Inspired from Multilanguage generalization of "What number is that? Asks Grandpa" and What number is that ? Asks Grandpa
In this puzzle we will use the English alphabet of 26 letters. We create a language that uses it and we call it $LANG$. Previously in What number is that ? Asks Grandpa the issue was:
What is the smallest positive integer $N$ for which:
If you take its WORD anagram and subtract the number $N$ itself, you get some positive integer quantity.
Anagram of the number - Number > 0"?
- Hence the smallest legal number in English would be 67 — SIXTY-SEVEN— since 76 — SEVENTY-SIX — is also a legal number, and 76 - 67 is positive.
For $LANG$ things are quite different as each number is written this way:
- $0$ can not be written in $LANG$.
- $i>0$ written in $LANG$ is at least of length $i$.
- $i>0$ written in $LANG$ is of length $i+k$ with $0 \le k\le i$ with probablity $\dfrac{1}{i+1}$.
- All letters of any $i>0$ written in $LANG$ are equally randomly drawn from the English alphabet. Each letter having a probability of $\dfrac{1}{26}$ to be drawn.
Example $i=3$
- $i$ will be composed by $3$ letters at least.
- $i$ will be composed by $3$ letters with probability $\dfrac1{i+3}=\dfrac14$. $i$ could also be composed of $4$, $5$, or $6$ letters with the same probability of $0.25$.
- If after a first draw, $i$ was to be of length $4$, $i$ could be: $aaaa$ or $love$ both with probability $\dfrac{1}{26^4}\simeq 0.0000022$.
Puzzle
You have the choice to select $N > 0$ such that you will have a draw on $I=\{1,\dots ,N\}$. There is a success if and only if there is an anagram on at least one $(i,i')\in I^2$ with $i$ and $i'$ representing two different $LANG$ numbers.
What smallest $N$ would you select to ensure that your probability of success is greater than $0.26$?