help-0

[mortal siren]

final amount=P

cant remember

[alpine sable]

Uhh it’s not that

sorry if my previous comment was rude

[mortal siren]

nah its fine

lol

i got this as my answer

[alpine sable]

The formula is

Where p is principal amount or 5000

1 is 1

R is interest rate

N is number of periods

T is time I think

[mortal siren]

thank you thank you

[alpine sable]

Wait

I might have given u wrong formula

[mortal siren]

oh

[alpine sable]

Been 5 years since I did this stuff XD

Lemme get the actual one

[mortal siren]

lol i get it

ty

[alpine sable]

Oh yeh it’s right formula

n is basically 12 in your question

T is 4

12 for 12 months since it’s compounded monthly

[mortal siren]

ah i see

i got 224207754.3

i just need to put it in the nearest £

[alpine sable]

There is is something wrong

I can feel it

[mortal siren]

okay

[alpine sable]

Wait lemme do dis

[mortal siren]

ty

[alpine sable]

Nvm it’s right

Lol

[mortal siren]

ah k could u help me put it in the nearest £ pls

[alpine sable]

I forgot how huge the compounding rate is and how big is that principal

[mortal siren]

yh ikr

[alpine sable]

That is pounds

[mortal siren]

it says round to nearest pound?

[alpine sable]

Oh

224207754

That’s ur answer

[mortal siren]

ty

just need help on this one

[alpine sable]

BRB

[lone heart]

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[copper tangle]

@near oriole Only occupy one channel pls

#❓how-to-get-help

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[pine root]

Any short method to factorize the numerator

[pine root]

Ik how to do factorizeing normaly but still if there is a short method it will help me a lot

I need for my competitive exams currently it's just practice

[echo socket]

Here I immediately noticed that x^2 - x - 6 can be factoried as (x + 2)(x - 3), using vietta's formulas

Vietta's formulas state that product of roots, in this case, will be -6 and sum of them will be +1

So I tried to guess pair of numbers for which product is -6 and sum is +1

And I came up with -2 with 3

[pine root]

K

Then u just square those

?

[echo socket]

Yes and (x + 2)^2 cancels out

And you'll be left with (x - 3)^2

[pine root]

K

Thnx

👍

[echo socket]

np

[pine root]

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[frank plover]

Since differentiability implies continuity, does that mean that if a function isn't differentiable at some point a, that the function isn't continues there?

[frank plover]

So i guess im asking wether the implication is "causal"

[covert agate]

yes

a implies b means (not b) implies (not a)

like sunny day implies school = no school implies not sunny

very useful logic thing

[oak chasm]

That's the opposite of what they have.

[frank plover]

Im confused lol

[oak chasm]

No, a function can be nondifferentiable at a point for more reasons than noncontinuity.

[frank plover]

But if its non differentiable does that mean its non continious?

[oak chasm]

No, a function can be nondifferentiable at a point for more reasons than noncontinuity.

[frank plover]

oke thanks

[frank plover]

Are you sure about this btw?

[covert agate]

absolutely

[oak chasm]

For example, |x| is nondifferentiable at x = 0, but it's continuous there.

[covert agate]

wait oh i thought you mean does non-continuity imply non-differentiability

my bad

pretend i said nothing

[oak chasm]

Here are some more examples: https://www.analyzemath.com/calculus/continuity/non_differentiable.html.

[frank plover]

Since if its not sunny but there is school then the implication still holds. But the negation of the elements, if its sunny but there is no school, does not hold

so (a implies b) does not imply (not a implies not b)

[frank plover]

so this isnt true right? (in general)

[covert agate]

this is

it’s absolutely true

[oak chasm]

When you not both of them and switch places, it's still true.

[covert agate]

i simply read your first question wrong

[frank plover]

OOHHH fuck i read your question wrong too

or well your explenation

[oak chasm]

If a → b is true, then
¬a → ¬b is not necessarily true
¬b → ¬a is necessarily true

[frank plover]

you said not b implies not a, i thought u ment not a implies not b

Anyway, thanks for the help both of you!

.close

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[lean dawn]

dont roast me im bad at math

[echo socket]

Here two functions would be the same if they have same pairs of x and y values

[lean dawn]

can someone show me how to solve

[echo socket]

I think all the functions are different here

[lean dawn]

yea

[gray isle]

this is a poorly structured question

[lean dawn]

ik right

omg i knew i was not the only one who thought that

sooo?

[gray isle]

was there anything else in the question above that

[lean dawn]

nope ):

Select all the correct answers.
Which two tables represent the same function?

thats all it say

[bleak ridge]

Maybe they want you to assume theyre linear?

Idk though

[gray isle]

seems like that's what they want

[lean dawn]

has to be that

im really behind in math so i need help

[covert agate]

yea

[lean dawn]

help pls

with the pic

i send

in the top

i need to solve dont know how

[covert agate]

bruh what the hell is this question

[lean dawn]

why does my fricking scores be so high on test

im not really that good at math

[lean dawn]

idk

[covert agate]

this question makes no sense

functions need not be ‘neat’

are there other requirements?

such as function being linear

if yes there’s a chance

[lean dawn]

this is the assinment name

[covert agate]

if no data insufficient

[covert agate]

aight you’re 14 right

[lean dawn]

yea

[covert agate]

i assume linear functions

then just calculate the slopes

i assume you know that

[lean dawn]

oof ):

lol told u

[covert agate]

you do know slopes right..?

