A Doubly Balanced Table

★ 9/10combinatorics number theory Modular Arithmetic Induction

Problem

Prove that infinitely many positive integers $n$ admit an arrangement of $1, 2, \dots, 3n$ into an $3 \times n$ table

\begin{matrix} a_1 & a_2 & \cdots & a_n \\ b_1 & b_2 & \cdots & b_n \\ c_1 & c_2 & \cdots & c_n \end{matrix}

such that

1. all column sums $a_i + b_i + c_i$ are equal and divisible by $6$ , and 2. all row sums are equal and divisible by $6$ .

Answer

Solution

Difficulty	9/10
Topics	combinatorics, number theory, Modular Arithmetic, Induction

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