Number Theory and UNO

Origins

Much has been written about the role of chance and fate in UNO, but it was Carl Friedrich Gauss who first introduced the concept of formal mathematical notation as a way to express the probabilistic models of the game. In his book (image right) UNO und Arithmeticae, Gauss clearly described that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree.

However, his inability to predict (let alone win) games of UNO strained his comprehension and sanity, ultimately leading to dementia and death. Ultimately it was Kurt Hensel who described the notation widely used today, known as p-adic number system.

P-adic expansions in UNO

Hensel’s notation is easy to follow.

If p is a fixed prime number (in UNO, the primes are 1,3,5 and 7), then any positive UNO number card can be written in a base p expansion in the form

Close followers of the subject will recognize a more familiar fashion in which this description can be generalized; by mapping it to the larger domain of the rational UNO numbered cards (all of them) (and, incidentally, all of the real numbers, but of course they are included here). This can be accomplished by including the sums of the form

In UNO, we needn't extend the base p expansions by allowing infinite sums of the form because of the limited number of integers.

From there you can clearly deduce that all UNO numbered cards are rational, and only four of them are prime.

NOTE: readers who wish to debate endlessly the advantages or disadvantages of the fact that representations can be much larger by simply storing the numerator and denominator in binary may do so in the forum. Please do not email the UNOtips team directly: your mails will go unanswered.

The seminal work by Gauss

A page from Kurt Hensel's Uber eine neue Begrundung der Theorie der Algebraischen Zahlen.

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