i’m developing an incremental game and I need help with something
Given a non negative number N and a base(>1 but can be noninteger) b
how would I find what the largest number in the form b^0+b^1+b^2+b^3…
that’s not greater than N
#math problem for game dev
languid fossil
cursive stratusBOT
i’m developing an incremental game and I need help with something
Given a non...
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languid fossil
for example if N is 17 and B is 2
then the largest number would be 2^0+2^1+2^2+2^3
sturdy holly
Let (S = 1 + b + \ldots + b^k).
Then
\begin{align*}
b S &= b + \ldots + b^k + b^{k + 1} \
S - 1 + b^{k + 1} &= b^1 + \ldots + b^k + b^{k + 1}
\end{align*}
so (b S = S - 1 + b^{k + 1} ). Solving for (S) yields
[S = \frac{b^{k + 1} - 1}{b - 1} .]
So we want the largest (k) such that
[\frac{b^{k + 1} - 1}{b - 1} \leq N.]
languid fossil
yep decimals also work for b>=2
vocal bluffBOT
invariance.
sturdy holly
given (N) and (b), you can solve (\frac{b^{k + 1} - 1}{b - 1} = N) (using a logrithm at the end)
languid fossil
lol i actually got that formula too (the end formula)
vocal bluffBOT
invariance.
languid fossil
maybe I was mistaken that it didn’t work for b<2
sturdy holly
then you can floor to get your result
sturdy holly
maybe I was mistaken that it didn’t work for b<2
yeah it should work fine
languid fossil
hm
k
i actually only know algebra 2 so some of this stuff is beyond me but surprisingly my knowledge on different number bases helped me in this case
cause numbers are essentially the same idea
sturdy holly
you want the biggest number of the form 111...111 in base b, you mean?
languid fossil
no no
sturdy holly
wdym then?
languid fossil
like for b = 3 and n = 26 i would want
3^0+3^1+3^2
something i don’t get is
(1.1^4-1)/(0.1) is not equal too 1.1^0+1.1^1+1.1^2+1.1^3?
shouldn’t it be equal
sturdy holly
,w 1.1^0+1.1^1+1.1^2+1.1^3
vocal bluffBOT
sturdy holly
,w (1.1^4-1)/(0.1)
vocal bluffBOT
sturdy holly
they are equal
(idk why you're saying they're not, but remember that === is very unreliable with floating point numbers)
languid fossil
oh i’m stupid
i forgot to add 1.1^3
anyways, why not I might use it, how would this problem be solved for 0<b<1
would it work the same?
sturdy holly
anyways, why not I might use it, how would this problem be solved for 0<b<1
the infinite series converges
vocal bluffBOT
invariance.
languid fossil
yea what about different fractions?
sturdy holly
so the series sums to 2
sturdy holly
yea what about different fractions?
it converges if and only if |b| < 1
languid fossil
i see
let me try to get some code working for it once done I’ll close this
btw let’s say it starts at b^2+b^3…
it would be the equivalent to the answer of the same problem starting at b^0
but the answer is divided by b^2
right?
sturdy holly
(b^2 + b^3 + \ldots + b^k = b^2 (1 + \ldots + b^{k - 2}))
languid fossil
yea
vocal bluffBOT
invariance.
languid fossil
btw the whole idea of this is that a price of an upgrade in my game scales multiplicatively but the game will eventually reach a point where that upgrade must be bought really fast
and this just calculates that in bulk
sturdy holly
I had guessed that
it's neat that you're making games and finding an application for this stuff at the middle-school level
languid fossil
thanks!
honestly it’s nice that I covered logarithms in school
since i have been finding lots of uses
ok so this works but for technical reasons I also need to find k and return that too
how would I do that
noooo it doesn’t work ):
yea honestly not sure how to approach this
