A Curious Weighing Problem
You have a bag with 101 coins identical in appearance, but 50 of these coins are fake, and the other 51 coins are genuine.
You know that a fake coin has a different weight from a genuine coin. The difference is only 1 gram, but you don't know if the fake coins are heavier or lighter than the genuine coins.
You received a curious scale to measure the weight of these coins. This scale has two pans, and instead of compare weighs, this is a digital scale that will show to you the modulus of the difference between the weights in the two pans. For example, if a pan has 67 grams, and the other pan have 55, the scale will show to you \(|67-55|=12\) or \(|55-67|=12\) (the same value). Also, the scale only works if there are something in both pans.
The problem is:
If you take at random a coin from this bag, it is possible to know if this coin is fake or not, using the scale only 1 time to weigh the coins?