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## Imputing Out of Mixtures, or Un-Stirring Spicy Soup

Here is a fun combinatorial puzzle. I’ve probably seen this used to teach before, but let’s try to define or work this one from memory. I would love to hear more solutions/analyses of this problem.

Suppose you have `n` kettles of soup labeled `0` through `n-1`. For our problem we assume that `k` kettles of soup are extremely spicy. We want to figure out which kettles contain spicy soup.

Image source: Mad Dog 357 / Amazon

This presents an interesting puzzle when `k` is much smaller than `n`. We are assuming that spicy is a rare event we want to detect. We are also assuming the spicy soups are so spicy, that they remain spicy even when combined with other soups. So when we prepare mixtures of soups we experience the union of the spiciness of the included soups.

The question is: if we prepare tasting bowls that are mixtures of samples from the kettles- how many bowls do we have to prepare to reliably identify all of the spicy soup kettles? This is hopefully in the spirit of the “counterfeit gold coin puzzle” as seen in the Columbo detective show (though I end up using a bit more math).

## What is a win vector?

From time to time we are asked “what is the company name Win-Vector LLC referring to?” It is a cryptic pun trying to be an encoding of “we deliver victory.”

The story is an inside joke referring to something really only funny to one of the founders. But a joke that amuses the teller is always enjoyed by at least one person. Win-Vector LLC’s John Mount had the honor of co-authoring a 1997 paper titled “The Polytope of Win Vectors.” The paper title is obviously mathematical terms in an odd combination. However the telegraphic grammar is coincidentally similar to deliberately ungrammatical gamer slang such as “full of win” and “so much win.”

If we treat “win” as a concrete noun (say something you can put in a sack) and “vector” in its non-mathematical sense (as an entity of infectious transmission) we have “Win-Vector LLC is an infectious delivery of victory.” I.e.: we deliver success to our clients. Of course, we have now attempt to explain a weak joke. It is not as grand as “winged victory,” but it does encode a positive company value: Win-Vector LLC delivers successful data science projects and training to clients.

Winged Victory: from Wikipedia

Let’s take this as an opportunity to describe what a win vector is. Continue reading What is a win vector?