[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

Edited November 8, 2008 by unreality

[dms172]

A=total bots

B=total drones

C=total producers

1.producers growth rate=add A each time

drones growth rate = add C each time

total growth rate = add 2C + B

that's the best way i can explain it

[unreality]

Yes but you can't recursively use your 'total growth rate' formula to define the next step, since it doesn't distinguish between C & B for the next round.

A very ancient and awesome sequence is involved here...

[unreality]

No other attemptees?

[bonanova]

Trying to make sure I'm understanding what's being asked.

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,

[a] at what rate do they grow? How many swarmbots are added each minute?

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? how many swarmbots, drones and producers are added?

[c] Why does it work this way?

(2a) After x generations, how would you define the TOTAL number of swarmbots in ALL generations what is the swarmbot population, 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? what is the swarmbot population?

(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 the swarmbot population more than doubles every two minutes?

[bonanova]

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.

Notation: S = swarmbot; D = drone; P = 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?

Every S produces at least one additional S every minute. [Ps produce 2 Ss, Ds produce 1 S].

So the growth rate is more than 100% per generation.

The growth rate is 100 x [s+P]/S %.

When the OP says generation, it sometimes means minutes and other times means either births [that minute] or population [after that minute].

If it means births, then each minute P Ds are born and [D+P] Ps are born: [D+2P] Ss are born.

If it means population, take the sum of these terms for {generation} minutes.

(2a) After x generations, how would you define the TOTAL number of swarmbots in ALL generations, assuming they receive no casualties?

As the sum {for 1 = 1 to X} of the above terms.

(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?

I'll look at this later.

(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

Since the growth rate exceeds 100%, the population more than doubles each minute.

That means more are born each minute than existed before that minute.

Perhaps the OP means to say prove [or not] both of the two most recent populations exceed the sum of the populations for all of the minutes that precede that last two.

A simple table shows this is true.

Consider the case where the population only doubles each minute.

That is, each minute the following number of Ss are born: 1 2 4 8 16 32 63 128 256[let's stop here]

The populations each of those minutes is the cumulative sum of these numbers: 1 3 7 15 31 63 127 255 511

The cumulative sums of these populations are 1 4 11 26 57 120 247 502 1013.

247 would be the sum of population of all the minutes that precede the last two.

It is exceeded by both 255 and 511 - the populations of the last two minutes.

It's clear this condition exists for all the minutes tabulated and will persist indefinitely.

These sums have closed forms, but given the uncertainty of the question, I'll leave it at that.

[unreality]

Nobody so far I'll rephrase the OP if you want:

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, how would you define their growth (ie, the new number of swarmbots based on previous generation(s))? In each generation, how would you define 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 (the total population of swarmbots), 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... ie, say there are 10 generations. Call the total population of the first 8 generations 'x', and the total population of the last two generations 'y'. Prove (or disprove) that y>x (independent of any specific generation number)

It has nothing to do with 2^x at all

[Guest]

So each generation would be of one minute correct?? as each minute goes by, a new generation comes to life.... am i right?

[unreality]

Yep, that's correct

[Guest]

Well this is really confusing, there's too much to keep track of and they multiply so freaking fast! hehe.

But I do know this much:

Ok, the numbers will represent the minutes. The numbers all the way to the left are like the generations, but are also the minutes. the other numbers represent only minutes.

So its kinda like, what happens to generation 2 on the 3rd minute, then 4th minute, etc. Then its the 3rd generation on the 4th minute, and so on.

(1 Drone)

1. P

2. D,P 3. D,P + D,P 4. D,P + D,P +D,P

3. P, D, P 4. P, D,P + P, D,P + P, D,P 5. P, D,P ( 6 of these in total)

4. D,P, P, D,P 5. D,P, P, D,P (4 sets of these) 6.D,P, P, D,P (10 of these in total)

5. P, D,P, D,P, P, D,P 6. it will have 5 total of the P, D,P, D,P, P, D,P (sets of kids)

But anyway, I think you get my drift.

And this is without counting the other 2 Producer kids the first drone created. (He will only be able to make 3 since by the 3rd minute he will have great grandchildren, therefore he'll die.

I know that by the 5th minute the the first D,P pair of kids that were born on the second minute will die. The next pair (born on the 3rd minute) will die on the 6th and so on for the last pair.

I also know that for the 3rd generation, they'll have 8 sets of the P, D,P kids by the 6th minute.

The 4th generation will have 10 sets of the D,P P, D,P kids by the 6th minute as well.

The 5th generation will have 5 sets of its own kids by the 6th minute and 15 in minute 7.

I don't know how much sense that makes, but since you must know the answer it should make sense to you unreality.

I haven't really found a pattern yet, but i would think im on the right track...

[unreality]

There is a deep underlying pattern to answer at least #1 & #3 and will probably help you a lot on 2a and 2b

[Guest]

Well, am I at least correct in my assumptions so far?

[Guest]

Man, I don't even know.... even if everything i have so far is correct, I still can't find any logical pattern. It must be really deep. I give up lol. This one is too hard.

I can figure out how many bots there will be, but I don't know any pattern.

