# Metric

A **metric** (more specifically, **move count metric** or **turn metric**) is a convention for counting moves. The same sequence of moves can have different move counts depending on the metric used.

## Contents

## List of metrics

### HTM

The **half turn metric** (HTM), also known as the *face turn metric* (FTM) is a metric for the 3x3x3 where any turn of any face, by any angle, counts as 1 turn; thus it is different from the quarter turn metric because half turns count as one move instead of two. It is important to note that in HTM a slice move actually counts as two turns, since the centers are assumed to be fixed. Cube rotations do not count as turns, so a 'double layer' turn such as r would count as one turn.

Thus the following algorithm would count as 11 turns HTM: r2 U R' U' M U R U' R' r' (an edge 3-cycle on the front face).

God's Number in Half Turn Metric is 20 moves.

### QTM

The **quarter turn metric** (QTM) is a metric for the 3x3x3 where any turn of any face, by 90 degrees clockwise or counterclockwise, counts as 1 turn; thus it is different from the half turn metric because half turns count as two moves instead of one. It is important to note that in QTM a slice move can count as either two turns (if it is a quarter turn like M) or four turns (if it is a half turn like M2), since the centers are assumed to be fixed. Cube rotations do not count as turns, so a 'double layer' turn such as r would still count as one turn.

Thus the following algorithm would count as 12 turns QTM: r2 U R' U' M U R U' R' r' (an edge 3-cycle on the front face).

God's Number in Quarter Turn Metric is 26 moves.

### STM

The **slice turn metric** (STM) is a metric for the 3x3x3 where any turn of any layer, by any angle, counts as one turn. This differs from HTM in that a slice move counts as one turn, not two. And it differs from the ATM in that it does not include anti-slice turns and other axial turns.

Cube rotations do not count as turns. STM is a very popular metric for those who use methods with many slice turns (such as Roux) although the official Fewest Moves event uses HTM.

The following algorithm would count as 10 turns STM: r2 U R' U' M U R U' R' r' (an edge 3-cycle on the front face).

God's Number in Slice Turn Metric is between 18 and 20 moves, but the exact number has not yet been proven.

### QSTM

**QSTM**, short for **Q**uarter **S**lice **T**urn **M**etric, is a move count metric for the 3x3x3 in which any clockwise or counterclockwise 90-degree turn of any layer counts as one turn, and rotations do not count as moves.

This differs from QTM in that a slice move counts as one turn, not two; and it differs from STM in that 180-degree turns in any layer count as two moves, not one.

God's Number in QSTM is likely around 23-26 moves, but the exact number has not yet been proven.

This metric is also known as **SQTM**.

### ETM

The **execution turn metric** (ETM), is a metric for the 3x3x3 where any perceived movement counts as a turn; this includes rotations. Therefore, the interpretation of this metric can be somewhat subjective. For example, U2 can be either 1 or 2 moves.

ETM was designed for measuring 'true' TPS by David Woner. It is intended mainly for reconstructions, where videos are available to see how many movements were actually performed.

#### Example reconstruction ETM count

http://www.youtube.com/watch?v=puk3bbS86NE

Scramble: L2 R2 U2 L2 U F2 L2 F2 U L B2 F' R B2 U' B F L F2 D'

Cross: L2 u R' F' (4 ETM, obvious)

F2L 1: L' U' L d R' U R (7 ETM, obvious)

F2L 2: y' R U' R' U2 R U' R' (8 ETM, rotation requires regrip so it is counted)

F2L 3+4: y L U2 L' R U R' (7 ETM, rotation counts, L' R counts as two moves not one, because it is performed as such)

OLL: M U' M' U' U' M U' M' (8 ETM, Slice moves are performed as slices here. The U2 is two flicks with the left index, thus two moves.

PLL: R' U2 R U' U' R' F R U R' U' R' F' R2 U' (15 ETM, the first U2 is an index-middle double flick with the right hand, thus one move. The second is two left indexes again, thus two. R2 is a single wrist turn, thus one.)

