December 2023 - Challenge
This riddle was proposed by Hugo Pfoertner - thanks Hugo!
We consider the infinite grid of equilateral triangles with sides length 1, such that (0,0) is a grid point.
In this grid, we consider all circles passing through at least one grid point. The smallest circle has radius 1, hitting the grid point (1,0) and 5 other grid points.
We are interested in all the circles with non-integer radii. In the following illustration, all the circles are drawn, and a blue mark denotes the intersection point of a circle with a non-integer radius along the positive direction of the x-axis.
For every positive integer m, f(m) denotes the number of circles that pass through at least one grid point, with a radius of x such that m < x < m+1.
For example, f(11)=5 and f(42)=19.Your goal: Find a value of m such that f(m)=1,000,000.
Hint: There is such an m which is a base 10 palindrome, but other solutions are accepted as well.
A Bonus "*" will be given for finding **all** the values of m such that f(m)=1,000,000.
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:
28/11/2023 @ 19:10 PM EST
Solution:
05/01/2023 @ 12:00 PM EST
List Updated:
16/01/2024 @ 10:20 AM EST
*Dan Dima(28/11/2023 10:07 PM IDT)
Lazar Ilic(29/11/2023 3:15 AM IDT)
*Alper Halbutogullari(29/11/2023 4:22 AM IDT)
*Bert Dobbelaere(29/11/2023 9:12 AM IDT)
*Franciraldo Cavalcante(29/11/2023 4:01 PM IDT)
*Stéphane Higueret(29/11/2023 6:19 PM IDT)
*Yan-Wu He(29/11/2023 8:11 PM IDT)
*Bertram Felgenhauer(29/11/2023 8:32 PM IDT)
*Richard Gosiorovsky(29/11/2023 9:19 PM IDT)
Reda Kebbaj(29/11/2023 10:00 PM IDT)
Giorgos Kalogeropoulos(30/11/2023 12:28 AM IDT)
David F.H. Dunkley(30/11/2023 1:39 PM IDT)
*Martin Thorne(1/12/2023 4:06 AM IDT)
Victor Chang(1/12/2023 6:50 AM IDT)
*Sanandan Swaminathan(1/12/2023 7:02 AM IDT)
Maksym Voznyy(1/12/2023 8:17 AM IDT)
*Julien Pradier(1/12/2023 8:59 AM IDT)
*Yi Jiang(1/12/2023 9:07 AM IDT)
Dieter Beckerle(1/12/2023 11:52 AM IDT)
*Peter Taylor(1/12/2023 12:39 PM IDT)
*Arthur Vause(1/12/2023 1:46 PM IDT)
Guillaume Escamocher(1/12/2023 3:14 PM IDT)
*Dominik Reichl(1/12/2023 6:25 PM IDT)
*Jeffrey Harris(1/12/2023 7:03 PM IDT)
*Vladimir Volevich(2/12/2023 1:46 AM IDT)
Gary M. Gerken(2/12/2023 3:50 PM IDT)
Daniel Chong Jyh Tar(2/12/2023 3:53 PM IDT)
Laurence Warne(2/12/2023 8:06 PM IDT)
*Ivan Lazaric(2/12/2023 9:45 PM IDT)
Kurt Foster(3/12/2023 12:31 AM IDT)
*John Tromp(3/12/2023 1:21 AM IDT)
Christopher Marks(3/12/2023 6:06 AM IDT)
*Andrew Gauld(3/12/2023 8:19 AM IDT)
Kang Jin Cho(3/12/2023 7:50 PM IDT)
*Vamsi Talupula(3/12/2023 8:28 PM IDT)
*Karl Mahlburg(3/12/2023 8:30 PM IDT)
*Amos Guler(3/12/2023 10:15 PM IDT)
*John Spurgeon(3/12/2023 11:29 PM IDT)