[lean dawn]

my scores be high but im not good in math oof

[lean dawn]

yea

[covert agate]

yea just calculate it

seems like this is the question

a very poorly specified one

[lean dawn]

that would take so long

omg

[covert agate]

nah

[lean dawn]

yes

[covert agate]

best case scenario in 2 tries

[lean dawn]

lol

what u mean?

[covert agate]

idk

how much time you take literally depends on luck

gl lol

[lean dawn]

bruh i have to make a chart to do that

off

ooof

that gonna take 10mins

at least to make the chart

oof

[covert agate]

no

why do you need the chart??

[lean dawn]

bc slope r rise over run

[covert agate]

is it in the question requirements

[covert agate]

then just calculate it

no graphing needed

[lean dawn]

so with the numbers i have to rise then run

lol

[lean dawn]

how?

[covert agate]

each function is like 10 seconds

[lean dawn]

WHAT

[covert agate]

given two points find the slope

you don’t need a graph

[lean dawn]

bro that took me like 20mins yesterday

with no help

[covert agate]

is that some dumbass requirement from teacher

to make you chart every question

[lean dawn]

yes

[covert agate]

jesus fuck

[lean dawn]

lol

[covert agate]

do you have printer or something

[lean dawn]

its takes so long

[covert agate]

or if you’re on google classroom

[lean dawn]

yes

not in my room

[covert agate]

just screencap desmos

[lean dawn]

so i cant use

[covert agate]

just use desmos lol

saves you forever

[lean dawn]

blocked

[covert agate]

desmos is blocked?????

[lean dawn]

my sister messages up something on my chrome book so everything is blocked but 2 things

[covert agate]

how do you use discord if desmos is blocked

[lean dawn]

idk

[covert agate]

use phone then..?

or geogebra

[lean dawn]

dont have rn

[covert agate]

any graphing software

[lean dawn]

nope

):

oof

[covert agate]

huh

no graphing websites can be accessed?

[lean dawn]

no ):

[covert agate]

wtf

[lean dawn]

can u show me how to calculate it

[covert agate]

slope between two points is easy

it’s algebra 1

[lean dawn]

ik ):

[covert agate]

if you don’t know that how do you do calculus??

[lean dawn]

my friend

my scores was high so they give me calc

):

whitch made it more hard

[covert agate]

ok you can’t learn calculus lol

you should deal with basics first

[lean dawn]

my friend was surprised I am

[covert agate]

which it seems like you don’t know

[lean dawn]

he was like no way

[covert agate]

that you’re what

[lean dawn]

in calc

lol

he said no way

[covert agate]

sin a + sin b

[lean dawn]

they i showed him

lol

[covert agate]

sum to product, go

[lean dawn]

?

[covert agate]

you don’t know trig formulae for sum to product?

[lean dawn]

wth

[lean dawn]

what the fuck is that

[covert agate]

ok you’re not equipped enough to deal with calc

sorry to say

[lean dawn]

OOF

[covert agate]

this is literally the basis of proving ‘the derivative of sin x’

[lean dawn]

ok i will just got to next question

[covert agate]

which, conceptually, derivative ITSELF is slope

i assume you don’t know it well enough lol

[lean dawn]

this one

😛

[last island]

F

[covert agate]

point slope form

find slope, kick ass

[last island]

(F)ind slope, kick ass

[covert agate]

bruh

[lean dawn]

im dumb

in only math

[median dirge]

Oh

Hi.

[lean dawn]

LOL

[covert agate]

huh

[lean dawn]

im a tard

[median dirge]

Well, two guys are helping out already.

[covert agate]

what did i miss

alright too much

couldve gotten you banned

[lean dawn]

?

wjat

[covert agate]

you know it

[lean dawn]

know what

[covert agate]

nth

[lean dawn]

dm me it

[covert agate]

anyway, apply point slope form

[lean dawn]

.close

[lone heart]

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[karmic hare]

I found that the answer was option a. But I found it by substituting x=0 which was not the correct way. How to solve this using actual algebra?

[lone heart]

@karmic hare Has your question been resolved?

[chrome salmon]

Tou need help?

[karmic hare]

yes bro

[chrome salmon]

What does bro mean?

[karmic hare]

brother

[chrome salmon]

Oh then in that case call me sis

Why don't u know the answer

[karmic hare]

I don't know how to find the square root since it was complex, so I substituted x=0 in the question and the options and found that option a matched with the question.

[chrome salmon]

You can try squaring the options

You have to check when a²+b²=1+x²

Like for first option
a²=1+x+x² , b²=1-x-x²
So a²+b²=2/2 = 1

And 2ab isn't free of square root

So first option is wrong

[karmic hare]

But when you square the options and check, option a yields the question

[chrome salmon]

Oh yes I made a sign mistake there

[karmic hare]

So I guess the fastest method would be to try squaring the options to see which one would yield the question, right?

