"Find which coin is fake and which is genuine?"-A coin riddle on parity of numbers

Question

Of $101$ coins, $50$ are counterfeit, and differ from genuine coins in weight by $1$ gram. Peter has a scale in form of balance which shows the difference in weights between the objects placed in each pan. He chooses one coin and wants to find out in one weighing whether it is counterfeit. Can he do this? How?

This question was posed by my teacher. He told me that answer was "Yes, he can find out the nature of coin".
Now I'm stuck on "how he could do that?".
A hint will also help me out as, 'I don't know where to start?'.

Answer 1

Let's suppose the fake are $1g$ different, but always on the same side (lighter or heavier).

You have one coin. You keep it on the side.

Split the remaining $100$ coins into two groups of $50$.

If the coin is a good one, you have then $50$ fake ones. Meaning that on the right pan, you will have $50$ coins, $50-x$ being good, $x$ being fakes, and on the left pan, it is the reverse.

So the DIFFERENCE between the two pans is in the form $(50-x)k+x(k+1)-(50-x)(k+1)+xk=2N$, $N \in \mathbb{Z}$, meaning an even number.

If you have it in the form $(2N+1)$, $N \in \mathbb{Z}$, meaning an odd number, then the coin was fake...

Answer 2

Let's call the coins G for genuine and F for fake.

Imagine instead of 101 coins there were 5 coins: 2 Gs and 3 Fs

Pick one out and there are two options

Option 1. You picked a G so there are 1 G and 3 Fs left

If you split these across the two scales you would have FG/FF or FF/FG and the scales would either read +1 or -1

Option 2. You picked an F so there are 2 Gs and 2 Fs left

If you split these across the two scales you would have FG/FG or FF/GG and the scales would either read 0 or 2.

We can therefore say if we pick one coin and split the remainder: if the scales read + /-1 we picked a genuine coin, if the scales read 0 or 2 we picked a fake one.

Extend this thinking for 7 coins, 9 coins....