[unreality]

I'm not sure... your diagram is pretty confusing. Here's a hint:

first generation - one drone

second generation - one producer

third generation - one drone, one producer

etc

the total number in each generation is 1,1,2,3,5,8,13...

that should be a major help

Edited November 9, 2008 by unreality

[Guest]

I'm not sure... your diagram is pretty confusing. Here's a hint:

first generation - one drone

second generation - one producer

third generation - one drone, one producer

etc

the total number in each generation is 1,1,2,3,5,8,13...

that should be a major help

I don't think that works out right...shouldn't the third generation be one drone and two producers? The second generation has a drone and a producer, which would make a producer, and a drone and a producer, respectively. That means that the Fibonacci sequence doesn't follow the same pattern.

[Guest]

ok so i think i got it... the pattern at least.

you see, i misunderstood the way they reproduced.

I thot that the drone would keep making more and more producers every minute and so each generations would actually keep growing and growing. I was thinking waaay too much.

Ur saying they only reproduce once, right?

If so, the pattern is the following:

For every new generation, you add the previous one and you'll have the new one. for example, that first drone made one producer, that producer made 2 drones (so it was 1+1=2). Those two made three (that was 2+1=3). The next became 3+2=5 and so on. THATS the pattern in that case.

now i gotta go back and read the q's again hehe

[Guest]

I don't think that works out right...shouldn't the third generation be one drone and two producers? The second generation has a drone and a producer, which would make a producer, and a drone and a producer, respectively. That means that the Fibonacci sequence doesn't follow the same pattern.

yeah, i thot that every single drone would keep making more and more every minute (thats how I understood it with the wording). Which is y had said earlier that they reproduce so "freaking fast" but it wasnt like that at all apparently.

[Guest]

So I think I may have the specific answers to the questions.

1. Their growth would be defined by adding the two latest generations together, resulting the new generation. The breakdown of how many drones/producers are in each generation is the same number as the latest 2 generations, producers having the largest number because Drones only make producers, and producers make both. So the producers are being made twice as fast.

2. This question I don't really understand because for one, you say "assume no casualties", but then u say "they're guaranteed to die after seeing their great grandchildren." So, what's the deal here? But to answer that question as literal as I hear it I would say: you just add every single generation to have the TOTAL number of swarmbots.

2b. There will always be only 3 generations present (alive) because as soon as the 4th one is born, the first one dies. So basically the total number of swarmbots alive will be the last 3 generations.

4. The addition of the first 8 generations (as stated in your own example) would come very close to the population of the 10th generation, but adding the 9th to it would always clear any doubt that previous generations can be larger than the most recent two. Y will always be greater than X.

So, how did i do?

[unreality]

oh yeah, duh, I forgot the most important part: after they assemble once (ie, make either 1 producer [if they're a drone] or 1 of each [if they're a producer]) then for the rest of their minutes they do other swarmbot stuff ;D (If they just reproduced constantly, nothing would get done )

Sorry for omitting that crucial part

1) correct in that the total population of each generation follows the Fibonacci sequence; ie, the previous two added together. But you're wrong about the Producers being twice as abundant in each generation... in fact the breakdown of 'producers' and 'drones' within the generation itself is just as interesting as the pattern that defines the TOTAL in the generation

2) two is two DIFFERENT problems. For 2a, you assume no casualties and find the TOTAL population of ALL generations... ie, you make a formula to define it based on x (x = number of generations). For 2b, you make another, different, formula for x generations after assuming that they die after whatever time it was. Your answers for 2 are very vague... what if I want to know the total after 100 generations... you couldn't tell me without counting from the beginning, could you?

3) well you made an assertion but you didn't prove it. The purpose is to PROVE it ;D

great job so far, guys

Edited November 10, 2008 by unreality

[unreality]

Answer time

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?

It follows the Fibonacci pattern. The generation numbers are like this:

1,1,2,3,5,8,13,21,34,55,89,etc

furthermore, the number of Producers & Drones in each generation is a breakdown of the previous two total populations. For example, in the 9th generation, which has 34 swarmbots, the breakdown is: 21 producers, 13 drones. The previous two numbers in the series, AND the previous TOTAL populations

I'll leave you to figure out why it works this way

(2a) After x generations, how would you define the TOTAL number of swarmbots in ALL generations, assuming they receive no casualties?

The Second Identity of the Fibonacci sequence states:

The sum of the first n Fibonacci numbers is the (n+2)nd Fibonacci number minus 1.

for example, the 7th generation has 13 swarmbots. So the total of all the swarmbots in the 1st to the 7th is 1+1+2+3+5+8+13 = 33, which is 34-1. 34 is the Fibonacci number two higher than 13

(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?

in this case, each swarmbot is only alive for three generations, thus each minute only has 3 generations: the current generation, the previous generation, and the generation before that. Because the two previous generations add to the number of the current generation, just double the number in the current generation

(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

You could prove it like this:

the first 'n' Fibonacci numbers sum to the (n+2)th Fibonacci number minus one (the second idendity used earlier. You can prove this by adding Fibonacci numbers, adding this to the next one, etc, it's a simple proof), so say that n+2 is the number of generations alive right now. Thus we are trying to prove that the sum of n generations is greater than the sum of the (n+1)th generation and (n+2)th generation added together.

But the first 'n' generations add to the (n+2)th generation minus 1, so the (n+1)th generation added on just has to be 2 or more (which it will be) to make the proof true. Ie:

sum of first n generations = F(1)+F(2)+..+F(n) = F(n+2)-1

thus F(n+2) + F(n+1) [as long as F(n+1) is greater than 1] will always be greater than F(1)+F(2)+..+F(n)

[unreality]

Did you like this one? I'm thinking of making a similar one, but better & harder & more clarified in what it's asking for