You have 101 coins, and you know that 50 of them are counterfeit.
Every true coin has the same weight, an unknown integer, and every false coin has the same weight, which differs from that of a true coin by 1 gram.
You also have a two-pan pointer scale that will show you the difference in weight between the contents of each pan.

You randomly oick up a coin.
Find in a single weighing whether it’s true or false.

Source:Sergei Aleksandrovich Genkin, Dmitrii Vladimirovich Fomin, and Ilia Itenberg, Mathematical Circles: (Russian Experience), 1996.)

Set aside the selected coin.  Split the remainder into 2 piles of 50 and compare the piles on the balance.  If the selected coin is real, then you have 50 real and 50 counterfit.  No matter how you constuct the piles the difference will be an even number or zero.  If the selected coin is fake, the difference will lways be odd.  

Posted by Kenny M on 2018-10-22 08:33:29