Subpalindromes question | Mathematics | Page 1

pale aurora Apr 26, 2025, 10:47 PM

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This is a question a friend showed me:
A palindrome is any sequence of 2 or more letters that reads the same

forwards as it does backwards. For example, MM, EVE, NOON, and

ABABA are all palindromes.

A subpalindrome of a palindrome is any palindrome it contains. Notice

that this includes the palindrome itself.

For example, ABBBA has four subpalindromes, as underlined below:

ABBBA

Note that we count the subpalindrome BB twice since it appears in two

different positions.

a Show how two letters can be added to ABBBA to create a seven-letter

palindrome that has exactly five subpalindromes.

b Find a palindrome of length 30 that has exactly 30 subpalindromes,

or explain why no such palindrome exists.

c Find a palindrome of smallest possible length that has at least 30 sub-

palindromes.

d Find a palindrome of smallest possible length that has exactly 30 sub-

palindromes.

What I got so far:

So far, I can't even get A through trial and error method. For example, I tried AABBBAA which has too many then I have CABBAC which I think reduces it. I need a methodical method to continue the question - also it will be needed in further questions.

inner ridgeBOT Apr 26, 2025, 10:47 PM

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pale aurora This is a question a friend showed me: A palindrome is any sequence of 2 or more...

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whole topaz Apr 26, 2025, 11:15 PM

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pale aurora This is a question a friend showed me: A palindrome is any sequence of 2 or more...

doesn't CABBBAC work?

pale aurora Apr 26, 2025, 11:57 PM

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It should work except the bigger questions i dont know how to do?

flint jay Apr 27, 2025, 12:08 AM

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pale aurora It should work except the bigger questions i dont know how to do?

Here's a simpler question; what is the smallest number of subpalindromes that a palindrome of n characters can have?

pale aurora Apr 27, 2025, 12:17 AM

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I think it should be 0? if less than 26 (not really sure)

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Also I would like anyone to review my approach to the first part

flint jay Apr 27, 2025, 1:39 AM

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pale aurora I think it should be 0? if less than 26 (not really sure)

...of n characters?

pale aurora Apr 27, 2025, 1:41 AM

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yes the answer to the simpler question

flint jay Apr 27, 2025, 1:48 AM

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pale aurora yes the answer to the simpler question

Show me a palindrome with two characters with zero subpalindromes.

pale aurora Apr 27, 2025, 1:53 AM

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AB?

flint jay Apr 27, 2025, 1:55 AM

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pale aurora AB?

...that's not a palindrome.

pale aurora Apr 27, 2025, 1:56 AM

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So AA?

flint jay Apr 27, 2025, 1:56 AM

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That has one subpalindrome; itself.

pale aurora Apr 27, 2025, 1:57 AM

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I dont really think it is possible

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Anyways is this correct to part (a) CABBBAC

flint jay Apr 27, 2025, 1:57 AM

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Well, yes, but why?

pale aurora Apr 27, 2025, 1:58 AM

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because a subpalindrome has to have at least 2 right?

flint jay Apr 27, 2025, 1:58 AM

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...what?

pale aurora Apr 27, 2025, 1:59 AM

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i mean it has to have at least 2 letters right?

flint jay Apr 27, 2025, 1:59 AM

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Okay, and?

pale aurora Apr 27, 2025, 2:04 AM

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Does this mean that CABBBAC is correct in (a)?

flint jay Apr 27, 2025, 2:19 AM

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pale aurora Does this mean that CABBBAC is correct in (a)?

No, it doesn't mean that.

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Okay. ABBBA has four subpalindromes, right?

pale aurora Apr 27, 2025, 2:19 AM

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Yes, itself and the combinations of B

flint jay Apr 27, 2025, 2:20 AM

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You want to add two characters to give it five.

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So you want to add two characters to add one subpalindrome.

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So prove that adding the C's on the end adds one subpalindrome.

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It trivially adds at least one.