Total solve: 56 QTM/49 HTM/44 STM/49 ETM

### ATM

The **axial turn metric** (ATM) is a metric for the 3x3x3 where any movement within the same axis counts as a single move. This includes quarter turns, half turns, slice turns, and antislice turns.

### Snyder Metric

The **Snyder Metric** is a move count metric in which every parallel simultaneous movement of the puzzle is counted as one turn, regardless of the puzzle shape or the complexity of the turn. It is more or less equivalent to the Axial Turn Metric, where a movement of any layers (on the same axis) in any direction(s) counts as one turn. Snyder Notation may be used to represent turns using this metric.

The Snyder Metric was invented by Anthony Snyder in 1983. He argues for it as follows: "I have never understood why the turn counting rules/standards follow a 'range of motion metric' rather than an 'efficiency metric'. Solving for fewest turns is a challenge in efficiency to start with, so the metric should also be based on efficiency. In my opinion the most sensible way to count turns is to figure that any parallel simultaneous movement is one turn. This would also make the rules far simpler. Another point is that there are many ways to fine-tune solves by adding more anti-slices. Examples: I far prefer solving the U-Twist (headlights) with R L U2 R' U' R U' R' L' U2 L U L' U, which works out to just 12 very easy to perform turns once you define the anti-slice into the metric (using Snyder Notation the same algorithm: R+o' U2 R' U' R U' R'o+ U2 L U L' U). This requires only 12 parallel simultaneous movements, which is in my opinion more efficient than the 13 turn F U' R2 U R2 U F U' F2 D R2 D' R2. Another example is the H-PLL, which can take just 6 turns using the Snyder Metric."

#### Examples

The following algorithm for a corner 2-twist uses only 12 turns in this metric, fewer than most popular algorithms. In normal and Snyder Notation:

- R L U2 R' U' R U' R' L U2 L U L' U (14 turns HTM)
- R+o' U2 R' U' R U' R'o+ U2 L U L' U (12 turns Snyder Metric)

And the H perm can be solved in 6 turns in the Snyder Metric:

- R L U2 R' L' F' B' U2 F B (10 turns HTM)
- R+o' U2 R'o+ F'o+ U2 F+o' (6 turns Snyder Metric)

### PTM

NOTE* This article discusses historical events for which no primary sources exist.

In 2017, the **Pacelli Turn Metric** (PTM) was a metric for the 3x3x3 that was very similar to ATM, except with rotations and wide moves being discounted from the movecount, to encourage old-style cube turning.

On December 8, 2017, User:Martinss overruled the metric, pursuant direct orders from the divine ankh of Pharaoh Sekhemib-Perenma´at, peace be upon him.

### OBTM

**OBTM**, short for **O**uter **B**lock **T**urn **M**etric, is the official turn metric system of the World Cube Association.

OBTM defines 1 move as any non-slice move done on any twisty puzzle. Outer layer moves and outer block moves (outer layer plus adjacent inner layers) turned once or more is considered as 1 turn. Rotations are considered as 0 moves.

### BTM

The **block turn metric** (BTM), is a metric for bigcubes where any group of contiguous slices moving the same way is counted as one move.

## Comparison

Move sequence | Description | HTM | QTM | STM | QSTM | ETM | ATM | PTM | OBTM | BTM |
---|---|---|---|---|---|---|---|---|---|---|

R | One quarter move | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

R2 | One double turn | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 |

M | One quarter slice move | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 |

M2 | One double slice move | 2 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 1 |

(U D') | Two simultaneously executed (quarter) turns | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 |

(U' D') | Two simultaneously executed (quarter) turns/An anti-slice move | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 |

x2 | A rotation | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

(x y) | Two simultaneously executed rotations | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

Fw' | One wide (quarter) move | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |

2R | Slice move on a big cube | - | 2 | 1 | ||||||

Rw | Outer move on a big cube | - | 1 | 1 |

## Uses

Metrics are used to measure the length of move sequences such as Algorithms or Reconstructions. God's number is known to be 20 in HTM and 26 in QTM. From a mathematical standpoint, however, QTM seems a more natural choice; half-turns are visibly redundant as generators.

## Debate over metric in FMC

The WCA uses HTM in FMC. There has been a number of debates arguing for different metrics.