Grandfather problem

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Grandfather problem
Grandfather problem image
Pattern type Miscellaneous
Number of cells 298
Bounding box 24×23
Discovered by mtve
Year of discovery 2016

The grandfather problem is the following question, posed by John Conway in 1972 in Lifeline Volume 6:

Is there a configuration which has a father but no grandfather?

Although a 50$ cash prize was offered by Conway, the problem remained open until May 2016, when mtve presented such a pattern.[1]

Generation

The pattern was found with Mark Owen's picosat, using an algorithm similar to the one employed by Nicolay Beluchenko's OEISicon light 11px.pngA196447:[2]

  1. Try setting the next cell on and off;
  2. Skip first level gardens of eden;
  3. Choose the minimum number of grandparents, where "minimum number" is not the exact number but a minimal distance between first and last.

The final configuration has a total of 17920 parents, but no grandparents.

Verification

mtve's discovery was confirmed to solve the grandfather problem by Matthias Merzenich with JLS, using the following steps[3]:

  1. Run a search to find all 1-generation predecessors of mtve's pattern. Copy the cells that had the same setting in all solutions to the pattern.
  2. Mark all of unset cells with "X", increase the period by 1 and shift the pattern to the future by 1 generation. Set the outer cells to "X" in generation 0 and run another search, which eventually gives "Search finished: 0 solutions found".

Optimization

It's possible to reduce the size of this pattern by removing some cells, and also some other patterns with the same property exist[4].

Next levels

"A father and grandfather, but no great-grandfather" pattern[5], and a "father, grandfather and great-grandfather, but no great-great-grandfather" pattern[6], have also been constructed by the same method.

No further levels have been found yet, but they almost certainly do exist.

References

  1. mtve (May 5, 2016). "Re: Thread for your unsure discoveries". ConwayLife.com forums. Retrieved on June 22, 2016.
  2. mtve (May 6, 2016). "Re: Thread for your unsure discoveries". ConwayLife.com forums. Retrieved on June 22, 2016.
  3. Matthias Merzenich (Sokwe) (May 6, 2016). "Re: Thread for your unsure discoveries". ConwayLife.com forums. Retrieved on June 22, 2016.
  4. mtve (June 2016). "Garden Of Eden 2G". Retrieved on July 8, 2016.
  5. mtve (June 2016). "Garden Of Eden 3G". Retrieved on July 8, 2016.
  6. mtve (December 12, 2016). "Garden Of Eden 4G". Retrieved on December 14, 2016.