[chrome salmon]

Yeah I guess so

I can't see any other fast method here

[karmic hare]

Thank you very much for your help sis!

.close

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[covert agate]

laws of indices

fourth root is just 1/4-th power

apparently 3⁴ = 81

the other i let you figure out

no lol

do you know laws of indices?

$(ab)^n = a^n \cdot b^n,$ if $a^n$ and $b^n$ are well defined

[ocean seal]

Chromium

[covert agate]

in this case, n = 1/4

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[sweet vessel]

how do I find a reasonable initial guess to the newton method for an nth order equation that might have complex roots

[pale kestrel]

sketch?

[sweet vessel]

without graphs

I wanna do this programmatically btw

well one thing I've thought of is finding the roots of the highest order term using De Moivre's Theorem and using first one or two iterations of the secant method

is there a better way to do what I want?

or is that reasonable enough

I don't wanna face the issue where newton's method never converges

[pale kestrel]

sorry i dont have any immediate ideas

[sweet vessel]

I don't wanna keep this channel occupied, should I take this to #math-discussion?

[pale kestrel]

you could 👌 or

#computing-software maybe?

[sweet vessel]

alright

.close

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[alpine sable]

I’m not sure how to use the integer clue

[echo socket]

So let's say we have consecutive even numbers 2n, 2n + 2 and 2n + 4

[alpine sable]

Ohh

I took n, n+2 and n+4

[echo socket]

That'd be correct as well if n is even

[alpine sable]

Well n ended up not being even

I used this

[echo socket]

So the problem states that if we subtract 2n + 4 from 4*(2n), the result will be 6 more than 2(2n + 2)

Ye

Btw it should be the other way

8n - (2n + 8) = 2(2n + 2) + 6

We can start by dividing both sides by 2

3n + 4 = 2n + 2 + 3

n = 1

So the consecutive even integers were 2, 4 and 6

[alpine sable]

Oh let me read it again and try to solve it

[alpine sable]

Shouldn’t it be “8n-(2n+4)=2(2n+2)+6”?

[echo socket]

Oh right mb

[alpine sable]

So n=3?

And the consecutive even integers are 6 , 8 , 10

[echo socket]

8n - (2n + 4) = 2(2n + 2) + 6
6n - 4 = 2(2n + 2) + 6
3n - 2 = (2n + 2) + 3
n = 7

[alpine sable]

Yes thank you

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[fair osprey]

how would I do this?

cirlces are confusing

[covert agate]

that’s 2 semi circles and 1 smaller quarter circle

[lone heart]

@fair osprey Has your question been resolved?

[fair osprey]

yeah but how would I find the area and perimiter

[covert agate]

do you know circumference of a circle?

area of a circle?

[fair osprey]

like how to find them? no

[covert agate]

ok

if a circle has radius r

its area is πr² and its circumference (perimeter) is 2πr

not hard to memorise

(its proof requires advanced shit)

[fair osprey]

how do I multiply 2 by pi if pi is never ending

[covert agate]

π is a number

1/3 is never ending, you can do 3 • 1/3 right?

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[covert agate]

so are most fractions one can think of

.reopen

[lone heart]

✅

[fair osprey]

yeah

so show pi as a fractoin???

[covert agate]

so can you do 2 • π?

[fair osprey]

how do you do that

[covert agate]

π is not a fraction

it’s been mathematically proven to not be a fraction

non fractions are termed ‘irrational’

not everything has to be a fraction

[fair osprey]

wait I'm now confused. so how would this work?

[covert agate]

do you accept that some numbers aren’t fractions and never will be?

[fair osprey]

yeah ig?

[covert agate]

but they’re numbers

ie they have value

we can work with them

and they exist

so π exists, so does 2 • π, π • π (which π²), and other things involving π

[fair osprey]

this is confusing but is it kinda like a varaible of sorts then?

[polar barn]

@fair osprey

π can not be written down as a fraction

if you want to represent 2 multiply by π

write it as 2π

circle area is πr^2

ok good luck

[fair osprey]

oh you can do that?

[polar barn]

yes

[fair osprey]

so 2π2/2 for the bigger semi circle?

[polar barn]

you mean π2^2/2?

[fair osprey]

yeah that

[polar barn]

yes

[fair osprey]

so π4/2?

would it just be 2π

[polar barn]

yes

[fair osprey]

then the saller one is 1π^2/2 so just π??

[polar barn]

π1^2/2 *

and thats π/2

[fair osprey]

so the end could be 2π and π/2?

it says thats wrong

[polar barn]

2π+π/2=?

[fair osprey]

3π/2?

[polar barn]

5π/2

[fair osprey]

how do you get 5?

[polar barn]

2π + π/2 = 4π/2 + π/2 = 5π/2.

[fair osprey]

OH I forgot to convert the units

so area 5π/2, circunference is 3π+2?

