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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.

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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).

Continue reading Imputing Out of Mixtures, or Un-Stirring Spicy Soup

Posted on Categories Expository Writing, math programming, Mathematics, TutorialsTags , , , ,

Unspeakable bets: take small steps

I was watching my cousins play Unspeakable Words over Christmas break and got interested in the end game. The game starts out as a spell a word from cards and then bet some points game, but in the end (when you are down to one marker) it becomes a pure betting game. In this article we analyze an idealized form of the pure betting end game. Continue reading Unspeakable bets: take small steps

Posted on Categories Expository Writing, MathematicsTags , , , ,

Some puzzles about boxes

This article is a break from data-science, and is instead about the kind of problem you can try on the train. It is problem 70 in Bollobas’s “The art of mathematics” (though I forgot that and re-worked the problem crudely from memory when writing this article).

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One of the many irritating things about airlines is the fact that the cary-on bag restrictions are often stated as “your maximum combined linear measurement (length + width + height) must not exceed 45 inches” when they really mean your bag must fit into a 14 inch by 9 inch by 22 inch box (so they actually may not accept a 43 inch by one inch by one inch pool spear as your carry-on). The “total linear measure” seems (at first glance) “gameable,” but can (through some hairy math) at least be seen to at least be self-consistent. It turns out you can’t put a box with longer total linear measurements into a box with smaller total linear measurements.

Let’s work out why this could be problem and then why the measure works. Continue reading Some puzzles about boxes