Counterfeit coin problem, but harder
You have nine identically looking coins, one of which is counterfeit and weighs less than the original. You also have three balance scales two of which work properly and one is broken.
Broken balance scale can show the result randomly, not correlated with the weight of the coins (it can be identified correctly or it may show the weight is lighter (when it's heavier) or heavier (when it's lighter)). Which balance scale is broken is unknown .
What is the minimum number of weighing needed to guarantee that the counterfeit coin is identified?
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1 solution
Let the coins be c_1, c_2, ..., c_9 , and let the balance scales be A, B, C
We take a pile of coins c_1, c_2, c_3 and weigh them against c_4, c_5, c_6 in balance scale A . If one pile is lighter we put the pile aside, if they are equal that means, according to scale A , the pile c_7, c_8, c_9 is lighter, so we take it and put it aside.
Now we take the pile c_1, c_4, c_7 and weigh them against c_2, c_5, c_8 in balance scale B . We, again, determine which pile is lighter, according to scale B , and put it aside (if the weight of c_1, c_4, c_7 is equal to the weight of c_2, c_5, c_8 , we take the pile c_3, c_6, c_9 and put it aside (because it is lighter, according to scale B)
WLOG, suppose A showed the pile c_1, c_2, c_3 is lighter, B showed the pile c_1, c_4, c_7 . Now we take c_2, c_3 and weigh them against c_4, c_7 in scale C. If scale C shows that c_2, c_3 < c_4, c_7 that means one of scales showed wrong result, but in this case we know for sure that scale A is working properly and we easily determine the counterfeit coin with scale A out of c_1, c_2, c_3 . If c_2, c_3 > c_4, c_7 , the the scale B is for sure working properly and we identify the counterfeit with the help of B out of c_1, c_4, c_7
Now, if the scale C shows the weight c_2, c_3 = c_4, c_7 that means the counterfeit coin is c_1 .
Note : if 2 results from any two different balance scales correlate or correspond to each other that means the result is correct.