# Result of the optimal solution search for Checkerboard Origami Triangle

## Kazuki Ohshima†, Ryuhei Uehara‡ and Jun Mitani†

University of Tsukuba
Japan Advanced Institute of Science and Technology

Last updated on January 04 2020

The Origami Checkerboard Triangle Puzzle is a puzzle with the goal of folding a single one-sided colored equilateral triangle sheet of paper into a pattern in which nine equilateral triangles (we call these triangles "cells") are stacked in 3 with as few steps as possible, which was proposed by Serhiy Grabarchuk. We show an example of solution of this puzzle in Figure 1.

In this puzzle, only simple folds can be applied. A single simple fold in a procedure is counted as one step. A simple fold is a folding that transforms a flat state of a paper to another flat state with respect to a single crease. There is no restriction on the number of layers at the folding (it is called some-layers simple fold).

As the goals of this puzzle, there are 59 patterns duplication by flip, rotate, and mirror. Solutions have been found for all of them (we call a procedure which reach a certain pattern "solution"). When multiple solutions exist for one pattern, we consider the solutions with shorter steps are better. Further, when there are multiple solutions with the same number of steps, we consider the solutions with smaller initial size of triangle are better. The known best solutions have been published on the webpage by Ishino.

We enumerated solutions of this puzzle using super computer situated at the Japan Advanced Institute of Science and Technology (JAIST). The range of the search is up to six steps for initial triangle sizes 4 to 7, and up to five steps for initial triangle sizes 8 and 9. "Size n" indicates that stacked equilateral triangles are in n rows. As the result we obtained better solutions for several patterns than the known solutions.

Click a pattern in the following table to display the procedures of the selected pattern.