# Another 2016 coins and a balance

This continues the puzzle "2016 coins and a balance".

Again, there is a balance with two pans on the table. Again, the display on the balance tells the difference between the weight in the left ban and the weight in the right pan, measured in gram.

Again, there are $2016$ coins on the table, of which exactly $99$ are fake. All genuine coins have the same weight. Some fake coins weigh one gram less than the genuine coins, and the remaining fake coins all weigh one gram more than the genuine coins.

Cosmo points at the rightmost coin $y$ and asks Fredo to determine whether $y$ is fake.

Question: Can Fredo decide whether coin $y$ is fake by using the balance only once?

Part a: If the coins have rational weights.
Part b: If the coins have integer weights.

## 1 Answer

Part A:

If Fredo doesn't weigh coin $y$ and also doesn't weigh some other coin, then one could be real and one could be fake, and it would be impossible to tell which was which.

If Fredo doesn't weigh coin $y$ and does weigh all 2015 other coins, then one side will have $x>0$ coins more than the other (because 2015 is an odd number; if there was an even number here, weighing half against half and checking the parity of the result would work). The weight shown can vary freely by changing the weight of every coin, whether or not $y$ is fake. For example, increasing the weight of all coins by $\frac{1}{x}$ increases the reading by 1 gram. Because the reading is consistent with $y$ being real or fake, Fredo cannot determine whether it is fake.

If Fredo does weigh coin $y$ and puts at least one coin in the same pan, one could be real and one could be fake, and again he couldn't tell the difference.

If Fredo weighs coin $y$ alone against exactly one other coin, they could both be real or both be heavy (or both light), and the scale would show zero in both cases.

If Fredo weighs coin $y$ alone against nothing or against more than one coin, there is a difference in the number of coins, and the reading can vary freely like in the second case, so he can't determine whether the coin is genuine.

These are all the possible weighings (assuming the coins cannot be cut), and none of them allows Fredo to determine whether coin $y$ is real or fake.

If coins can be cut, then Fredo should weigh coin $y$ against $\frac{1}{2015}$ of every other coin. The result will be at most $\frac{99}{2015}$ if $y$ is genuine, and at least $\frac{1917}{2015}$ if it is fake.

Part B:

Fredo should set aside coin $y$ and weigh all 2015 other coins together. The result will necessarily be within 99 grams of a multiple of 2015, which represents 2015 times the weight of a genuine coin. If the reading differs from that by an even number, then an even number of fake coins were weighed, so $y$ must be fake. On the other hand, if the reading differs by an odd number, then an odd number of fake coins were weighed, and $y$ must be genuine.