Quad Squares

TL;DR

This paper explores the structure and enumeration of special 4x4 squares formed from the EvenQuads deck, introducing new types of magic squares based on card attributes and mathematical properties.

Contribution

It defines and analyzes Latin, semimagic, magic, and strongly magic quad squares, providing formulas for counting these squares for any deck size and linking them to integer XOR properties.

Findings

Characterization of quad squares using XOR of four integers

Formulas for counting semimagic, magic, and strongly magic quad squares

Introduction of new types of magic squares based on card attributes

Abstract

We study 4-by-4 squares formed by cards from the EvenQuads deck. EvenQuads is a card game with 64 cards where cards have 3 attributes with 4 values in each attribute. A quad is four cards with all attributes the same, all different, or half and half. We define Latin quad squares as squares where the cards in each row and column have different values for each attribute. We define semimagic quad squares as squares where each row and column form a quad. For magic quad squares, we add a requirement that the diagonals have to form a quad. We also define strongly magic quad squares. We analyze types of semimagic and strongly magic quad squares. We also calculate the number of semimagic, magic, and strongly magic quad squares for quad decks of any size. These squares can be described in terms of integers. Four integers form a quad when their bitwise XOR is zero.