A random problem I suddenly thought which I haven't think of a solution yet

[fallen slate]

An evil villain wants to destroy the world. In order to achieve his goal, he created an AI that can create another AI. The AI he created initially has a power of 10, meaning it has the power to devastate 10 cities before it self-destruct. Everytime the AI creates a new AI, it's power decreases by 10%, and the newly created AI is 2 times as powerful as it's creator. Once it's power reaches 0, it will be dead. Every newly created AI can also create another AI and the same rule holds. Assume that the total number of cities in the whole world is 10⁶. What is the minimum number of AI the evil villain must create in order to achieve his goal?

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[gaunt ember]

One is enough, as it can create several more powerful AI's

[stark prawn]

💀💀

[fallen slate]

yea you're right lol, just realized it turns out to be a trick question lmao

[stark prawn]

WHAT

oh yeah

because the AI they produce further reproduce

[fallen slate]

yea

[spice lagoon]

lets see
given -

• The initial AI has a power of 10.
• Every time an AI creates another AI, its power decreases by 10%, and the new AI is 2 times as powerful.

• cities - 10⁶

[stark prawn]

nah the best strategy the could use is
1AI produces 10AIs
10AIs produce 100AIs
....

100AIs get 1000AIs
1000AIs get 10000AIs
Each AI destroys 10 cities

10^6 cities destroyed starting from 1AI

[spice lagoon]

10×(0.9)^n ×(2^n ) ≥ 10^6 (n = number of AI's created)

[lyric siren]

an evil supervillain creates AIs starting with 5 fireballs and 10 power. at any moment, each AI has 3 options:
-spend a power to make a new fireball, if it has no fireballs
-once it has n fireballs, it can devastate n+1 towns then self destruct
-they also have the ability to imagine new fireballs into existence with the following procedure, which uses 1 power

1. they can only imagine versions of themselves that have fewer fireballs, due to heisenberg's uncertainty principle, it is impossible for an AI to imagine a version of itself with as many fireballs as itself.
2. they then imagine that AI devastating various towns.
3. however many towns the imaginary AI devastates, is how many fireballs are magically inside the AI at the end of the procedure

how many towns can one super AI devastate

[stark prawn]

trivial

[spice lagoon]

now solving
-> 0.9^n × 2^n ≥10^5

-> 1.8^n ≥ 10^5

(taking log on both sides )

now , -> n ≥ log(10^5)/log(1.8)
​aprox - 19.58

rounding off to 20
the creator would need to make 20 AI's to achieve his goal

[stark prawn]

no.

i proved it for when the AI doesnt even double its power

[spice lagoon]

where did i go wrong smh

[stark prawn]

since if the AI doesnt double its power, i proved the answer to be 1AI, and since 1 is the minimum you can go, that remains the same for when the AI doubles its power

[spice lagoon]

makes sense

[lyric siren]

1AI will produce an AI of strength 20, an AI of srength 18, of strength 16.2, ... which has a total of 200 i guess

[clear river]

The AI's power never reaches 0, only approaches it.

[fallen slate]

yea you're right

[analog atlas]

The number of cities devastated by each creation of AL decrease in a geometric sequence with a common ratio of 0.9.
To find the sum of a infinite geometric sequence, we can use the formula:
SUM= a/(1-r)
Where "a" is the value of the first AL power and "r" is the common ratio.
In this case the first AL power is 10 and the common ratio is 0.9. using the formula.
SUM= 10/(1-0.9)=100
Therefore the sum of series is 100 which means that the number of A L creations needed to destroy 10 million cities is 100.

I hope this answers you question.

[lyric siren]

Artificia Lintelligence

[fallen slate]

you didn't take into account the info that the newly created AI is 2 times as powerful as it's creator, if you do it that way then the newly created AI is 10% less powerful
also 100 is the number of cities destroyed by creating one AI, not the minimum number of AI which needs to be created in order to destroy all 10 million cities

[analog atlas]

OK.

[analog atlas]

Um I tried and found out the answer is 2.752424849×10^13AI . Here is how I did it.
To find the total number of cities to be destroyed, we can use the formula:
T(C)=2^n ×90 where T is the total number of the cities and n is the generation value.
10-(10% of 10)= 9
9 × 10= 90
In this case T(C) is a million.
10^6=2^n × 90
Applying logarithm and simplifying the equation,
n= log(11111.11111)/ log(2)
n= 13.43971547th generation.
Now to find the total number of AI created from the first generation up to this generation can be calculated using the formula:
T(AI)= 10^n where n is having its stated meaning
T(AI)= 10^13.43971547
T(AI)= 2.752424849× 10^13
Therefore the total number of AI created by the evil villain to destroy a million cities is 2.752424849×10^13.

[analog atlas]

This answer could be gotten from very long algebraic summation which is very complex.

[lyric siren]

is this chatgpt

[analog atlas]

What is chatgtp?

[fallen slate]

how did you get those formulas?

[lyric siren]

fr an english speaker hasnt heard of it

[analog atlas]

I figured it out

It was not easy.

[fallen slate]

what was the reasoning?

I still don't quite understand

[analog atlas]

Have I answered your question?

[lyric siren]

because there was no reasoning because it was ai generated

[fallen slate]

by logic the answer is actually just 1 AI needed cuz a single AI can create another AI which can also create another AI and so on and each of the newly created AI can destroy twice more cities then it's creator

[analog atlas]

I thought of that also but thought twice.

From that large value, you would get the number of AI to destroy the cities and the number of AI needed to create the AI which would destroy the cities. That was the idea I used and I think its correct.

[lyric siren]

?

what exactly is that large value

if you only need 1

[analog atlas]

2.752424849×10^13 AI

[lyric siren]

i already knew that

what does it mean

[analog atlas]

Yeah

What further explanation do you need?

[lyric siren]

i mean i would like the question to be answered

[analog atlas]

One AI can destroy only 10 cities.

[lyric siren]

it can duplicate and make more

[stark prawn]

No

You need 1

and heck

it just reproduces infinite AIs

because

it never really reaches 0 power

only approaches it

[analog atlas]

I said, one AI can destroy only 10 cities.

[lyric siren]

^

[analog atlas]

Why?

[stark prawn]

what if the first AI doesnt even destroy cities

it just reproduces

[analog atlas]

That is what I am talking about. They keep on reproducing until the reach the number of AI which can destroy a million cities.

[stark prawn]

yh

only one needed

cuz one AI can reproduce as long as it wants

[analog atlas]

So 2.752424849×10^13 AI should be right right?

[stark prawn]

no

why would there be more AIs than number of cities

[analog atlas]

OK you win.

[stark prawn]

it just needs to produce 10^5 AIs

[analog atlas]

But I would not accept yours.

[stark prawn]

what 😭

[analog atlas]

I also took that into consideration but later found out that it was wrong.

I divided the total number of cities by the number of cities 1 AI can destroy.

You can only use that idea if the power of the AI do not increase.

[analog atlas]

What do you mean?