A riddle - or, if you prefer, the KYR puzzle

[Kyriakos]

Edit: Some important edits. Sorry about that :/

Ok, here is a riddle. It is actually a relatively famous one (although only if you are interested in a specific order of knowledge), and so I did change the symbols in which it was originally expressed (but it means the same).



Let’s call this the KYR puzzle.

We have tree letters: K, Y and R. We can only use those.



Rules:

1.For some reasons it is true that any sentence which is “KY” (uses just those two letters) is true.



2.(Edit, which is final- I double checked in the book...) We also know that if the last letter of a sentence is “Y”, we are allowed to add an R to that sentence, therefore produce true sentences of the form xYR (for example, since we know the sentence “KY” is true, and there “K” is the x which precedes “Y”, we also know the sentence “KYR” is true). Another example, showing that indeed the part before the Y can be any number of symbols: you could go (if you had formed that sentence first as a true one) from KYYYY to KYYYYR.
More examples: If you had formed a sentence KR, you could then form KRR. From KYRRY (if possible to form it), you'd get KYRRYR. Etc.



3.We are also told that if any true sentence begins with “K”, what follows it (the x part; regardless of how many symbols it contains) can be doubled. For example: Since KY is a true, so is KYY, from which follows that KYYYY is true etc. Also, since KYR is true, so is KYRYR, KYRYRYRYR etc. If you had KRR, you'd get KRRRR. In other words, whatever follows K, regardless of how many symbols it has, can be doubled and stay true (assuming the original sentence was true).



4.They tell us, moreover, that if at some part of those sentences, we get thrice the letter “Y” (ie YYY), we can replace it (if we want to) with “R”. Careful: you can't go the other way around: if you have an R somewhere in your sentence, you are not allowed to replace it for III. For example, we saw we can come up with the true sentence “KYYYY”, which (if we wish) we can rewrite as “KRY” or “KYR”. If we had the sentence KYYYYYYYY, it could (eg) become KRRYY or KYYRR.



5. Finally, we are also told that if at some point in those sentences, we get twice the letter “R” (RR), we are allowed to erase it. So (eg) KYYRR becomes KYY.



Now, the riddle! Can you ever come across, using the above rules, the (true, ie produced using the rules given) sentence “KR”?

 

[The_J]

From what do we start?
Because I'd just start from KR, and would be done ^^.

 

[Kyriakos]

You can start from any true sentence You haven't shown KR to be such.

 

[Sarin]

Yes.

Because (x)YR is true, a sentence KYYYYYYR where x=KYYYYY is true, and KYYYYYYR can be rewritten as KRRR, and then shortened to KR.

 

[Samson]

Is there a difference between "we know" and "we are told"?

 

[Kyriakos]

Yes.

Because (x)YR is true, a sentence KYYYYYYR where x=KYYYYY is true, and KYYYYYYR can be rewritten as KRRR, and then shortened to KR.

Nice going I was actually thinking of that, I made the mistake to say that if you have xy, x can be "an entire series of letters". But of course this is not so... If it was, indeed you could do what you said, eg go from ky to kyy to kyyy (which you cannot, sadly...). From kyyy, kr is reachable due to one of the rules. But you can never get kyyy!

By the way, the original creator of the puzzle was very coy in saying "xY" becomes "xYR", since as you may have noticed, x (when it is only a single symbol) can only be K, nothing else This is because you start from KY, and there is no rule to allow K being taken out, so it will be first forever.

And a more important "by the way", with a serious clue: Notice that the Ys (prior to diminishing by replacing them with Rs) are either 1 or a power of 2 (2,4,8 etc).

 

[Kyriakos]

Is there a difference between "we know" and "we are told"?

I was trying to be informal
Basically the first point is the axiom: "KY is true" (set as axiom), while the rest of the points are rules by which you arrive at a theorem, from the starting axiom.

 

[red_elk]

I think it's impossible.
If we set Y=1 and R=3, we'll need to go from K(1) -> K(3), where number in brackets is sum of all Y,R letters after K
2-nd rule is K(X) -> K(X+3)
3-rd: K(X) -> K(X*2)
4-th doesn't change X
5-th: K(X) -> K(X-6)

If we start with KY, then X=1 at start and by applying these rules you can't get X mod 3 = 0, therefore X=3 is unreachable.
X mod 3 will always be 1 or 2. All rules don't change it, except 3-rd rule which flips 1,2 values.

