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# Ponder This

Welcome to our monthly puzzles.
You are cordially invited to match wits with some of the best minds in IBM Research.

## August 2023 - Challenge

This puzzle was suggested by Michael Brand - thanks Michael!

A square number is a number of the form n^2 for some n.

A triangular number is a number of the form 1+2+3+\ldots+m for some m.

The number 1225 is both square and triangular: 1225=35^2=1+2+3+\ldots +49. Such a number is called square-triangular.

It is easy to see that there are exactly 7 positive square-triangular numbers below 10^{10}.

Your goal: Find the number of positive square-triangular numbers below 10^{100} (googol).

A bonus "*" will be given for finding the number of positive square-triangular numbers below 10^{10^{100}} (googolplex).

A "**" will be given for finding the exact solution to the "*" part; a good approximation (5 significant digits) is enough for a "*"

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: 31/07/2023 @ 12:30 PM EST
Solution: 05/09/2023 @ 12:00 PM EST
List Updated: 28/09/2023 @ 13:00 PM EST

**Lazar Ilic(31/7/2023 2:56 PM IDT)
Prashant Wankhede(31/7/2023 5:27 PM IDT)
*Serino Dario(31/7/2023 5:49 PM IDT)
**Kang Jin Cho(31/7/2023 6:13 PM IDT)
*Reiner Martin(31/7/2023 7:11 PM IDT)
*Lorenzo Gianferrari Pini(31/7/2023 7:27 PM IDT)
Fabio Michele Negroni(31/7/2023 7:49 PM IDT)
**Evan Semet(31/7/2023 7:59 PM IDT)
**Yan-Wu He(31/7/2023 8:42 PM IDT)
*Serino Dario(31/7/2023 8:42 PM IDT)
**Bert Dobbelaere(31/7/2023 8:43 PM IDT)
**Alper Halbutogullari(31/7/2023 11:20 PM IDT)
Richard Gosiorovsky(31/7/2023 11:26 PM IDT)
Sanandan Swaminathan(31/7/2023 11:44 PM IDT)
**David F.H. Dunkley(1/8/2023 12:08 AM IDT)
**Hansraj Nahata(1/8/2023 3:19 AM IDT)
**Jeffrey Harris(1/8/2023 3:41 AM IDT)
**Karl Mahlburg(1/8/2023 4:53 AM IDT)
**Hok Leung Nip(1/8/2023 4:54 AM IDT)
*Daniel Chong Jyh Tar(1/8/2023 5:06 AM IDT)
Gary M. Gerken(1/8/2023 5:17 AM IDT)
**Dan Ismailescu(1/8/2023 6:21 AM IDT)
**Blaine Hill(1/8/2023 8:14 AM IDT)
Alex Fleischer(1/8/2023 8:57 AM IDT)
Kandala Viswakanth(1/8/2023 9:38 AM IDT)
Hakan Summakoğlu(1/8/2023 11:22 AM IDT)
Giogos Kalogeropoulos(1/8/2023 12:57 PM IDT)
**Reda Kebbaj(1/8/2023 2:13 PM IDT)
Christian Pfaffenzeller(1/8/2023 3:26 PM IDT)
*Cameron and John Ritson(1/8/2023 3:39 PM IDT)
Thomas Egense(1/8/2023 4:18 PM IDT)
**Armin Krauß(1/8/2023 6:17 PM IDT)
Evert van Dijken(1/8/2023 6:44 PM IDT)
**Uoti Urpala(1/8/2023 6:46 PM IDT)
**José Eduardo Gaboardi de Carvalho(1/8/2023 6:49 PM IDT)
Victor Chang(1/8/2023 7:46 PM IDT)
*Kindulov Mikhail(1/8/2023 8:02 PM IDT)
**Dominik Reichl(1/8/2023 8:50 PM IDT)
**Kipp Johnson(1/8/2023 9:34 PM IDT)
Michał Franczak(1/8/2023 10:09 PM IDT)
