Sumplete is Hard, Even with Two Different Numbers

TL;DR

The paper proves that deciding solvability for Sumplete remains NP-complete even when the grid contains only two values, $1$ and $3$. It achieves this via a reduction from XSAT on $3$-CNF$_{+}^{3}$ to a $(1,3)$-Sumplete instance of size $(n+1) imes n$, carefully aligning row sums and column sums to capture clause satisfaction and variable consistency. The construction uses entries $a(i,j)\in\\{1,3\}$ with $a(n+1,j)=3$, row hints $R(i)$ and column hints $C(j)$ to force the necessary logical constraints. This work clarifies the computational hardness of a popular grid puzzle and shows that even highly restricted instances are computationally intractable in the worst case.

Abstract

Sumplete is a logic puzzle famous for being developed by ChatGPT. The puzzle consists of a rectangular grid, with each cell containing a number. The player has to cross out some numbers such that the sum of uncrossed numbers in each row and column is equal to a given integer assigned to that row or column. In this paper, we prove that deciding solvability of a given Sumplete puzzle is NP-complete, even if the grid contains only two different numbers.

Figures (3)

• Figure 1: An example of a $5 \times 5$ Sumplete puzzle (left) and its solution (right)
• Figure 2: An XSAT problem for $\text{3-CNF}_{+}^{3}$ (left) and its solution (right)
• Figure 3: A $(1,3)$-Sumplace instance transformed from the XSAT problem in Figure \ref{['fig2']} (left) and its solution (right)

Theorems & Definitions (1)

• Theorem 1