 

[Sarin]

Nice going I was actually thinking of that, I made the mistake to say that if you have xy, x can be an entire series of letters. But of course this is not so... If it was, indeed you could do what you said, eg go from ky to kyy to kyyy (which you cannot, sadly...). From kyyy, kr is reachable due to one if the rules. But you can never get kyyy!

By the way, the creator of the puzzle was very coy in saying "xY" becomes "xYR", since as you may have noticed, x (when it is only a single symbol) can only be K, nothing else

Hm, then this is weird:

3.We are also told that if any true sentence begins with “K”, what follows it (the x part) can be doubled. For example: Since KY is a true, so is KYY, from which follows that KYYYY is true etc. Also, since KYR is true, so is KYRYR, KYRYRYR etc.

The thing is...we're having a bit ambiguity here. Does x mean only the entirety of sentence following K after first application of multiplication as implied by part in italics, only the letter immediately preceeded by K as implied by Rule 2, or arbitrary section of sentence following K as implied by bolded part?

 

[Kyriakos]

Hm, then this is weird:

The thing is...we're having a bit ambiguity here. Does x mean only the entirety of sentence following K after first application of multiplication as implied by part in italics, only the letter immediately preceeded by K as implied by Rule 2, or arbitrary section of sentence following K as implied by bolded part?

You are right. I will have to take 5 min off to actually read if you are allowed to double groups (batches) of different symbols or not, and get back to you!

 

[Kyriakos]

Hm, then this is weird:

The thing is...we're having a bit ambiguity here. Does x mean only the entirety of sentence following K after first application of multiplication as implied by part in italics, only the letter immediately preceeded by K as implied by Rule 2, or arbitrary section of sentence following K as implied by bolded part?

Ok... I now read the thing again, and rewrote parts of the rules... Multiplication is about the entire group of symbols (from 1 to however many) following the (at the absolute start of the sentence!) symbol K. I also wrote new examples. You can (eg) go from KY to KYY. But you also can go from KYR to KYRYR etc.

 

[Kyriakos]

I think it's impossible.
If we set Y=1 and R=3, we'll need to go from K(1) -> K(3), where number in brackets is sum of all Y,R letters after K
2-nd rule is K(X) -> K(X+3)
3-rd: K(X) -> K(X*2)
4-th doesn't change X
5-th: K(X) -> K(X-6)

If we start with KY, then X=1 at start and by applying these rules you can't get X mod 3 = 0, therefore X=3 is unreachable.
X mod 3 will always be 1 or 2. All rules don't change it, except 3-rd rule which flips 1,2 values.

Indeed
We notice that the number of Ys is always a power of 2, since they can only be doubled each time and start from 1. No power of two is perfectly divisible by 3, which is how Ys are taken out (if you have 3Ys, you substitute with an R). So you will always be left with at least one Y, regardless of how many new sentences you form, so will never get a KR.

And while it is very possible I am responsible for leaving less room to others (such as @Sarin ) for answering this (also added the important clue rather too fast...), let me try to make amends by revealing that this is the so-called MU-system puzzle, by Hoftstadter: https://en.wikipedia.org/wiki/MU_puzzle

It was presented in the course of examining typographical systems (systems made of symbols, following some rules) as being something you can form by natural number-laws. While we find the answer to the question: "Is MU a theorem of the MIU system?" (KYR system in my phrasing) by going outside the system (thinking of number theory), it is crucial to notice that the inside of the system is already translatable to number theory, which is what Goedel used (for a more advanced typographical system) to come up with his theorems.

 

[Sarin]

I suspected it to be this way. It was plain to see, as you need odd number, with minimum 3, of Ys to form R, but for the sentence to be true, there has to be either 1 Y or even number of them. But the way you wrote the rules introduced too much ambiguity.

 

[Gori the Grey]

We have tree letters: K, Y and R

Kyr has sucked you all in to thinking this is a logic puzzle, when it's in fact a word puzzle. You guys are looking straight past the obvious answer:

There are no such thing as tree letters!

 

[Kyriakos]

Numbers are supposed to be a tree - I am sure that's what I was thinking

 

[red_elk]

Is there a reason why K,Y,R letters were used? Just random arbitrary choice?

 

[Kyriakos]

No, those are known as the Imperial letters.