December 2024 - Challenge
Denote by \omega_d(n) the size of the set of all natural numbers between 1 and n that have the digit d in their usual decimal representation. For example, \omega_1(1456)=728 and \omega_7(1456)=352.
We see that for n=1456, we have \frac{\omega_1(n)}{n} = \frac{1}{2}, i.e., exactly 50% of the numbers up to 1456 contain the digit 1. For d=7, we have \frac{\omega_7(n)}{n} = \frac{22}{91}.
Your goal: For each d between 1 and 9, find the maximum value of n for which \frac{\omega_d(n)}{n} = \frac{1}{2}.
An optional goal: Find a simple formula f(d) giving those maximum values for as many values of d as possible.
A bonus "*" will be given for finding the maximum value of n for which \frac{\omega_1(n)}{n} = \frac{3}{4} (i.e., only solve the case d=1. but for \frac{3}{4} this time).
We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!
We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.
If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com
Challenge:
26/11/2024 @ 11:10 AM EST
Solution:
05/01/2025 @ 12:00 AM EST
List Updated:
08/12/2024 @ 15:10 PM EST
Alex Fleischer(2/12/2024 5:59 PM IDT)
*Lazar Ilic(3/12/2024 10:46 AM IDT)
*Lorenz Reichel(3/12/2024 11:10 AM IDT)
*Serge Batalov(3/12/2024 12:30 PM IDT)
Daniel Chong Jyh Tar(3/12/2024 12:34 PM IDT)
*Alper Halbutogullari(3/12/2024 1:50 PM IDT)
*Bertram Felgenhauer(3/12/2024 2:24 PM IDT)
Paul Revenant(3/12/2024 2:40 PM IDT)
*Yi Jiang(3/12/2024 3:12 PM IDT)
Hugo Pfoertner(3/12/2024 4:04 PM IDT)
*Dieter Beckerle(3/12/2024 8:06 PM IDT)
*Dan Dima(3/12/2024 9:47 PM IDT)
Nadir S.(3/12/2024 11:39 PM IDT)
*Yan-Wu He(4/12/2024 2:01 AM IDT)
Victor Chang(4/12/2024 6:04 AM IDT)
Loukas Sidiropoulos(4/12/2024 7:23 AM IDT)
*Guangxi Liu(4/12/2024 8:36 AM IDT)
*Sanandan Swaminathan(4/12/2024 9:21 AM IDT)
Blaine Hill(4/12/2024 11:44 AM IDT)
Evert van Dijken(4/12/2024 12:31 PM IDT)
Michael Liepelt(4/12/2024 12:58 PM IDT)
Gary M. Gerken(4/12/2024 3:38 PM IDT)
Michael Vahle(4/12/2024 4:22 PM IDT)
*Sam Fischer(4/12/2024 4:33 PM IDT)
*Ashfaque Shaikh(4/12/2024 5:41 PM IDT)
Jeffrey Harris(4/12/2024 5:59 PM IDT)
*Hakan Summakoğlu(4/12/2024 6:49 PM IDT)
Karl-Heinz Hofmann(4/12/2024 7:45 PM IDT)
King Pig(5/12/2024 1:56 AM IDT)
William Han(5/12/2024 2:54 AM IDT)
*Arthur Vause(5/12/2024 11:10 AM IDT)
*Juergen Koehl(5/12/2024 11:58 AM IDT)
*Vladimir Volevich(5/12/2024 12:50 PM IDT)
*Latchezar Christov(5/12/2024 3:18 PM IDT)
Cameron Ritson(5/12/2024 11:25 PM IDT)
Mark J. Murphy(6/12/2024 12:13 AM IDT)
*John Spurgeon(6/12/2024 4:04 AM IDT)
Thomas Egense(6/12/2024 12:04 PM IDT)
Fabio Michele Negroni(6/12/2024 7:57 PM IDT)
Noam Zaks(6/12/2024 9:34 PM IDT)
Tamir Ganor & Shouky Dan(7/12/2024 12:48 PM IDT)
Janko Sustersic(7/12/2024 3:49 PM IDT)
Ganghun (Jason) Kim(7/12/2024 6:58 PM IDT)
Amos Guler(7/12/2024 9:17 PM IDT)
Reiner Martin(8/12/2024 2:29 AM IDT)
*Julian Ma(8/12/2024 2:56 AM IDT)
*Abdul Isik(8/12/2024 8:20 AM IDT)