Help please

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[idle seal]

do yk the game where you have like some stones and each player takes up to 3 stones

and whoever takes the last one wins

this is like, the same concept

[paper frost]

ummm

i don't

i haven't even heard of it 😭

[paper frost]

@idle seal this is gpt response, but this is a bit vague and i don't know if it's correct

[idle seal]

dont use gpt ever

[paper frost]

i don't know what to do 😭

[idle seal]

so think about this

whats the smallest number where alice is guaranteed to win

so basically

wait no

the secfond person

where the second person is guaranteed to win

whats the smallest number where

no matter what alice does

charlie wins

[paper frost]

umm

110

?

[paper frost]

if the starting number is 110 then charlie is guaranteed to win ig

that's the smallest i can think of

[idle seal]

nice

ok so now find the 2nd smallest number

[paper frost]

111?

wait no

umm

[paper frost]

ye i think it is 111

the second smallest number

[idle seal]

what if alice plays k=1

[paper frost]

the new number becomes 110

?

[idle seal]

and now who has the winning advantage when 110 is the number and someoen goes first

[idle seal]

@paper frost ?

[paper frost]

Charlie

@idle seal but what abt part a?

Isn’t that a bit vague?

[idle seal]

it is chariles move at 110, no?

[paper frost]

^?

^^^

[idle seal]

can charlie win

if it is his turn

at 110?

[paper frost]

No?

Bcuz he has only one option to subtract

Which is 1

He can’t subtract 0 Bcuz it’s not a non-zero digit

[paper frost]

So if he subtracts 1, then 110-1=109 and now it’s Alice’s turn and she can subtract 9 from 109 and win

[idle seal]

so 111 doesnt guarantee a charlie win right

what does?

[paper frost]

Yes

[paper frost]

Yes because Alice always starts first

It says in the q

[idle seal]

well we need the 2nd smallest number such that charlie is guaranteed to win

a hint is

how can charlie force alice to move at 110?

[paper frost]

He can’t?

[paper frost]

Alice wins every time, no?

[idle seal]

yup

so that’s not a forced win for him

try finding smthn else

a number for charlie

@paper frost sorry i just woke up

[paper frost]

I did part a

I’m now doing part b

But I don’t know how to find a general formula

[idle seal]

how’d you do it

[paper frost]

I’m out rn I’ll be back in 30 mins and I’ll show u

[idle seal]

alr

[paper frost]

@idle seal

Can u help me do part b

[idle seal]

why can’t charlie subtract any number greater than 2

anyways you should notice a pattern

the force 120 is good

[paper frost]

wdym?

[idle seal]

why is charlie forced to subtract 1 at 110

and why can charlie subtract only 1 or 2 at 120

[paper frost]

because there's only one non-zero digit he can subtract from 110, which is one

[idle seal]

?

what about 2

why doesn’t 2 work

[paper frost]

ok by 'digits' i mean the digits of the three digit number

that's how i understood the question

[idle seal]

oh

[paper frost]

is it right?

or am i wrong?

[idle seal]

oh it does say of n

nvm

[paper frost]

'a move consists of choosing a non-zero digit, k, of n and replacing n with n-k on the blackboard'

yes

[idle seal]

ok that’s fair

the solution is the same still

[paper frost]

for part two we want a general proof

[idle seal]

think about if k is any digit from 1-9

[paper frost]

i don't want to do it by logic

i want to do it systematically

[idle seal]

yea there’s a systematic way

[paper frost]

ok

[idle seal]

notice how you were forcing charlie to play on a multiple of 10

consider that

[paper frost]

ye

so should i force Alice asw?

[idle seal]

try it out

[paper frost]

so if the starting digit is 101, then Alice loses because in the first move Alice can only subtract one

wait no

nvm

it's not working

[idle seal]

hmm i guess what we can do is play the game ourselves

pick a number and pick who you wanna be

[paper frost]

Alice

[idle seal]

this might help

ale

[paper frost]

lets say that the starting number is 150

wait that will make it longer

[idle seal]

we can do 150

[paper frost]

idk what the starting num should be

ohh

[idle seal]

it’s fine

[paper frost]

so if i subtract 5 then im gonna write 145

[idle seal]

i subtract 5

140

[paper frost]

136

[idle seal]

i subtract 6 130

[paper frost]

127

[idle seal]

i subtract 7 120

[paper frost]

118

[idle seal]

sub 8
110

[paper frost]

109

[idle seal]

sub 9

[paper frost]

ohh so charlie wins

[idle seal]

so that’s an interesting case right

now let’s try uhhh

138

ig

[paper frost]

kk

130

[idle seal]

129

[paper frost]

120

[idle seal]

118

[paper frost]

ohh i see the pattern

110

[idle seal]

yay

109

[paper frost]

100

[idle seal]

alright try to write the proof now

[paper frost]

if the player ends up in a three digit where the tens place is {0,1,2...,9} and the units place is 0 then this player wins

[idle seal]

try a systematic explanation for alice’s winning strategy

her moves follow a simple pattern

[paper frost]

ok i'll let you know when im done

meanwhile can you help me with this - https://discord.com/channels/624314920158232616/1351647642312114246

pls

[paper frost]

@idle seal sry i forgot to send my proof for part b

i 'll send it now

[idle seal]

alr

[paper frost]

@idle seal there^^^

[idle seal]

nice

that is very good

[paper frost]

ty 😄

alr so can we move on to the cuboid q 😭

[chilly oracle]

@paper frost

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