Let's consider a concrete example to illustrate the proof. Suppose we have the sequence
�
=
[
3
,
1
,
4
,
6
,
5
,
2
]
a=[3,1,4,6,5,2].

Identify Upsequence (U):

Longest Upsequence: [1, 4, 5], with length
�
=
3
U=3.
Identify Downsequence (D):

Longest Downsequence: [3, 1], with length
�
=
2
D=2.
Construct Labels (u, d):

Possible labels: (1,1), (1,2), (2,1), (2,2), (3,1), (3,2).
Assign Labels to Elements:

Assign labels based on the length of the ending upsequence and downsequence.
Element 1: (1,1)
Element 2: (1,2)
Element 3: (2,1)
Element 4: (3,1)
Element 5: (3,2)
Element 6: (2,2)
Verify Uniqueness:

Ensure that distinct elements get distinct labels.

but when you have the up or down sequence of two elements, for example

1,3    or   5,4

what you play all the missing elements to complete it

up    4,5,6                  down   3,2,1

for the next spin ?

6,5 for yours example is another down sequ.

I think we don't take U and D from main lines as we have A and B so one will be for U and the other for D!!!

 

I really can't understand the application of Erdos. After the 9th spin on the example it has to be 4 thats only the proof. That 4 can be any number!!!

With Erdoz you cannot predict!
Db

Quote from: donik7777 on Jan 02, 09:51 AM 2024With Erdoz you cannot predict!
Db

We know that! Who said about predict!

Ok bare with my poor trying on this,if we have N^2+1 variables; and let the N^2+1 variables denoting the length of the longest increasing subsequences be pigeons and the (n) length values be holes then using the PHP we can conclude that there exist n+1  length with the same value;

The example below is selected carefully so we don't have an increasing subsequence of length n=3 (4) so 10#;
   
After finding all the LU sequences what the PHP say , at least n+1 off them must be the same;

Alex, the problem is that here we won't have other answers, whoever gave, you'll perhaps only have other riddles to bang your head who knows where... As if by helping, in the end, they would win less. no one doubts the help given, but the reality is that this help consists only of continuous riddles. For me this discussion ends here, after only starting in 2018.. I will no longer follow this post

I agree with you Leo, I just wanted to share that as it has something interesting but the guru's won't help except turn in and piss some puzzle for us.. At the and life goes on with or without this.. This thread died...

giuseppe, excellent choice 😅 I have been following this page for some time although as you know I don't play kabbalistic or mathematical methods, out of curiosity I followed to see where the desire for protagonism of certain subjects who live by riddles goes... so I understand you..👍
good evening 😉