# Introduction

Welcome to my interactive FTO tutorial guide! This page is aimed at new solvers interested in learning my FTO method, Bencisco. No prior FTO knowledge is needed and all of the material can be taken at your own pace. Regardless if your goal is to go for speed or simply a casual solve, I highly recommend learning this method over others as it has been proven more efficient than older approaches, and being mostly intuitive, is relatively simpler to learn.

As I stated, this tutorial is interactive! Utilizing the Twizzle applet developed by Lucas Garron and Tomas Rokicki, this offers a brand-new learning experience which allows the move sequences and algorithms to be more clearly illustrated than ever.

Have a play with the applet below to get a feel for how it works!

This interactive 3D applet makes it easier to see how the puzzle moves. You can click and drag on the puzzle to rotate it in 3D space!

Playing moves works similarly to playing a video. Click the play button at the bottom-center to play the whole sequence, or play one move at a time using the arrow buttons. You can also adjust the play head to any point within the sequence of moves.

Most of the applets will show the puzzle in this form, with just the front view and some of the stickers on the posterior faces made visible. In all the applets for this solution guide, the color scheme from the Lanlan Octahedron will be used and referenced, which can be seen here.

In some instances the applet may go invisible while rotating the puzzle around. This isn't frequent, but refreshing the page will fix this problem.

The Basics

Before we begin, you should already be familiar with the standard Rubik's cube and some of the universal concepts that apply to solving twisty puzzles in general (e.g. solving whole layers as opposed to individual colors, and using algorithms to preserve completed sections). This also includes the move notation from the cube, as that will help with following many of the sequences used for the FTO. Experience with some higher-order cubes (4x4 and so on) is recommended as well, though not required.

The first thing to understand are the individual types of pieces present in the puzzle. An FTO has corner pieces and edge pieces (somewhat analogous to those on a 3x3 cube) but also has a kind of moving center piece, which we will call a "triangle" to keep things simple. There are a total of 6 corner pieces, 12 edge pieces, and 24 triangle pieces in the whole puzzle. The corners have four stickers each on the FTO (since four faces share a single vertex) while the edges and triangles have 2 stickers and 1 sticker each, respectively. The corners and edges are all unique in coloring, but the triangles come in 8 sets of 3 identical colors, meaning some triangles are identical when solving. One crucial detail is that the FTO does not have any fixed center pieces, and therefore the color scheme needs to be kept in mind when solving to avoid parity issues. However this is a similar setup to larger NxN cube puzzles (4x4 and up) so it shouldn't be too unfamiliar.

While the FTO shares many of the fundamental properties of most twisty puzzles, there are some unique behaviors inherent to the geometry of the puzzle itself to know about. The most significant of these being, that the 8 colors of the puzzle are always divided into two "orbits" of 4 non-adjacent colors, no matter how it is scrambled: the {U, F, BR, BL} faces (white, red, gray, and orange) form one orbit, and the {R, L, B, D} faces (green, purple, blue, and yellow) form the other orbit. This is referred to as the FTO's "half scrambling" property, which is virtually unique from all other twisty puzzles (you can see what this looks like in one of the applets below). It also directly implies that the corner pieces can only twist by 180 degrees, and the edge pieces cannot be flipped in place at all, which limits how they can be oriented. Further, the triangles behave as two separate sets of 12 pieces, and triangles in different sets cannot scramble with each other. All of this takes some getting used to, but it comes with familiarizing yourself around the movements of the FTO.

How The Method Works

By taking advantage of the aforementioned "half scrambling" property, we can build and preserve larger "groups" of smaller pieces when we solve the puzzle. There are two types of useful groups in particular, and they are very important for understanding the Bencisco solution method. These are "centers", and "triples".

A center consists of 3 edges and 3 triangles, grouped such that they form an inscribed hexagon on one face of the puzzle (fig 1, left). These groups have three possible orientations each, and behave similar to Pyraminx corners. There are four of these in total and together utilize all the edge pieces in the puzzle. Notice that the edge stickers overlap into the 3 adjacent faces; this is why only four separate center groups will ever be possible at once.

A triple consists of two triangles and a single corner, grouped such that the triangles lie on opposite sides of the corner (fig 2, right). These groups have two possible orientations each, and behave similar to Pyraminx edges. There are six of these in total and together utilize all the corner pieces in the puzzle.

Fig 1: a single center group.

Fig 2: a single triple group.

The piece groups serve both as a way to make cases simpler to recognize and as a preservation tool to help keep large sections of the puzzle solved in later steps of the solution. Essentially, by only moving certain layers of the FTO, we can keep the center and triple groups completely intact relative to each other, and this is a nice property which allows this method to work. Once we get into the solution, this will become more apparent.

Getting Started

To illustrate the steps of the method, I will describe each of the steps and the goals for completing them, and reference an example solve using the applets to visually show the move sequences used to do so. At certain points, I will also use the applets to separately demonstrate any algorithms needed or when unusual/difficult cases may occur during a solve.

The scramble I will use for the example solve is provided below:

BL U BR U' R L' D' L BR F' BL' F' L' F' L' B' BL' B' BL BR U' BL L' R U R' B F L U

If you would like to follow the example solve directly, you can scramble your puzzle from the solved state by using the above sequence of moves. But you definitely don't have to, as the steps of the Bencisco method work with any scramble and are completed in exactly the same ways. So if your puzzle is already mixed up, no problem. Perhaps after completing your first solve on your own, you may want to go back and follow along with the example solve directly, as that will help reinforce many of the concepts in the solving method.

With everything introductory out of the way, we are finally ready to jump into the solution!

This will be split into a total of three sections, with each section covering two steps of the Bencisco method. While there is no reasonable way to cover every single possible position that may arise from each step, I will include all the important cases to know and recognize along the way, and how to deal with them. You can navigate through the various parts of the guide through the navigation bar at the top of the page, or with the links below.

First Block - Steps 1 & 2 Equator Centers - Steps 3 & 4 Last 4 Triples - Steps 5 & 6