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Which you would know if you actually tried to answer the question I asked you.

pale aurora Apr 27, 2025, 2:21 AM

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wdym

flint jay Apr 27, 2025, 2:22 AM

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How. Many. Subpalindromes. Does. A. Palindrome. Of. Length. N. Have. At. Minimum?

pale aurora Apr 27, 2025, 2:24 AM

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that would be 1

flint jay Apr 27, 2025, 2:24 AM

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Okay, show me a palindrome of length 4 with exactly 1 subpalindrome.

pale aurora Apr 27, 2025, 2:26 AM

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I dont actually think it is possible

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Ifyou try it has to be something like ABBA which has two subpalindromes

flint jay Apr 27, 2025, 2:27 AM

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Right. So what is the minimum number of subpalindromes that a palindrome of n characters can have?

pale aurora Apr 27, 2025, 2:29 AM

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I know that ABA is a palindrome which should fulfil n=3 where it is still has 1 palindrome which is the minimum

flint jay Apr 27, 2025, 2:30 AM

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I'm. Not. Asking. What. The. Minimum. Possible. Number. Of. Subpalindromes. Is. For. A. Palindrome. Of. Any. Length. I'm. Asking. What. The. Minimum. Number. Is. For. A. Palindrome. Of. Length. N.

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For. Arbitrary. N.

pale aurora Apr 27, 2025, 2:31 AM

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Ah. that would be 2?

flint jay Apr 27, 2025, 2:31 AM

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Okay, show me a two-character palindrome with two subpalindromes.

pale aurora Apr 27, 2025, 2:32 AM

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That isnt possible since you cant further divide it

flint jay Apr 27, 2025, 2:32 AM

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So then the answer isn't 2.

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Stop guessing, start thinking.

pale aurora Apr 27, 2025, 2:33 AM

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so you're asking the minimum to create subpalindromes?

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That would be 4

flint jay Apr 27, 2025, 2:33 AM

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I don't even know what the hell that means.

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No.

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IF. YOU. HAVE. A PALINDROME. OF N CHARACTERS.

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HOW MANY SUBPALINDROMES MUST IT HAVE.

pale aurora Apr 27, 2025, 2:34 AM

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At least 1

flint jay Apr 27, 2025, 2:34 AM

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WHAT IS THE NUMBER OF SUBPALINDROMES THAT IT CANNOT HAVE LESS THAN.

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AND THAT IT CAN HAVE EXACTLY AS MANY AS.

flint jay Apr 27, 2025, 2:34 AM

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pale aurora At least 1

SO SHOW ME A FOUR CHARACTER PALINDROME WITH ONE SUBPALINDROME.

pale aurora Apr 27, 2025, 2:35 AM

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thats not possible

flint jay Apr 27, 2025, 2:35 AM

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SO THEN THE ANSWER ISN'T 1.

pale aurora Apr 27, 2025, 2:36 AM

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2?

flint jay Apr 27, 2025, 2:36 AM

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SHOW ME A TWO-CHARACTER PALINDROME WITH TWO SUBPALINDROMES.

inland sigil Apr 27, 2025, 2:36 AM

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techie. Have a chill pill. 🧊 💊

flint jay Apr 27, 2025, 2:36 AM

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inland sigil techie. Have a chill pill. 🧊 💊

This person isn't trying. They're just guessing.

inland sigil Apr 27, 2025, 2:37 AM

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flint jay This person isn't *trying.* They're just *guessing.*

yea, but hopefully they will learn something after you lead the way to the promised land (knowledge) like moses

flint jay Apr 27, 2025, 2:37 AM

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inland sigil yea, but hopefully they will learn something after you lead the way to the promi...

They're not taking my leadership! They keep making the same guesses that I've already disproven.

inland sigil Apr 27, 2025, 2:37 AM

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just a little patience,
or not you getting angry/annoyed is funny

inland sigil Apr 27, 2025, 2:38 AM

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pale aurora 2?

just so techie doesn't die,
A two letter palindrome can ONLY have 1 subpalidrome, which is itself, because you can only arrange two of the same letters in one way, and that happens to be a palidrome.

pale aurora Apr 27, 2025, 2:39 AM

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yeh

flint jay Apr 27, 2025, 2:39 AM

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pale aurora 2?

Do you even have any paper? How is your memory for wrong answers this bad?

pale aurora Apr 27, 2025, 2:40 AM

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I can write down everything you say if you want

flint jay Apr 27, 2025, 2:41 AM

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pale aurora I can write down everything you say if you want

Just write down what you know the answer isn't so you stop repeating guesses you already know are wrong.

