Coin weight puzzle Let us assume that we have a box of coins which contains 10 normal coins and 11 counterfeit coins. We know that each counterfeit coin is 1 gram less than each of the normal. One coin was taken from the box.
If we have a regular scale and we can only use it once (for one single measurement), how can we find out what was the coin which was taken out of the box (figure out whether it was a counterfeit or normal one)?
We can assume that the coins' weight can be any real number.
What I have considered so far:


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*I know that if we measure the coin alone, which was taken out of the box, we have no way of figuring out whether it is a counterfeit or a normal one without measuring the whole box - and we cannot measure it since we will be breaking our rule of having to use the scale just once. 

*I do not know how will measuring just the box would help either. Correct me if I am wrong but since we do not know how much is the weight of an individual coin, the total weight wouldn't be of any use to us, right? So I am stuck. 

 A: For the purposes of this answer, I am going to assume that the weight of a coin is an integer. For convenience, I'll denote the weight of a normal coin by $N$.
Once you remove one coin from the box, one of two events will occur:


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*You removed a normal count. In which case, there are now $9$ normal coins and $11$ counterfeit coins in the box. The total weight of these coins is given by $$9N + 11(N-1)=20N-11\tag 1$$

*You removed a counterfeit coin. In this case, there are $10$ normal coins and $10$ counterfeit coins left in the box. The total weight is there given by $$10N + 10(N-1) = 20N - 10. \tag 2$$
It should not be hard to see that regardless of the value of $N$, case (1) will give an odd number for the total weight, while case (2) will given an even number.
Therefore, you can determine whether the coin you selected was normal or counterfeit simply by measuring the total weight of all of the remaining coins (one measurement) and checking whether the resulting weight is even or odd. An even weight corresponds to a counterfeit coin, while an odd weight will be indicative of a normal coin.
