Preparing for a big end-of-semester party at Hogwarts, you crack open the Gryffindor cellar and count n bottles of fine drink.

Dumbledore has previously warned you that exactly k of these bottles have been poisoned (he wouldn't go into detail as to how exactly this came to be), and consuming poisoned drink will result in an unpleasant death.

The party starts in one hour, and you do not want to poison any of your guests. Luckily, a family of t docile rats occupies a corner of the cellar, and they have graciously volunteered to be test subjects for identifying the poisoned bottles.

Let l = o(n) and k = 1, and assume it takes just under one hour for poisoned drink to kill a rat. (Hence, you only get one shot at solving this problem.)

Describe a scheme by which you can feed the drink to rats and identify with complete certainty the poisoned bottle, prove that the scheme is correct, and give a tight bound on the number of rats l necessary to solve the problem.