[polar barn]

circumrefence seems to be 6pi

[fair osprey]

ok

for this do I find the top semicircle and hte bottom as arecangel

[polar barn]

if you don't need to write down your doings

theres some trick you want to use to quickly find the answer

for example

take the semicircle with base CD outside of ABCE and put it into the blank semi circle with base CB in ABCE

[fair osprey]

I got hte area but the perimiter is being kidna wired. Isnt is the 2 sides, then the inverted semicircle and the other semi circle?

[polar barn]

you can't use the same trick that you used for area

like imagine it like this

the arc CD when placed in ABCE is in ABCE

and won't be calculated

what part you are calculating instead is CD

so just do it the normal way

[fair osprey]

yeah I tried doing it speatly but got that

[polar barn]

yeah

[fair osprey]

how is that wrong?

wait nvm figured it out

.close

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[visual plinth]

I done this question and got 1)z=0.03 2)x-0.25 3)y=0.09

I would appreciate if someone could check this

[wintry lotus]

Looks good! (assuming you mean x=0.25 instead of x-0.25)

[visual plinth]

@wintry lotus thank you

Could you look at part a)

The video im watching mentions it would be 0.53 (0.36 + 0.08 +0.09)

but I got 0.47 because I took it away from 1

Wouldn't "proportion" mean out the whole or am I wrong?

[wary stream]

If you subtracted from 1, that would mean that a student is in two or more societies

[wintry lotus]

What ^ said. You are just needing exactly 1, so .36, .08, and .09

[wary stream]

But it's saying exactly in one society, so you just sum up the percentage of the student being in just one

[visual plinth]

thanks I think I was over complicating it.

!close

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[humble creek]

how would i do this

[humble creek]

i have P(x <= 4)

[placid zinc]

What's x here?

[humble creek]

probability of rolling a 6

[placid zinc]

x is the number of 6s that are rolled

[humble creek]

yeah ok that makes sense

but what do i do from there is there a formula?

[placid zinc]

P(x ≤ 4) is the probability that, 4 or less times, this happens

This follows a binomial distribution, so you'd look to that for your formula

[humble creek]

yeah i have to find the probabilty

ok

[placid zinc]

,w binomial distribution

[ocean seal]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Kaynex#7520

[placid zinc]

Okay that's a lot of info haha, but that second picture at the top is the formula

If that's new to you, a YouTube video is the place to get a full explain

[humble creek]

the cdf right

[placid zinc]

Oh good point, I was forgetting we do want the CDF

Problem is, there's no closed formula for it.

P(x ≤ 4) = P(x = 0) + P(x = 1) + P(x = 2)...

You can save yourself some effort by realizing the problem is symmetric. That is, P(x ≤ 5) = 0.5

[humble creek]

wait how does that work

[placid zinc]

Whoops no that's definitely not true, forget I said anything

Yeah sadly I think summing 5 binomial pdfs is the way to go here

[humble creek]

alright ty appreciate it

[dim holly]

would you happen to know how to go aboutdoing this as well

same problem just different question

P (X >= 7) where x is probability of rolling a six

[frigid geyser]

it's the same, P(X=7) + P(X=8) + P(X=9) + P(X=10)

[dim holly]

ok thanks so much

[frigid geyser]

wait it says more than 7

so i guess it's P(X > 7)

[dim holly]

ye sorry wrote it out wrong

thanks thou

[lone heart]

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[warped ferry]

I need an explanation of what the mod means in there

[warped ferry]

I'm learning the RSA cryptosystem algorithm

learned about phi and how phi of a prime number works

n is a prime

but I have no clue what the mod does

or why its there

if u need a wider view of the equation here:

n is pxq

[frigid geyser]

it means de % phi(n) = 1 % phi(n)

https://en.wikipedia.org/wiki/Congruence_relation

[lone heart]

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[warped ferry]

ty

[frigid geyser]

np!

[lone heart]

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[loud thorn]

Hi guys! I am currently writing a paper where I want to model the math of cutting off the ring, a boxing technique in which an aggressor pursues a defender as the aggressor attempts to cut off the opponents path gradually and corner them into one of the 4 corners of the square ring. I have already made my own triangular model (using law of cosines and vectors) to solve this problem and I am going to use the curves of pursuit model, but I was wondering if anyone knew of similar "pursuit" and "cornering" problems/equations in math I could consider. Thank you!

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[lost steeple]

I'm having trouble setting up this problem

this is the diagram from what i understand

I'm a bit confused on how to appraoch tihs

[loud thorn]

<@&286206848099549185>

[pale kestrel]

reopen new channel

[loud thorn]

sorry! wasnt sure how to re ask my question w/o spamming

[pale kestrel]

just new channel

cus it closed already

[pale kestrel]

diagram looks ok (I would just exaggerate the 2 degrees to make it clearer)

You have to resolve vectors here

The airplane vector + wind vector needs to go towards the fire

[lone heart]

@lost steeple Has your question been resolved?

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[muted cedar]

I’m trying to solve the above but a bit stuck. Anyone can help?

[tacit arch]

you have R^2 in the plot. do you know how to solve for R?

