Inverting the 10-Coin Triangle Puzzle and Other Shapes: A New General Solution

TL;DR

This paper introduces an intuitive formula for the minimum number of coin moves needed to flip triangular coin arrangements and explores similar inversion problems for other shapes, providing a more elegant solution than previous methods.

Contribution

The paper presents a direct, intuitive formula for solving the coin inversion puzzle for any number of rows and extends the analysis to other shapes like rhombuses.

Findings

A new formula always yields an integer result for the minimum moves.

The formula simplifies the inversion process for any number of rows.

Connections between triangle and other shape inversions are identified.

Abstract

This puzzle, often called the "Reverse the Triangle Puzzle," appears regularly in puzzle books. Four rows consisting of 1 coin in row 1, 2 coins in row 2, 3 coins in row 3, and 4 coins in row 4 form the shape of a triangle. What is the fewest number of coins you need to move to flip the triangle 180 degrees so that it points in the opposite direction? In the case of 4 rows, it requires moving just 3 coins. The general solution for any number of rows can be calculated by dividing the number of coins by 3 and ignoring any remainder. A triangle of 4 rows has 10 coins, so 10/3 = 3.333 leads to 3 moves. However, this calculation is fairly nonintuitive as to why it works and requires an inelegant maneuver of periodically discarding a remainder. In this paper, we present a more intuitive formula that always calculates the answer directly and never produces a decimal answer. Further, we explore inverting other shapes (e.g., rhombus) and find an interesting connection to the formula for triangles.