Swarmbots are invading! They work by building copies of themselves. Swarmbots come in two genders: drones and producers. In one minute, a drone can assemble a single producer. However, a producer, given the same amount of time, will assemble two Swarmbots: a drone and a producer.
(1) The swarm begins with one drone... assuming that all swarmbots work at their optimal speeds and work continuously, at what rate do they grow? In each generation, how would you define the total number of swarmbots in that generation and the breakdown of how many drones and how many producers make up that generation? Why does it work this way?
(2a) After x generations, how would you define the TOTAL number of swarmbots in ALL generations, assuming they receive no casualties?
(2b) Now a swarmbot is guaranteed to die right after seeing its "great-grandchildren" get assembled. After x generations, how would you define the TOTAL number of swarmbots?
(3) Prove (or disprove) that, given that no casualties or "swarmbot deaths" occur, the final two generations will always have a bigger total of swarmbots than ALL preceding generations
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unreality
Swarmbots are invading! They work by building copies of themselves. Swarmbots come in two genders: drones and producers. In one minute, a drone can assemble a single producer. However, a producer, given the same amount of time, will assemble two Swarmbots: a drone and a producer.
(1) The swarm begins with one drone... assuming that all swarmbots work at their optimal speeds and work continuously, at what rate do they grow? In each generation, how would you define the total number of swarmbots in that generation and the breakdown of how many drones and how many producers make up that generation? Why does it work this way?
(2a) After x generations, how would you define the TOTAL number of swarmbots in ALL generations, assuming they receive no casualties?
(2b) Now a swarmbot is guaranteed to die right after seeing its "great-grandchildren" get assembled. After x generations, how would you define the TOTAL number of swarmbots?
(3) Prove (or disprove) that, given that no casualties or "swarmbot deaths" occur, the final two generations will always have a bigger total of swarmbots than ALL preceding generations
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