[muted cedar]

No

[tacit arch]

If you know R^2, you can take the square root of that value to solve for R

If R^2 = 0.9, then R = sqrt(0.9).

[lone heart]

@muted cedar Has your question been resolved?

[swift shore]

Weird how that works out

Wait isn’t R^2 the sum of squared variances or some such thing

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[sturdy glen]

i'm trying to find the following

[sturdy glen]

and I got stuck when doing u-substitution

I got

$u=1 + 9r^2$

[ocean seal]

Josephh

[sturdy glen]

so then I got

$\int_{1}^{10} r \sqrt{u} du$

[ocean seal]

Josephh

[sturdy glen]

but I can't figure out what to change the r to

I looked up the answer and it turned out to be

[alpine sable]

But du is 18r dr

[pale kestrel]

rearrange?

[sturdy glen]

wait how come?

[pale kestrel]

but i dont think u substituted properly yeah

[sturdy glen]

it's been a year since I took calc 2 so i forgot

[alpine sable]

u= 1+9x^2

Differentiate

And boom

[sturdy glen]

thats just du = 18x then right, or do i need to add dx

[pale kestrel]

let me show

its basically chain rule

[sturdy glen]

i forgot how weird u sub was 😭

[pale kestrel]

$$dr = \frac{du}{du}dr = \frac{dr}{du}du$$

[ocean seal]

Shuri2060

[pale kestrel]

You 'multiply by 1'

then you 'swap the tops'

which is the idea behind chain rule

[sturdy glen]

ahhhh ok

so i need to multiply $$\sqrt{u}$$ by dr/du?

[ocean seal]

Josephh

[pale kestrel]

uhh if u go back to your original integral

when you substitute

you are swapping dr

for this

[pale kestrel]

.

$$I = \int^{r=1}_{r=0} r\sqrt{1+9r^2} ,dr$$

[ocean seal]

Shuri2060

[pale kestrel]

when i substitute, i always write that in the limits

$$ = \int^{r=1}_{r=0} r\sqrt{1+9r^2} ,\frac{du}{du}dr$$

[ocean seal]

Shuri2060

[pale kestrel]

That's the steps I'd show in my working

$$ = \int^{r=1}_{r=0} r\sqrt{1+9r^2} ,\frac{dr}{du}du$$

[ocean seal]

Shuri2060

[pale kestrel]

that way u dont make mistakes

[sturdy glen]

ah ok so then the

$\frac{dr}{du}du = \frac{dr}{18rdr}du$?

[ocean seal]

Josephh

[sturdy glen]

$=\frac{du}{18}$

[ocean seal]

Josephh

[sturdy glen]

because the drs cancel and the r in the integrand cancels the one on the bottom?

[pale kestrel]

let me see

so du/dr = 18r

[pale kestrel]

well i mean writing this is dodgy

[sturdy glen]

is it not 18rdr?

[pale kestrel]

but yeah

[sturdy glen]

fair

[pale kestrel]

$$ = \int^{r=1}_{r=0} r\sqrt{1+9r^2} ,\frac{dr}{du}du$$

[ocean seal]

Shuri2060

[pale kestrel]

$$ = \int^{r=1}_{r=0} r\sqrt{1+9r^2} ,\frac{1}{18r}du$$

[ocean seal]

Shuri2060

[pale kestrel]

$$ = \int^{r=1}_{r=0} \sqrt{1+9r^2} ,\frac{1}{18}du$$

[ocean seal]

Shuri2060

[pale kestrel]

Thats wht u meant right

[sturdy glen]

yeah yeah

ok bet thank you so much

i completely forgot about having to do that step

appreciate it

[pale kestrel]

and dont forget to sub the limits to u = ???

👌

[sturdy glen]

oh yeah i already got that part

.close

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[flat tangle]

hi how do i read this equation?

[flat tangle]

its purpose is to calculate simultaneous multiple of n numbers from 1 to 100

i tried using it but got 12 when finding the multiples of both 2 and 3

when instead it shouldmbe 16

[swift shore]

Read it like, you don’t understand notation or what

[flat tangle]

basically

[swift shore]

You mean x_i is between 1 and 100

[flat tangle]

yes

hence the 100/summ xi

mmy assumption was that it would be 100/(2^3)

[swift shore]

1 to 100 inclusive?

[flat tangle]

ye

whole question looks like this

for some reason my math just isnt checking out

[swift shore]

Ah Nx is the number of multiples of x less than 100

Do you understand the first equation @flat tangle?

[flat tangle]

yes

[swift shore]

Lit

The second equation is just replacing x with a product of numbers

Basically, the lcm of that sequence

That should make sense

[flat tangle]

what does lcm stand for?

