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User talk:Shalomkeshet

Revision as of 13:35, 17 March 2025 by Shalomkeshet (talk | contribs) (→‎Problem 2)
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Welcome to Shalom Keshet's

Mathematical Challenge of Christmas Cheer (MCCC) [2025]

Merry Christmas ladies and gentlemen, today I have procured yet another set of Jolly Problems for you to solve this year, good luck!


Problem 1

$\mbox{An } n\mbox{-digit number }N \mbox{ is considered the BEST }n\mbox{-digit number if it follows the following conditions:}$

$\bullet\ \mbox{Consider a function }f(x) = \left\lfloor{\sqrt{\sqrt{\sqrt{x}}}}\right\rfloor \\\bullet\ \mbox{Now, let }f(N) = a\mbox{ , then, if you remove the last digit of }N\mbox{ and apply }f\mbox{ on it, you get }a-1 \\\bullet\ \mbox{Repeat the same process with the new number, and you get }a-2 \\\bullet\ \mbox{Then }a-3 \\\qquad\ \ \ \vdots \\\bullet\ \mbox{Keep doing this until the cycle stops.} \\ \\N\mbox{ is considered the best }n\mbox{-digit number if it is the greatest }n\mbox{-digit number that has the longest cycle.}$

$\mbox{For example, 9999 is the best 4-digit number because }f(9999)=3, f(999)=2, f(99)=1,\mbox{ and }f(9)=1, \\ \mbox{yielding a cycle length of 3, which is the longest possible for 3-digit, and it is the greatest 3-digit number} \\ \mbox{satisfying it.}$

$\mbox{Additionally, what is the smallest }n\mbox{-digit number that follows these conditions?}$

$\mbox{For example, 6561 is the smallest 4-digit number satisfying this because }f(6561)=3, f(656)=2, f(65)=1, \\ \mbox{and }f(6)=1, \mbox{again yielding a cycle length of 3.}$


Problem 2

Test

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