The 15-puzzle is a puzzle game played with 15 square tiles numbered from 1 to 15 on a 4×
4 board. It has been popular for generations because of its simplicity and challenge. The (w× …

Suppose that a sequence of n cards, numbered 1 to n, is placed face up in random order.
Let k be the number on the first card in the sequence. Then take the first k cards from the …

A zero-knowledge proof (ZKP) allows a prover to prove to a verifier that it knows some
secret, such as a solution to a difficult puzzle, without revealing any information about it. In …

Card-based cryptography uses a physical deck of cards to achieve secure computations. To
evaluate the performance of card-based protocols, the numbers of helping cards and …

In this paper, we propose zero-knowledge proof (ZKP) protocols using physical objects for
four pencil-and-paper puzzles: the well-known Sudoku as well as Makaro, Futoshiki, and …

Decomposition puzzles are pencil-and-paper logic puzzles that involve partitioning a
rectangular grid into several regions to satisfy certain rules. In this paper, we construct a …

In this paper, we propose a physical protocol to verify the first nonzero term of a sequence
using a deck of cards. The protocol lets a prover show the value of the first nonzero term of a …