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That's what's pissing me off the most here.

ashen olive Apr 27, 2025, 2:42 AM

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Then take a break.

pale aurora Apr 27, 2025, 2:42 AM

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A 2 letter has one subpalindrome, 4 letter has to have at least 2

flint jay Apr 27, 2025, 2:43 AM

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pale aurora A 2 letter has one subpalindrome, 4 letter has to have at least 2

Right. So then the minimum number of subpalindromes isn't 1 and isn't 2, and in fact isn't constant at all.

pale aurora Apr 27, 2025, 2:44 AM

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so it isnt constant at all for any n numbers. is there a pattern?

flint jay Apr 27, 2025, 2:45 AM

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See, now you're thinking.

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And there must be a pattern, right? After all, a palindrome is a very well-defined structure.

pale aurora Apr 27, 2025, 2:46 AM

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so is the approach to find the pattern and extrapolate?

flint jay Apr 27, 2025, 2:46 AM

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...yes.

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Which is to say, the approach is to do math, because that's what math is.

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Finding patterns in logical structures.

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Here's a simple question about the logical structure of a palindrome; how is one constructed?

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@pale aurora That is to say, if I want to make a palindrome, how would I do it?

inland sigil Apr 27, 2025, 3:06 AM

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is help still needed?

flint jay Apr 27, 2025, 3:06 AM

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Help is still here.

inland sigil Apr 27, 2025, 3:07 AM

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oh

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hi techie

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i'm a big fan of your's

flint jay Apr 27, 2025, 3:07 AM

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It's been literally eleven minutes.

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And it's OP's turn to speak.

inland sigil Apr 27, 2025, 3:07 AM

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ahem. @pale aurora

pale aurora Apr 27, 2025, 3:12 AM

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So far, ive been trying to find a pattern

inland sigil Apr 27, 2025, 3:12 AM

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SPEAK OF THE DEVIL

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JUST IN TIME

tough blaze Apr 27, 2025, 3:12 AM

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5 mins late

inland sigil Apr 27, 2025, 3:12 AM

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pale aurora So far, ive been trying to find a pattern

Are there any patterns you see so far?

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remember adding a new letter to the start and end of abbba only raised the amount of sub palindromes by 1

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perhaps there is a pattern there you could state

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is this guy even still here?

flint jay Apr 27, 2025, 3:26 AM

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inland sigil is this guy even still here?

In the server? The thread would auto-close if they weren't.

tough blaze Apr 27, 2025, 3:28 AM

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I think he’s asking if the guy is still active in this thread

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Or if he just gave up and left or smth

tough blaze Apr 27, 2025, 3:44 AM

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pale aurora Anyways is this correct to part (a) CABBBAC

Or any other arrangement where you add C symmetrically around the centre.

tough blaze Apr 27, 2025, 6:14 AM

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@pale aurora, do you still need help?

pale aurora Apr 27, 2025, 6:15 AM

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Im back now
So far I have CABBAC as the answer to (a) - check if this is the answer
However, I have somehow found a pattern as mentioned before but I dunno how it helps

tough blaze Apr 27, 2025, 6:16 AM

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CABBBAC is correct, did you mistype?

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Did you get b?

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The 2nd sub question

pale aurora Apr 27, 2025, 6:18 AM

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Yeah i meant CABBBAC
I dont kno how to do (b)

tough blaze Apr 27, 2025, 6:19 AM

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Ok, at least how many sub palindromes should a palindrome of length 30 have?

pale aurora Apr 27, 2025, 6:21 AM

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Perhaps 15?

tough blaze Apr 27, 2025, 6:21 AM

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If you don’t get it, here’s a hint: there has to be atleast one axis of symmetry in a palindrome (at the centre).

tough blaze Apr 27, 2025, 6:21 AM

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pale aurora Perhaps 15?

Yes

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Now if I have another axis of symmetry, forming a palindrome at position x, there must be another one at…?

pale aurora Apr 27, 2025, 6:22 AM

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x/2

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but how does that really relate to proving it cannot be 30.

tough blaze Apr 27, 2025, 6:22 AM

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pale aurora x/2

No, not x/2

tough blaze Apr 27, 2025, 6:22 AM

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pale aurora but how does that really relate to proving it cannot be 30.