Michael Liepelt(1/8/2023 11:02 PM IDT)
Jordan Rinder(2/8/2023 1:29 AM IDT)
**Andrew Gauld(2/8/2023 2:15 AM IDT)
Matt Cristina(2/8/2023 3:21 AM IDT)
**Grant Boudreaux(2/8/2023 4:47 AM IDT)
Rashid Naim(2/8/2023 5:13 AM IDT)
**Xiao Liu(2/8/2023 6:38 AM IDT)
Kavi Raj(2/8/2023 7:59 AM IDT)
**Marco Bellocchi(2/8/2023 11:29 AM IDT)
**Bertram Felgenhauer(2/8/2023 11:31 AM IDT)
**Arthur Vause(2/8/2023 1:52 PM IDT)
**Luka Mushkudiani(2/8/2023 4:31 PM IDT)
Sri Mallikarjun J(2/8/2023 7:50 PM IDT)
**Martin Thorne(2/8/2023 7:51 PM IDT)
**Latchezar Christov(2/8/2023 10:22 PM IDT)
**K S(3/8/2023 12:14 AM IDT)
William Cho(3/8/2023 2:20 AM IDT)
*Alexander Daryin(3/8/2023 3:30 AM IDT)
*Daniel Bitin(3/8/2023 1:31 PM IDT)
*Amos Guler(3/8/2023 5:32 PM IDT)
Yannick Müller(3/8/2023 6:16 PM IDT)
**Guy Daniel Hadas(3/8/2023 7:27 PM IDT)
**Diane Pham(3/8/2023 8:18 PM IDT)
*Andras Varga(3/8/2023 10:20 PM IDT)
**Štěpán Šmíd(4/8/2023 12:17 AM IDT)
Abhinav Bana(4/8/2023 4:43 AM IDT)
**Dieter Beckerle(4/8/2023 7:55 AM IDT)
Daniel Lincke(4/8/2023 10:41 AM IDT)
**Asaf Zimmerman(4/8/2023 11:29 AM IDT)
**Franciraldo Cavalcante(4/8/2023 2:01 PM IDT)
*Christoph Schmidt(4/8/2023 2:08 PM IDT)
Michael Branicky(4/8/2023 8:32 PM IDT)
**Erik Hostens(4/8/2023 9:11 PM IDT)
Anders Bisbjerg Madsen(4/8/2023 9:27 PM IDT)
**Wolfgang Kais(5/8/2023 4:15 AM IDT)
**Sanandan Swaminathan(5/8/2023 7:02 AM IDT)
*Andreas Knüpfer(5/8/2023 11:34 AM IDT)
David Greer(5/8/2023 3:13 PM IDT)
*Evert van Dijken(5/8/2023 3:36 PM IDT)
**Harald Bögeholz(5/8/2023 8:07 PM IDT)
Joaquim Carrapa(5/8/2023 10:20 PM IDT)
*Cliff Amundsen(5/8/2023 11:14 PM IDT)
**Tim Walters(6/8/2023 12:57 AM IDT)
*Sachal Mahajan(6/8/2023 4:17 AM IDT)
*Tamir Ganor & Shouky Dan(6/8/2023 11:02 AM IDT)
JJ Rabeyrin(6/8/2023 1:45 PM IDT)
**Todd Will(6/8/2023 5:58 PM IDT)
Rashid Zia(6/8/2023 6:35 PM IDT)
**Petro Mudrievskyj(6/8/2023 7:58 PM IDT)
**Mark Beyleveld(7/8/2023 5:21 AM IDT)
Thomas Egense(7/8/2023 8:40 AM IDT)
Ralf Jonas(7/8/2023 3:22 PM IDT)
**A. Ramanujan(7/8/2023 5:37 PM IDT)
HyeongSik Hwang(7/8/2023 7:13 PM IDT)
**Nyles Heise(8/8/2023 4:53 AM IDT)
Paul Revenant(8/8/2023 7:26 PM IDT)
Phil Proudman(9/8/2023 12:54 AM IDT)
Stefan Wirth(9/8/2023 1:06 AM IDT)
**Maksym Voznyy(9/8/2023 6:14 AM IDT)
*Abhinav Bana(9/8/2023 10:37 AM IDT)
Dheeraj Reddy(9/8/2023 8:23 PM IDT)
**Chris Shannon(10/8/2023 8:40 AM IDT)
**Michael Liepelt(10/8/2023 7:50 PM IDT)
Roopa Rajaram(11/8/2023 12:59 AM IDT)
**Dan Mayer(11/8/2023 2:06 PM IDT)
Ismail Farach(11/8/2023 7:03 PM IDT)
**Alain Michiels(11/8/2023 7:23 PM IDT)
*Omid Gheibi(11/8/2023 9:39 PM IDT)
**Lorenz Reichel(11/8/2023 10:24 PM IDT)
**Peter Moser(11/8/2023 11:14 PM IDT)
Jean-Michel Torres(12/8/2023 7:22 PM IDT)
Ray Greiss(12/8/2023 9:57 PM IDT)
