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 Take the right one (Posted on 2013-10-18)
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

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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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