This is an extension of the problem posted here.
You have 14 coins. You know that 7 are real, and 7 are counterfeit. The real coins are 1000g, and the fake coins are 999kg.
You have a balance accurate enough to do the measurement. However, this balance is a peculiar balance. Assume that if you weigh multiple coins, you do not place them incrementally. That is, if you weigh 3 coins against another 3 coins, you would place all the coins in the balance scale at the same time.
1)If the weight one one side is heavier, the balance incinerates one coin selected at random from the heavier side. (Recall that the real coins are heaver than the fake ones). 2)If the weight is equal, you take back all the coins.
Find a methodology in which one can necessarily identify a single real coin that has not been incinerated. Either find the strategy, or prove that such a strategy does not exist.