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# Rubik’s Dice Superflips and God’s Quantity » Cleve’s Nook: Cleve Moler on Arithmetic and Computing

• How exhausting is it to unravel a Rubik’s Dice?
• When are you able to say that your Rubik’s Dice is totally scrambled?
• Why would possibly the reply rely upon the place you went to highschool?
• What attention-grabbing mathematical questions are concerned?

### Contents

#### Metrics

The issue of fixing any configuration of a Rubik’s Dice is the smallest variety of strikes required to return to the intial configuration the place every face is exhibiting a single shade.

However what, precisely, is a transfer? The so-called quarter-turn metric says a transfer is popping any face by 90 levels. The half-turn metric says turning any face by both 90 or 180 levels is a single transfer. For instance, utilizing Singmaster notation and the quarter-turn metric, the sequence “L L”, which turns the left face twice in the identical path, is 2 strikes. However within the half-move metric, the sequence turns into “L2” and counts as a single transfer.

Tomas Rokicki, who describes himself as a programmer from Palo Alto, offers some historical past at cube20.org.

Within the early days of dice arithmetic, two camps emerged on find out how to
measure the problem of a place. West coast and Stanford
mathematicians, free thinkers all, tended to choose the half-turn
metric, the place any twist of any face, whether or not 90 levels, 180 levels,
or 270 levels counted as a single transfer. The east coast crowd,
together with MIT, tended to choose the rigor of the quarter-turn metric,
the place a half-turn counted as two strikes, since in fact it may very well be
achieved by two consecutive quarter turns.

After I started improvement of a Rubik’s Dice simulator, Qube, I used to be unaware of this historical past and, despite the fact that I’m a religious West-coaster, I simply counted quarter-turns. Now a toggle swap in Qube permits use of both metric.

#### God’s quantity

Let Q denote a dice place,

| Q | = minimal variety of strikes to unravel Q,

Q = the set of all doable Q‘s, and

G(Q) = most over Q in Q of | Q |.

G(Q) is named “God’s quantity”. Q accommodates over 4.3*10^19 positions, so computing G(Q) is a formidable optimization downside. The definition of God’s quantity doesn’t require the optimum resolution itself, solely the variety of strikes.

G(Q) = 20

#### Q20

The second animation exhibits an answer of Q20 in 20 strikes obtained by reversing and complementing the producing strikes. Reid’s proof exhibits that every other resolution requires a minimum of 20 strikes. The excessive decision clip is: Q20solve.mp4

There are a number of different configurations that require 20 strikes. Any configuration with G(Q) = 20 will be thought to be fully shuffled within the half-turn metric.

#### Q26

In 2014, an enormous computation on the Ohio Supercomputer Heart by Rokicki and Morley Davidson proved that solely Q26 (and its two rotations) required 26 quarter-turn strikes All different configurations want fewer. cube20.org. So, this established that God’s quantity for the half-turn metric is

G(Q) = 26

The excessive decision clip is: Q26solve.mp4

#### Examine

Let’s examine Q20 and Q26 by alternating between the 2.

The kind 3 nook items are the identical in Q20 and Q26, and are within the right preliminary place.

The kind 2 edge items are additionally the identical in Q20 and Q26, however are reversed from their preliminary place.

All of the motion is with cubelets of kind 0 and sort 1. In an actual, bodily Rubik’s Dice, that is one stable piece that holds the Dice collectively.

Revealed with MATLAB® R2022b

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