[swift shore]

Least common multiple

Like, the lcm of 2, 3, and 4 is 12

[flat tangle]

oh right

[swift shore]

So actually the product is not always the lcm

But it does give a multiple of all of x_i

[flat tangle]

how should i be substituting numbers into the equation

given x = 2 and 3

[swift shore]

So I’m thinking, they want multiples of 2, 3, or 5

So you could first do # of all multiples of 2, then all multiples of 3, then all multiples of 5, and add them up. However, you’re overcounting multiples of 2 and 3, multiples of 3 and 5, multiples of 2 and 5, and multiples of 2, 3, and 5, so subtract those off

[flat tangle]

ye thats exactly it

[swift shore]

Does that make sense

Ok

[flat tangle]

its just that i dont know how to use the eqution

to get the multiples of 2 and 3

that im overcounting

[swift shore]

Oh

$$N_{2,3} = \left\lfloor \frac{100}{2 \cdot 3}\right\rfloor$$

[ocean seal]

abs_0

[flat tangle]

oh

that makes a lot more sense

thanks

[swift shore]

Np

Btw

[swift shore]

I’m pretty sure this is wrong

Unless x1, …, xn are all pairwise coprime, I don’t think that formula gets you the right number

For example 2,3,4 should give 100/12, not 100/24

[flat tangle]

oh abt it

[flat tangle]

checking wt u got ur supposed to add back multiples of 2,3,5 instead of subtract it

since that would lead to undercount

[swift shore]

Wait what

Multiples of 2 are 2, 4, 6, 8, …
Multiples of 3 are 3, 6, 9, 12, …

You’re counting 6 twice, so wouldn’t you need to subtract off 6 (and all the other common multiples of 2 and 3)?

[flat tangle]

it uses the inclusion exclusion rule

wait wrong image

[pale kestrel]

this question looks fun 👀

[swift shore]

Oh oh that makes sense actually

That’s really interesting

[pale kestrel]

im not immediately getting that hahaha

[swift shore]

When you count A + B + C, you’re overcounting their intersections

[pale kestrel]

Ok, i was assuming the diagram was representing

the thing above

but its the sum along the way

[swift shore]

Oh oh

[flat tangle]

.close

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[tribal sun]

I need some help i got 2 questions

[tribal sun]

The congruent shape repeats using a transformation, it creates a:

When a congruent shape repeats using a transformation, it creates a____

I am doing tessellations

[swift shore]

@tribal sun I don’t understand, aren’t those the same question

[tribal sun]

Yeah i thought it was similar too, my teacher just gave them to us

[swift shore]

Isn’t that just a definition of something

Can’t find anything online

I’m guessing it’s “tessellation”

[tribal sun]

i don't know that's why i cam here

are you sure? I thought of that but the answer is so easy it might be wrong

[swift shore]

Well the question is phrased horribly

[tribal sun]

yeah, I directly copied pasted it too into google

[pale kestrel]

what does repeats

even mean

zzz

heres some repeated approximately congruent circles >.>

[swift shore]

A “congruent shape”? The word “congruent” requires two different shapes to compare, it isn’t a descriptor of one shape

[wary stream]

I think they meant tessellation due to the topic they are on

[swift shore]

Yeah @tribal sun it’s probably tessellation

[tribal sun]

Yes so I made a tessellation, and after i finished I had to answer these questions

Ok.

.close

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[pale kestrel]

Problem: Count the number of possible paths if you take $2n$ steps starting and ending at the origin on the 2D plane. (each step takes you 1 unit along an axis direction)
Eg. 'Right Right Down Left Left Up' is a valid path for $n = 3$.

[ocean seal]

Shuri2060

[pale kestrel]

I reckon the solution to this is

[ocean seal]

Shuri2060

[pale kestrel]

But I do not have an intuitive explanation for why this is so, and am hoping it generalises to higher dimensions by changing the exponent

atm I am thinking for n = 2, you have a 4x4 grid and you choose 2 rows and columns. The 36 possible 4x4 grids should correspond to each path

If anyone has any ideas 🤷‍♂️

[bleak ridge]

Does it generalize to 1 dimension

[pale kestrel]

Yes.

This is just stars and bars problem in 1d

[vale wigeon]

oh hold on

[bleak ridge]

Oh no

[vale wigeon]

$\sum_{k=0}^n \frac{(2n)!}{k!^2 (n-k)!^2}$

[ocean seal]

Ann

[vale wigeon]

so this is $\sum_{k=0}^n \binom{2n}{n} \binom{n}{k}^2$

[ocean seal]

Ann

[pale kestrel]

yes this is how i got original answer

I can explain that way xd

[tacit arch]

I think these are the Dyck paths

[vale wigeon]

are they tho

[tacit arch]

No wait I'm wrong, they end at the x axis

[pale kestrel]

but really I am hoping for a 'direct' explanation

maybe not possible

[pale kestrel]

At least, can you generalise this explanation to higher dimensions? 🤔

[tacit arch]

I'm terrible at combinatorics. You need n ups/lefts and n downs/rights. The order doesn't matter so you're just counting how many ways to order them in a length of 2n

[hasty elk]

well, in higher dimensions you end up with a sum of multinomial coefficients

I'm not too experienced with those, but the underlying idea is the same: in $j$ dimensions, split $2n = 2(n_1 + \dots + n_j)$, and then there are $\binom{2n}{n_1,n_1,n_2,n_2,\dots,n_j,n_j}$ ways to arrange the sequence of steps. Then you have to sum over all possible tuples $(n_1,\dots,n_j)$, i.e. $\sum_{n_1+\dots + n_j = n} \binom{2n}{n_1,n_1,n_2,n_2,\dots,n_j,n_j}$.

