---
tag: B2DNP2N7WZ
kind: statement
role: proposition
name: Ultimate Gilbreath principle
sources:
  - wikipedia: en:38064572@r1348023430
    locator: Gilbreath shuffle, section Ultimate Gilbreath principle
    note: adopted
    mention: wikipedia://en/38064572#4J3C
    sections: [Statement]
inventory: [YZP67M459G]
---

## Statement

A theorem called "the ultimate Gilbreath principle" states that, for a permutation $\pi$ of $\{1, 2, 3,\dots, n\}$, the following four properties are equivalent:

- $\pi$ is a Gilbreath permutation.
- For each $j$, the top $j$ cards $\pi(1),\dots \pi(j)$ are distinct modulo $j$.
- For each $j$ and $k$ with $kj\le n$, the $j$ cards $\pi\bigl((k-1)j+1\bigr),\pi\bigl((k-1)j+2\bigr),\dots,\pi(kj)$ are distinct modulo $j$.
- For each $j$, the top $j$ cards are consecutive in $1,2,\dots, n$.