A Demigod's Number for the Rubik's Cube
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910768851582976 |
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| author | Merino, Arturo Subercaseaux, Bernardo |
| author_facet | Merino, Arturo Subercaseaux, Bernardo |
| contents | It is well-known by now that any state of the $3\times 3 \times 3$ Rubik's Cube can be solved in at most 20 moves, a result often referred to as "God's Number". However, this result took Rokicki et al. around 35 CPU years to prove and is therefore very challenging to reproduce.
We provide a novel approach to obtain a worse bound of 36 moves with high confidence, but that offers two main advantages: (i) it is easy to understand, reproduce, and verify, and (ii) our main idea generalizes to bounding the diameter of other vertex-transitive graphs by at most twice its true value, hence the name "demigod number".
Our approach is based on the fact that, for vertex-transitive graphs, the average distance between vertices is at most half the diameter, and by sampling uniformly random states and using a modern solver to obtain upper bounds on their distance, a standard concentration bound allows us to confidently state that the average distance is around $18.32 \pm 0.1$, from where the diameter is at most $36$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_00144 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Demigod's Number for the Rubik's Cube Merino, Arturo Subercaseaux, Bernardo Combinatorics Discrete Mathematics It is well-known by now that any state of the $3\times 3 \times 3$ Rubik's Cube can be solved in at most 20 moves, a result often referred to as "God's Number". However, this result took Rokicki et al. around 35 CPU years to prove and is therefore very challenging to reproduce. We provide a novel approach to obtain a worse bound of 36 moves with high confidence, but that offers two main advantages: (i) it is easy to understand, reproduce, and verify, and (ii) our main idea generalizes to bounding the diameter of other vertex-transitive graphs by at most twice its true value, hence the name "demigod number". Our approach is based on the fact that, for vertex-transitive graphs, the average distance between vertices is at most half the diameter, and by sampling uniformly random states and using a modern solver to obtain upper bounds on their distance, a standard concentration bound allows us to confidently state that the average distance is around $18.32 \pm 0.1$, from where the diameter is at most $36$. |
| title | A Demigod's Number for the Rubik's Cube |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2501.00144 |