case 4, Y<Z, all Z normal
Case 5, Y<Z, all Z normal   

  Posted by xdog on 2013-10-20 13:24:15
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