[ocean seal]

Camilleone

[pale kestrel]

Does this come out to the same answer 🤔

(2n c n)^j

I'm hoping

[hasty elk]

i...really doubt it tbh

[pale kestrel]

it feels like it should though

just feels haha

[hasty elk]

maybe there's something in Gould's manuscripts, but i'm a little lazy to check

as an add-on, it even fails in dimension 3

consider walks of length 2, then $\binom{2}{1}^3 = 8$ walks are predicted, yet simple counting shows only six walks are possible

[ocean seal]

Camilleone

[pale kestrel]

rip

rlly?

[hasty elk]

exercise! 😛

[pale kestrel]

12
21
34
43
56
65

ehhhmmm

huh

$$\binom{2n}{n}^k$$

[ocean seal]

Shuri2060

[pale kestrel]

But this is true for k = 1, 2 right?

At least I was pretty sure...

[hasty elk]

sure, yeah

[pale kestrel]

never thought to do this

very surprised.

just assumed it would still hold

[hasty elk]

yeah, lattice walks in j dimensions are a little bit special

for 1 and 2 dimensions everything is well-behaved

[pale kestrel]

what

never heard of this haha

[hasty elk]

for 3 and up, weird things happen

for example, if you try and do random walks in 1 or 2 dimensions starting from the origin, you return to the origin with probability 1

in 3 dimensions and above, you don't

[pale kestrel]

oh yh I've heard of this

this feels very disappointing ngl lol

[hasty elk]

HAHAHA

[pale kestrel]

So there might not end up being an intuitive explanation for k = 2

like directly from the formula

🤔

feels like there probably can't be

[hasty elk]

there should be though

[pale kestrel]

but wouldnt such a thing generalise up

[alpine sable]

what about using the probabilities

[hasty elk]

not if it only works in dimension 2

[pale kestrel]

hmmmmmmmm

[pale kestrel]

whats this

[hasty elk]

i believe i once read an explanation using some strange labelling that keeps changing, but i can't reproduce the argument offhand

[pale kestrel]

https://i.imgur.com/JShGgxT.png

lol i spent quite a bit of time yesterday

trying to assign each grid to a path

but failing

[hasty elk]

there is a way

just that i can't prove it at the moment

[lone heart]

@pale kestrel Has your question been resolved?

[hasty elk]

i think i have it

[alpine sable]

There was an incredibily similar problem to this in Sheldon Ross,which I learned during my probability course.It is similar to problem 21 in chapt 1 of that book.There they have point B from A which requires 7 steps to reach and no matter what you need to do 3 up and 4 right.So you just have to count number of 7 letter words in U,R with 3 words U and 4 words R

[pale kestrel]

diagonals 🤔

[hasty elk]

yes

there are 2nCn ways to pick the first set of diagonals

and 2nCn ways to pick the second

what you do is to combine those two together, then:
two diagonals means go up
one diagonal means go right or left depending on direction
no diagonals means go down

in this way you form a sequence of steps for your walk

each such sequence maps to a walk returning to the origin, and each such walk maps to such a sequence

so the walks and maps are in a one-to-one correspondence and the number of walks is (2nCn)^2

[pale kestrel]

wow. Gonna think about this

[alpine sable]

I'm sorry to interfere.But I will just give it a try.So since we have to return to origin,we need number of ups,U=number of downs ,D and number of lefts,L=number of rights R.Now So we have s U,we need s D and if we have t R then we need t L.So look for s and t with s+t=n

[pale kestrel]

go ahead, try the problem lol. It's fun

Big ty 💙

[hasty elk]

and now you have a problem because this method doesn't generalise to dimension 3

[pale kestrel]

Attempting a similar approach for a different formula...

Probably doesn't work?

[hasty elk]

in dimension 3 presumably you start with four diagonal directions, but you have no way of defining a direction for lone arrows

[pale kestrel]

urgh

damn this is so strange

[hasty elk]

i'd say welcome to flatland but well

[alpine sable]

So what we are looking for is words in 2n letters with s U,sD,t L,t R and with t=n-s.So the result is just sum of (2n!)/(s!)^2(n-s)!^2 for all possible s 's(I.e for s from 0 to n)

yeah true that

[alpine sable]

I'm Sorry if I made calculation mistake somewhere .But is this fine @pale kestrel

[pale kestrel]

latex possible?

it looks correct

[hasty elk]

it's right

[alpine sable]

I don't know unfortunately

[hasty elk]

that's the usual argument

[alpine sable]

Thank you!

[pale kestrel]

Are you trying to simplify that expression now?

[alpine sable]

Nope,I'm just keeping as it is.