*Tim Walters(4/12/2023 12:10 AM IDT)
*Thomas Ilsche(4/12/2023 12:47 AM IDT)
*Maksym Voznyy(4/12/2023 2:44 AM IDT)
Lorenzo Gianferrari Pini(4/12/2023 8:27 AM IDT)
*Christopher Marks(4/12/2023 3:47 PM IDT)
Adrian Neacsu(5/12/2023 1:31 AM IDT)
Ashfaque Shaikh(5/12/2023 12:50 PM IDT)
Karl D’Souza(5/12/2023 3:02 PM IDT)
Clive Tong(5/12/2023 10:46 PM IDT)
*Marco Bellocchi(5/12/2023 11:51 PM IDT)
*Guglielmo Sanchini(6/12/2023 3:50 PM IDT)
*Daniel Bitin(6/12/2023 4:13 PM IDT)
*Dieter Beckerle(7/12/2023 12:25 PM IDT)
*Mark Beyleveld(7/12/2023 9:54 PM IDT)
*Lorenz Reichel(8/12/2023 10:37 AM IDT)
*Phil Proudman(9/12/2023 3:45 PM IDT)
Paul "Lorxus" Rapoport(10/12/2023 1:29 AM IDT)
Evan Semet(10/12/2023 7:52 AM IDT)
Blaine Hill(10/12/2023 7:55 AM IDT)
*David Greer(11/12/2023 1:03 PM IDT)
*Latchezar Christov(11/12/2023 4:21 PM IDT)
Paul Revenant(11/12/2023 5:42 PM IDT)
Fabio Michele Negroni(11/12/2023 8:38 PM IDT)
Pieter Meijer(11/12/2023 10:23 PM IDT)
Fakih Karademir(12/12/2023 5:26 AM IDT)
*Andreas Knüpfer(12/12/2023 9:32 PM IDT)
*John Mandalas & David Yang(12/12/2023 10:30 PM IDT)
*Nyles Heise(13/12/2023 6:27 AM IDT)
Prashant Wankhede(13/12/2023 7:27 AM IDT)
*Fabio Michele Negroni(13/12/2023 7:40 PM IDT)
*David Dunkley(14/12/2023 11:52 PM IDT)
*Noam Zaks(15/12/2023 12:59 AM IDT)
*Paul Revenant(15/12/2023 3:22 PM IDT)
*Alexander Daryin(15/12/2023 3:29 PM IDT)
*Harald Bögeholz(15/12/2023 5:24 PM IDT)
*Ashfaque Shaikh(16/12/2023 7:10 PM IDT)
Shouky Dan & Tamir Ganor(17/12/2023 1:20 PM IDT)
*Matt Cristina(19/12/2023 11:55 PM IDT)
Cameron Ritson(20/12/2023 4:09 AM IDT)
*Lyes Belhoul(20/12/2023 4:05 PM IDT)
Arvind Venkatesan(20/12/2023 10:45 PM IDT)
Michael Liepelt(21/12/2023 12:19 AM IDT)
Shirish Chinchalkar(22/12/2023 4:11 AM IDT)
*Wenjie Yang(23/12/2023 12:34 PM IDT)
*Sergey Koposov(23/12/2023 5:12 PM IDT)
*Hakan Summakoğlu(25/12/2023 9:31 PM IDT)
Reiner Martin(27/12/2023 2:37 AM IDT)
Andrew Scherbakov(27/12/2023 4:35 AM IDT)
*Daniel Stadtmauer(27/12/2023 8:36 AM IDT)
Christoph Baumgarten(27/12/2023 12:16 PM IDT)
*Motty Porat(28/12/2023 2:41 AM IDT)
Andi Guo & Benjamin Wang & Henry Xian & Isaac Luo(28/12/2023 9:01 PM IDT)
Hamidreza Bidar(29/12/2023 6:12 PM IDT)
Jens Leskinen(30/12/2023 3:06 PM IDT)
*Alex Izvalov(30/12/2023 8:39 PM IDT)
*Armin Krauss(31/12/2023 1:11 AM IDT)
*Zoltan Haindrich(31/12/2023 4:15 PM IDT)
*Oscar Volpatti(31/12/2023 9:30 PM IDT)
Vaskor Basak(31/12/2023 11:06 PM IDT)
Radu-Alexandru Todor(1/1/2024 10:40 PM IDT)
*Daniel Lincke(2/1/2024 1:31 PM IDT)
Maayan Ehrenberg & Ohad Einav & Nir Rosenfeld & Nadav Rubinstein & Eden Saig & Yonatan Sommer(2/1/2024 3:19 PM IDT)
Alain Michiels(3/1/2024 11:40 PM IDT)
Todd Will(3/1/2024 11:55 PM IDT)
*Li Li(6/1/2024 8:16 AM IDT)