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Main Authors:	Huang, Eric, Khovanova, Tanya, Kilybayev, Timur, Li, Ryan, Ni, Brandon, Seidel, Leone, Sharma, Samarth, Sheffield, Nathan, Varanasi, Vivek, Yin, Alice, Yun, Boya, Zelevinsky, William
Format:	Preprint
Published:	2025
Subjects:	Number Theory Combinatorics Primary 11A99
Online Access:	https://arxiv.org/abs/2509.11395
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author	Huang, Eric Khovanova, Tanya Kilybayev, Timur Li, Ryan Ni, Brandon Seidel, Leone Sharma, Samarth Sheffield, Nathan Varanasi, Vivek Yin, Alice Yun, Boya Zelevinsky, William
author_facet	Huang, Eric Khovanova, Tanya Kilybayev, Timur Li, Ryan Ni, Brandon Seidel, Leone Sharma, Samarth Sheffield, Nathan Varanasi, Vivek Yin, Alice Yun, Boya Zelevinsky, William
contents	Various card tricks involve under-down dealing, where alternatively one card is placed under the deck and the next card is dealt. We study how the cards need to be prepared in the deck to be dealt in order. The order in which the $N$ cards are prepared defines a permutation. In this work, we analyze general dealing patterns, considering properties of the resulting permutations. We give recursive formulas for these permutations, their inverses, the final dealt card, and the dealing order of the first card. We discuss some particular examples of dealing patterns and conclude with an analysis of several existing and novel magic card tricks making use of dealing patterns. Our discussions involve 30 existing sequences in the OEIS, and we introduce 44 new sequences to that database.
format	Preprint
id	arxiv_https___arxiv_org_abs_2509_11395
institution	arXiv
publishDate	2025
record_format	arxiv
spellingShingle	Card Dealing Math Huang, Eric Khovanova, Tanya Kilybayev, Timur Li, Ryan Ni, Brandon Seidel, Leone Sharma, Samarth Sheffield, Nathan Varanasi, Vivek Yin, Alice Yun, Boya Zelevinsky, William Number Theory Combinatorics Primary 11A99 Various card tricks involve under-down dealing, where alternatively one card is placed under the deck and the next card is dealt. We study how the cards need to be prepared in the deck to be dealt in order. The order in which the $N$ cards are prepared defines a permutation. In this work, we analyze general dealing patterns, considering properties of the resulting permutations. We give recursive formulas for these permutations, their inverses, the final dealt card, and the dealing order of the first card. We discuss some particular examples of dealing patterns and conclude with an analysis of several existing and novel magic card tricks making use of dealing patterns. Our discussions involve 30 existing sequences in the OEIS, and we introduce 44 new sequences to that database.
title	Card Dealing Math
topic	Number Theory Combinatorics Primary 11A99
url	https://arxiv.org/abs/2509.11395

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