Poisoned Cookies

Here’s an interesting puzzle which was part of the application for SPARC 2016. (Applications are now closed)

Suppose you have 240 cookie jars, one of which has been poisoned, so all the cookies in it are poisonous.

You also have 5 grey pigeons, each of which is immune to the poison, but has an interesting property: if it eats even a tiny crumb from a poison cookie, it will turn white at exactly 9am the following day, and stay white forever. The pigeons also have hearty appetites, and will immediately eat anything you put in front of them (which could include mixtures from multiple jars).

It is currently 8am, and you’re hosting a huge party at 10am the day after tomorrow (50 hours from now). You’d like to serve out as many cookie jars as possible while knowing, with certainty, that they are non-poisonous.

How many cookie jars is that, and how do you know? To get full credit, you should state the maximum, explain a strategy that achieves the maximum, and then explain why the strategy is optimal. (Partial solutions that establish a large number of safe cookie jars but not the maximum will also be considered. Solutions that make erroneous arguments will be penalized, so it is better to submit a correct partial solution than an incorrect full solution.)

Think about it before you read on. This post will contain:

My Kinda Elementary but Long-winded Solution
Samyak’s Streamlined Solution
Multinomial Digression

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