Wait

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It’s 31-x

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Does that make sense? If I have another axis of symmetry at some position x, there has to be a corresponding one at 31-x

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Because the entire thing is symmetrical along the centre

pale aurora Apr 27, 2025, 6:25 AM

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So how would that fit in the question?

tough blaze Apr 27, 2025, 6:25 AM

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Be patient

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So now, if there is another sub palindrome, besides the aforementioned 15, there would be a corresponding copy of it, right?

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What does that tell you about the number of sub palindromes besides the aforementioned 15?

pale aurora Apr 27, 2025, 6:29 AM

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That it will be embedded?

tough blaze Apr 27, 2025, 6:29 AM

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No, the NUMBER

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Odd or even?

pale aurora Apr 27, 2025, 6:30 AM

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even

tough blaze Apr 27, 2025, 6:30 AM

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And 15 + an even number = ?

pale aurora Apr 27, 2025, 6:30 AM

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Odd

tough blaze Apr 27, 2025, 6:30 AM

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So, the total number of sub palindromes is always?

pale aurora Apr 27, 2025, 6:31 AM

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odd?

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also for part c) it would just be a string of the same letter
i found a pattern earlier
AA -> 1 (number of subpalindromes)
AAA -> 3
AAAA -> 6
AAAAA -> 10
AAAAAA -> 15
and so on
so the smallest length with at least 30 would be 8 i believe

is this correct?

tough blaze Apr 27, 2025, 6:33 AM

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No, but we’ll come back to that

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Did you understand b?

pale aurora Apr 27, 2025, 6:34 AM

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no

tough blaze Apr 27, 2025, 6:35 AM

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pale aurora odd?

You said the total number of sub palindromes is always odd

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What is 30?

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Even or odd?

pale aurora Apr 27, 2025, 6:35 AM

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so does that automatically disprove that it is 30

tough blaze Apr 27, 2025, 6:35 AM

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Yes

pale aurora Apr 27, 2025, 6:35 AM

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so the total amount applies to all palindromes?

tough blaze Apr 27, 2025, 6:36 AM

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Yes it is always 15 + some even number

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Some even number can also be 0

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So it’s always odd

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So never 30

pale aurora Apr 27, 2025, 6:36 AM

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Why 15?

tough blaze Apr 27, 2025, 6:36 AM

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pale aurora Perhaps 15?

.

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Those are the ones formed by the central axis of symmetry

pale aurora Apr 27, 2025, 6:37 AM

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so is it not always 15 or not?

tough blaze Apr 27, 2025, 6:38 AM

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The number of sub palindromes is always 15 + some even number, the even number CAN be 0, but needn’t be

pale aurora Apr 27, 2025, 6:38 AM

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why tho?

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15 is very specific

tough blaze Apr 27, 2025, 6:39 AM

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You said it yourself. It’s the least number of sub palindromes a palindrome of length 30 HAS to have

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Or it wouldn’t be a palindrome

pale aurora Apr 27, 2025, 6:39 AM

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so it applies to the 30

tough blaze Apr 27, 2025, 6:39 AM

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Yes

pale aurora Apr 27, 2025, 6:39 AM

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so where does the addition of the even number come from?

tough blaze Apr 27, 2025, 6:40 AM

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There CAN be additional sub palindromes, since 15 is the LEAST.

pale aurora Apr 27, 2025, 6:41 AM

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why is it always even

tough blaze Apr 27, 2025, 6:41 AM

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tough blaze Does that make sense? If I have another axis of symmetry at some position x, the...

Look at this message and read on

pale aurora Apr 27, 2025, 6:42 AM

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that doesn't say why you always add an even num

tough blaze Apr 27, 2025, 6:42 AM

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tough blaze So now, if there is another sub palindrome, besides the aforementioned 15, there...

.

