Yet another counterfeit coin weighting problem

Question

This is a problem from Tournament of Town competition taking place today.

There are 100 coins in a row. We know that there are $26$ counterfeit coins in that row. The counterfeit coins lay successively in the row (if I didn't explain it clear enough, then here is an example: $10$ genuine, $26$ counterfeit, $64$ genuine). The masses of genuine coins are equal. The mass of a counterfeit coin is less than the mass of a genuine coin. How can I find at least one counterfeit coin by using only one weighing?

Answer 1

I assume that you use a balance, to compare two coins. Take coins number 25 and 75. If one is lighter, it is fake. If both are the same, the 26 fake coins are in between. Then coin number 50 is fake.

A more detailed explanation: if both are the same, there can only be 24 coins before or 24 coins after, so that contradicts the successive condition. If fake coins start right after number 25, you go past 50. If fake coins just end before 75, you started having fake coins before 50. So 50 is a fake coin

Note that if the coins number 25 and 75 have the same weight, they cannot be both fake (too far apart)

Answer 2

Hint: I can do this by weighing two coins one against the other. More coins simply make noise. I need to make sure that my two coins cannot both be in the $26$ - otherwise I learn nothing. If they are equal in weight, I know they are both good, so I need to restrict the remaining options enough that I can tell where a fake coin must be. So I can't then have two disjoint runs of $26$ from which I have picked nothing - either could be all fake.