A short introduction to
the theory of inverted dice sums (
)
Abstract: This paper introduces the concept of inverted dice sums. Dice rolls are represented by multisets. For each roll, 𝑅, the regular dice sum, Σ(𝑅), is the sum of all dice values in 𝑅. The inverted dice sum, Σ̃(𝑅), is the sum of all dice values that are not present in 𝑅. Both standard 6-faced dice and generalized 𝑓-faced dice are discussed. The concept of distinct rolls is defined, and the images of the functions Σ and Σ̃ are explored. The problem of finding rolls where Σ(𝑅) = Σ̃(𝑅) is discussed. Some hints regarding the use of inverted dice sums in games are presented. Even though the probability distribution of inverted dice sums is not intuitively clear, some patterns are established here. But the main result is a formula for the probability of rolling the inverted sum 𝑥 with 𝑛 standard dice, using Stirling numbers of the second kind to count the number of outcomes corresponding to each inverted dice sum 𝑥.
Publiceringsår: 2023. Format: A4/PDF. Sidor: 20.
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