Satya Tamby(13/8/2023 3:13 PM IDT)
Laurence Warne(13/8/2023 9:47 PM IDT)
**John Tromp(14/8/2023 12:35 AM IDT)
Diogo Manuel Cavaco(14/8/2023 11:32 AM IDT)
*Shubham Rathi(14/8/2023 11:54 AM IDT)
Deep Kankaria(14/8/2023 12:49 PM IDT)
Rajesh Sathiyanarayanan(15/8/2023 2:01 PM IDT)
**George D. Logan(16/8/2023 4:21 AM IDT)
**Katie Liu(16/8/2023 11:04 AM IDT)
*Lyes Belhoul(16/8/2023 2:11 PM IDT)
**Govind Jujare(16/8/2023 3:55 PM IDT)
**Christoph Baumgarten(17/8/2023 11:24 PM IDT)
**Lazar Ilic(31/7/2023 2:56 PM IDT)
Prashant Wankhede(31/7/2023 5:27 PM IDT)
*Serino Dario(31/7/2023 5:49 PM IDT)
**Kang Jin Cho(31/7/2023 6:13 PM IDT)
*Reiner Martin(31/7/2023 7:11 PM IDT)
*Lorenzo Gianferrari Pini(31/7/2023 7:27 PM IDT)
Fabio Michele Negroni(31/7/2023 7:49 PM IDT)
**Evan Semet(31/7/2023 7:59 PM IDT)
**Yan-Wu He(31/7/2023 8:42 PM IDT)
*Serino Dario(31/7/2023 8:42 PM IDT)
**Bert Dobbelaere(31/7/2023 8:43 PM IDT)
**Alper Halbutogullari(31/7/2023 11:20 PM IDT)
Richard Gosiorovsky(31/7/2023 11:26 PM IDT)
Sanandan Swaminathan(31/7/2023 11:44 PM IDT)
**David F.H. Dunkley(1/8/2023 12:08 AM IDT)
**Hansraj Nahata(1/8/2023 3:19 AM IDT)
**Jeffrey Harris(1/8/2023 3:41 AM IDT)
**Karl Mahlburg(1/8/2023 4:53 AM IDT)
**Hok Leung Nip(1/8/2023 4:54 AM IDT)
*Daniel Chong Jyh Tar(1/8/2023 5:06 AM IDT)
Gary M. Gerken(1/8/2023 5:17 AM IDT)
**Dan Ismailescu(1/8/2023 6:21 AM IDT)
**Blaine Hill(1/8/2023 8:14 AM IDT)
Alex Fleischer(1/8/2023 8:57 AM IDT)
Kandala Viswakanth(1/8/2023 9:38 AM IDT)
Hakan Summakoğlu(1/8/2023 11:22 AM IDT)
Giogos Kalogeropoulos(1/8/2023 12:57 PM IDT)
**Reda Kebbaj(1/8/2023 2:13 PM IDT)
Christian Pfaffenzeller(1/8/2023 3:26 PM IDT)
*Cameron and John Ritson(1/8/2023 3:39 PM IDT)
Thomas Egense(1/8/2023 4:18 PM IDT)
**Armin Krauß(1/8/2023 6:17 PM IDT)
Evert van Dijken(1/8/2023 6:44 PM IDT)
**Uoti Urpala(1/8/2023 6:46 PM IDT)
**José Eduardo Gaboardi de Carvalho(1/8/2023 6:49 PM IDT)
Victor Chang(1/8/2023 7:46 PM IDT)
*Kindulov Mikhail(1/8/2023 8:02 PM IDT)
**Dominik Reichl(1/8/2023 8:50 PM IDT)
**Kipp Johnson(1/8/2023 9:34 PM IDT)
Michał Franczak(1/8/2023 10:09 PM IDT)
Michael Liepelt(1/8/2023 11:02 PM IDT)
Jordan Rinder(2/8/2023 1:29 AM IDT)
**Andrew Gauld(2/8/2023 2:15 AM IDT)
Matt Cristina(2/8/2023 3:21 AM IDT)
**Grant Boudreaux(2/8/2023 4:47 AM IDT)
Rashid Naim(2/8/2023 5:13 AM IDT)
**Xiao Liu(2/8/2023 6:38 AM IDT)
Kavi Raj(2/8/2023 7:59 AM IDT)
**Marco Bellocchi(2/8/2023 11:29 AM IDT)
**Bertram Felgenhauer(2/8/2023 11:31 AM IDT)
**Arthur Vause(2/8/2023 1:52 PM IDT)
**Luka Mushkudiani(2/8/2023 4:31 PM IDT)
Sri Mallikarjun J(2/8/2023 7:50 PM IDT)
**Martin Thorne(2/8/2023 7:51 PM IDT)
**Latchezar Christov(2/8/2023 10:22 PM IDT)
**K S(3/8/2023 12:14 AM IDT)
William Cho(3/8/2023 2:20 AM IDT)
*Alexander Daryin(3/8/2023 3:30 AM IDT)
