| 5. (a) Both CA give a live cell for DLL and LLD, so a live pair surrounded by dead cells will be alive in the next generation. |
| The only other configurations giving a live cell are LLL and DLD, both requiring the central cell already is alive. So there is no possibility of patterns evolving from other live cells colliding with the live pair. |
| (b) Crossing over at the middle configuration produces two children: |
| one with configurations LLD and DLL giving a live cell, |
| the other with LLL, LLD, DLL, and DLD giving a live cell. |
| The first is just the CA of problem 4, so it certainly has this property. |
| The second has this property for the reason given in part (a): both LLD and DLL give live cells, so a pair of live cells surrounded by dead cells survives in the next generation. The other configurations giving a live cell are LLL and DLD, both of which must have the central cell alive already. So no collisions are possible. |
| (c) All crossovers will satisfy this property. |
| First, both parents have LLD and DLL, so all crossovers give a live cell from both these configurations. Consequently, a live pair surrounded by dead cells will persist for all crossovers. |
| Second, the only other configurations giving a live cell are LLL and DLD, both of which require the central cell already alive. Regardless of how these are distributed among the children, dead cells never become alive and so all successive live cells must lie in vertical lines under the original live cells. Of course, not every initially live cell will stay alive. |
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