The previous solution assumed that you can only resolve 7 coins when you get down to two weighings. But when you're down to two weighings you have the traditional problem and you can resolve 9 coins. Split the 9 into three groups of 3, and use the two weighings to find the fake coin.
So with three weighings, you can actually resolve 9 + 5 + 5 = 19 coins. With four weighings you can resolve 19 + 7 + 7 = 33 coins, with five weighings you can resolve 33 + 9 + 9 = 51 coins, with six weighings you can resolve 51 + 11 + 11 = 73 coins, and with seven weighings you can resolve 73 + 13 + 13 = 99 coins, just as stated in the original problem!
In general, then, within k weighings, you can resolve up to 2k^2 + 1 coins.
Posted by tomarken
on 2014-07-02 14:56:37