I know how to approach this question but I'm having trouble finding what the sequences actually are

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[calm prism]

01 23 45 67 89
10 32 54 76 98
90 81 72 63 54
a few to start you off

[crude wing]

wait a minute

i have seen this question before

[twilit echo]

how do you know what they are

[calm prism]

it has been sent here before

you would’ve come across it

[crude wing]

yeah

[calm prism]

there are four with 0~4 in one digit and 5~9 in the other

[crude wing]

should we start with the amount of possibilities?

[calm prism]

another four with one digit being purely odd and the other even

[twilit echo]

arithmetic is increasing by the same amount each time right

[calm prism]

i have not found others

[calm prism]

yes

[twilit echo]

i dont understand how you know what the sequences are

[crude wing]

OH

WAIT

[calm prism]

what do you mean?

[crude wing]

for the tens digit at least

[twilit echo]

how do you come up with these like is the a rule

[calm prism]

i have found a rule that gives me some of the sequences, but i cannot prove it gives all of them

[crude wing]

you can only have 1 increasing or 0,2,4,6,8 and 1,3,5,7,9 for tens digit

[calm prism]

the rule is that if both the tens digit and the ones digit are in arithmetic progression, then the two-digit numbers will be

[crude wing]

for strictly increasing

[calm prism]

this gives me 8 numbers

[calm prism]

10 32 54 76 98

[crude wing]

oh yeah that too

[twilit echo]

so there are 8 sequences

[calm prism]

i would perhaps elaborate on why something like 0 2 4 7 9 fails

[crude wing]

eh

[twilit echo]

but it isnt arithmetic is it

[calm prism]

not for the tens digit, but a reason should be provided for why the whole number cannot be in AP

[twilit echo]

because it doesnt increase by same amount each time?

[calm prism]

12 24 36 48 60 has the 10s digits being 12346

yet it is arithmetic

[crude wing]

a “carrying over” rule, correct?

[twilit echo]

ok so these numbers are the tens digits

[calm prism]

yes; very annoying

[crude wing]

lol, this is what makes problems like these 10x harder

[twilit echo]

i still dont get how we find sequences which use all number 0 to 9

[crude wing]

i have a faint idea, but it probably will not work

we can define the first number of the sequence to be 10a+b

and the common difference d

[twilit echo]

what so second is 10a+b+d

[crude wing]

yes

[twilit echo]

and third os 10a+b+2d

and so on

[crude wing]

and so on

i’m figuring this out as i go

[twilit echo]

but this doesnt help with using all numbers 0 to 9

[crude wing]

well

we already have 2 numbers chosen, namely a and b, right?

[twilit echo]

yeah

[crude wing]

we can choose a d such that all the numbers are used

let’s go with the case d=9 since it is the easiest to show

we can start with a=0 and b=9

wait i think i just got an idea

d either has to be a multiple of 3 greater than 6 or a multiple of 11

!

[twilit echo]

why

[crude wing]

well, consider something like 6

06,12,18

[twilit echo]

uses 10 twice

[crude wing]

it repeats the tens digit, right?

so we can set a lower bound

namely d>8

[twilit echo]

yes ok

[calm prism]

why

[crude wing]

look at what happens when d<=8

[calm prism]

hypothetically, 08 16 24 32 40

[crude wing]

we are guaranteeing, with d=>8, that the tens digit is not repeating

[calm prism]

tens digit doesn’t repeat

[crude wing]

maybe geq then

[twilit echo]

+close