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2n is always even, where n is a whole number

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And there are always 2 copies of any other sub palindrome (other than the 15 that is)

pale aurora Apr 27, 2025, 6:44 AM

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so you're saying if there's a palindrome on one side, there is another one on the other?

tough blaze Apr 27, 2025, 6:44 AM

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Yes

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The same one too

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Because by definition of a palindrome, the entire thing is symmetrical along the centre

pale aurora Apr 27, 2025, 6:45 AM

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so i would i go writing my solution in an acceptable way

tough blaze Apr 27, 2025, 6:49 AM

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The centre is between the 15th and 16th characters.
So, the 15th character = 16th character and 14th = 17th and so on
So, there are 15 sub palindromes along this axis. (Of length 2,4,6,…,30)
Any other axis of symmetry forming a palindrome at say,x, would have a copy at 31-x.
So, all the other axes contribute an even number of sub palindromes.
So total number of sub palindromes = 15+some even number which is an odd number.

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Since 30 is an even number, the total number of sub palindromes can’t be 30

pale aurora Apr 27, 2025, 6:49 AM

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so i write it like that?

tough blaze Apr 27, 2025, 6:50 AM

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I mean, yeah, but do you understand?

pale aurora Apr 27, 2025, 6:50 AM

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ok yes

tough blaze Apr 27, 2025, 6:51 AM

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My answer doesn’t have the proper notation, since I don’t know what is expected of you, so you have to write it in your own words

pale aurora Apr 27, 2025, 6:51 AM

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so this is correct for (b) and CABBBAC is correct for (a)
could you check my solution for (c)
(lemme get it up)

tough blaze Apr 27, 2025, 6:51 AM

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It’s wrong, 8 is wrong

pale aurora Apr 27, 2025, 6:51 AM

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ok for part c) it would just be a string of the same letter
i found a pattern earlier
AA -> 1 (number of subpalindromes)
AAA -> 3
AAAA -> 6
AAAAA -> 10
AAAAAA -> 15
and so on
so the smallest length with at least 30 would be 9 i believe

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can you recheck the solution?

tough blaze Apr 27, 2025, 6:52 AM

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Yeah, 9

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What is it exactly?

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How many sub palindromes?

pale aurora Apr 27, 2025, 6:52 AM

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Its a bit like triangle

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Because you start with the whole then it adds on

tough blaze Apr 27, 2025, 6:54 AM

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Ok, but how many sub palindromes does a palindrome of length 9 have?

pale aurora Apr 27, 2025, 6:59 AM

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it is basically a triangle

tough blaze Apr 27, 2025, 6:59 AM

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Huh? I asked for the number

pale aurora Apr 27, 2025, 7:01 AM

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it should be the 8th triangular number which is 36
the one before that is 28

tough blaze Apr 27, 2025, 7:01 AM

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Yeah

pale aurora Apr 27, 2025, 7:02 AM

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so these are right answers for a-c?
a) CABBBAC
b) no such palindrome exists because a palindrome of length 30 must contain an odd number of subpalindromes (30/2 = 15 then add an even)
c) AAAAAAAAA (9)

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Please answer if so, please answer yes or no. For (d), i dont really get how to do it?

tough blaze Apr 27, 2025, 7:04 AM

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Correct

pale aurora Apr 27, 2025, 7:04 AM

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For D can you check this solution and suggest how i can back up this answer without myself just guessing it CBAAAAAAAABC

tough blaze Apr 27, 2025, 7:06 AM

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That’s actually correct

pale aurora Apr 27, 2025, 7:07 AM

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I randomly guessed it. can you help me write my solution so it is methodical and not guessing?

tough blaze Apr 27, 2025, 7:09 AM

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AAAAAAAA has 28 sub palindromes

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As you know

pale aurora Apr 27, 2025, 7:09 AM

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so we just slap two more on right?

tough blaze Apr 27, 2025, 7:09 AM

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So now you just wrap it in CB and BC to get two more sub palindromes

pale aurora Apr 27, 2025, 7:09 AM

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Nice

tough blaze Apr 27, 2025, 7:10 AM

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So, 12

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Is the answer

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It can also be something like ABCCCCCCCCAB

pale aurora Apr 27, 2025, 7:10 AM

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ok i'll review what you said. i think i should understand but i still need to collate this into my own words

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anyways the letters dont matter too much!

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I'll close this once I'm finished

#Subpalindromes question