[pale kestrel]

I suddenly have a feeling that arrow argument generalises to 2^n dimensions... but thinking about it

[hasty elk]

lemme know how it goes

i'm rather interested

[pale kestrel]

im rewriting the diagonals as 0 and 1

but the issue probably comes from ensuring i get back to the origin

hm no, I can't return to the origin --- can't create the same thing with parity. Unless I change the argument perhaps

[lone heart]

@pale kestrel Has your question been resolved?

[alpine sable]

@pale kestrel so it’s basically an infinity by infinity grid right?

On which we are moving

[pale kestrel]

yh

I misworded original question

Check the pinned post, not latex

[alpine sable]

Hmm I have a suggestion

Let’s use 0,1 for this problem

Binary representation

0 if u move along x axis

1 if u move along y axis

So for example

The string of digits for let’s say going up , up , right , left , down , down will be 110011

Now let’s check how many such string of digits is possible for a fixed 2n

@pale kestrel btw our trip ends when we reach origin right ?

Like for example

1100

[pale kestrel]

we have to reach the origin at the end

we have to make 2n steps regardless

[alpine sable]

Yes

[pale kestrel]

we can visit the origin multiple times

[alpine sable]

In our trip ?

[pale kestrel]

yes

im not sure how that binary representation helps unless u split into 2 cases

[alpine sable]

Like we visit the origin twice in our trip before coming back to it , it still counts as a trip right ?

[pale kestrel]

yes

you can consider 11111100

and then split the problem into 2.

That is the approach most people take

[alpine sable]

Hmm

For n=1 we have

00,11 as our two cases

For n=2

We have 1111, 0000

1010

0101

0011

1100

Do u see any more ?

[pale kestrel]

its 4c2

isnt it

[alpine sable]

How ?

[pale kestrel]

uhhjh

uhhhhh maybe not.

feels like it should be 4c2

[alpine sable]

It’s coming out to 6 cases

[pale kestrel]

but i cant explain how to count

[alpine sable]

Like for n=1

It is 2C1

Which is 2

Exactly what we got

For n=2

It seems like 4C2 ngl

But I only got 6 ways so far

The ones I have listed above

[pale kestrel]

no its not 4c2

u missed

1001

0110

[alpine sable]

4c2 is 12

[pale kestrel]

6

[alpine sable]

Oh shit

I’m dumb

[pale kestrel]

8

[alpine sable]

Yeh

So the 2nCn is wrong formula

[pale kestrel]

???

its almost certainly right - verified by 2 others

[alpine sable]

How

We get 8 ways so far

[pale kestrel]

youre not distinguishing between your ups and downs

[alpine sable]

Oh shit

So we do something like quaternary sequences ?

0,1,2,3

[pale kestrel]

i mean u can try 👀

I think however u want to call them

u will get to the same result

[tacit arch]

Can't you just square your result since up and down double count a 1

[pale kestrel]

its just a question of 'demonstrating' how the counting is done. It is not easy to show (2nc2)^2 is intuitive

[pale kestrel]

they had 8

[alpine sable]

Time to try quaternary sequences

Wait

It won’t work for n=1

Since n=1 only has two,path ways

[pale kestrel]

up down

left right

right left

down up

[alpine sable]

Fine 4

My point still stands

[pale kestrel]

what?

2c1 squared is 2 times 2

[alpine sable]

I can’t use quaternary for n=1

[pale kestrel]

???

[alpine sable]

It’s supposed to have 4 digits

[pale kestrel]

im not sure what u mean

[alpine sable]

Nvm I’m confused af

I’m confooz

@pale kestrel are u in Uni

[pale kestrel]

yh

[alpine sable]

This question from ur uni text book ?

[pale kestrel]

lol no

[alpine sable]

Then ?

[pale kestrel]

#discrete-math

someone gave it to me the other day

[alpine sable]

Is this problem uni level

[pale kestrel]

no

the ideas you need to solve it just apply combinatorics u see in high school

the question can be considered olympiad style

[alpine sable]

Hmm

How did u solve it

[bright hedge]

High school Olympiad is pretty much university

[pale kestrel]

This is the most elegant way of solving it I've seen.

[alpine sable]

i am very bad at counting problems

[pale kestrel]

The most common way to solve it is by counting like this.

[alpine sable]

I still don’t get it

[pale kestrel]

i cant write down the explanation from the top of my head

[alpine sable]

Which do we have 2n choose n

In the summation

[pale kestrel]

ok ok i have it

[alpine sable]

And n choose K whole squared

[pale kestrel]

First, you have 2n steps

[alpine sable]

Ok

[pale kestrel]

1 sec

[alpine sable]

Ohh

Is it something like u choose the initial n steps

And the rest will replicate it to follow back

[pale kestrel]

You can consider half your journey

to be going away

half to be coming back

ok?

[alpine sable]

Ok

[pale kestrel]

so then you choose

which n of the 2n steps

are going away

[alpine sable]

Ok

[pale kestrel]

thats 2n choose n

[alpine sable]

Ok

[pale kestrel]

Next you divide into cases

thats the summation

You choose how many of the steps are vertical

how many are horizontal

[alpine sable]

Ok ok

[pale kestrel]

That is k

[alpine sable]

Ahhh