*Daniel Bitin(3/8/2023 1:31 PM IDT)
*Amos Guler(3/8/2023 5:32 PM IDT)
Yannick Müller(3/8/2023 6:16 PM IDT)
**Guy Daniel Hadas(3/8/2023 7:27 PM IDT)
**Diane Pham(3/8/2023 8:18 PM IDT)
*Andras Varga(3/8/2023 10:20 PM IDT)
**Štěpán Šmíd(4/8/2023 12:17 AM IDT)
Abhinav Bana(4/8/2023 4:43 AM IDT)
**Dieter Beckerle(4/8/2023 7:55 AM IDT)
Daniel Lincke(4/8/2023 10:41 AM IDT)
**Asaf Zimmerman(4/8/2023 11:29 AM IDT)
**Franciraldo Cavalcante(4/8/2023 2:01 PM IDT)
*Christoph Schmidt(4/8/2023 2:08 PM IDT)
Michael Branicky(4/8/2023 8:32 PM IDT)
**Erik Hostens(4/8/2023 9:11 PM IDT)
Anders Bisbjerg Madsen(4/8/2023 9:27 PM IDT)
**Wolfgang Kais(5/8/2023 4:15 AM IDT)
**Sanandan Swaminathan(5/8/2023 7:02 AM IDT)
*Andreas Knüpfer(5/8/2023 11:34 AM IDT)
David Greer(5/8/2023 3:13 PM IDT)
*Evert van Dijken(5/8/2023 3:36 PM IDT)
**Harald Bögeholz(5/8/2023 8:07 PM IDT)
Joaquim Carrapa(5/8/2023 10:20 PM IDT)
*Cliff Amundsen(5/8/2023 11:14 PM IDT)
**Tim Walters(6/8/2023 12:57 AM IDT)
*Sachal Mahajan(6/8/2023 4:17 AM IDT)
*Tamir Ganor & Shouky Dan(6/8/2023 11:02 AM IDT)
JJ Rabeyrin(6/8/2023 1:45 PM IDT)
**Todd Will(6/8/2023 5:58 PM IDT)
Rashid Zia(6/8/2023 6:35 PM IDT)
**Petro Mudrievskyj(6/8/2023 7:58 PM IDT)
**Mark Beyleveld(7/8/2023 5:21 AM IDT)
Thomas Egense(7/8/2023 8:40 AM IDT)
Ralf Jonas(7/8/2023 3:22 PM IDT)
**A. Ramanujan(7/8/2023 5:37 PM IDT)
HyeongSik Hwang(7/8/2023 7:13 PM IDT)
**Nyles Heise(8/8/2023 4:53 AM IDT)
Paul Revenant(8/8/2023 7:26 PM IDT)
Phil Proudman(9/8/2023 12:54 AM IDT)
Stefan Wirth(9/8/2023 1:06 AM IDT)
**Maksym Voznyy(9/8/2023 6:14 AM IDT)
*Abhinav Bana(9/8/2023 10:37 AM IDT)
Dheeraj Reddy(9/8/2023 8:23 PM IDT)
**Chris Shannon(10/8/2023 8:40 AM IDT)
**Michael Liepelt(10/8/2023 7:50 PM IDT)
Roopa Rajaram(11/8/2023 12:59 AM IDT)
**Dan Mayer(11/8/2023 2:06 PM IDT)
Ismail Farach(11/8/2023 7:03 PM IDT)
**Alain Michiels(11/8/2023 7:23 PM IDT)
*Omid Gheibi(11/8/2023 9:39 PM IDT)
**Lorenz Reichel(11/8/2023 10:24 PM IDT)
**Peter Moser(11/8/2023 11:14 PM IDT)
Jean-Michel Torres(12/8/2023 7:22 PM IDT)
Ray Greiss(12/8/2023 9:57 PM IDT)
Satya Tamby(13/8/2023 3:13 PM IDT)
Laurence Warne(13/8/2023 9:47 PM IDT)
**John Tromp(14/8/2023 12:35 AM IDT)
Diogo Manuel Cavaco(14/8/2023 11:32 AM IDT)
*Shubham Rathi(14/8/2023 11:54 AM IDT)
Deep Kankaria(14/8/2023 12:49 PM IDT)
Rajesh Sathiyanarayanan(15/8/2023 2:01 PM IDT)
**George D. Logan(16/8/2023 4:21 AM IDT)
**Katie Liu(16/8/2023 11:04 AM IDT)
*Lyes Belhoul(16/8/2023 2:11 PM IDT)
**Govind Jujare(16/8/2023 3:55 PM IDT)
**Christoph Baumgarten(17/8/2023 11:24 PM IDT)
**Kai Guttmann(18/8/2023 2:22 AM IDT)
**Zhang Zhiqiang(18/8/2023 11:24 AM IDT)
Chiho Haruta(18/8/2023 11:57 AM IDT)
*Stéphane Higueret(19/8/2023 10:49 AM IDT)
*Florian Sikora(19/8/2023 1:32 PM IDT)
**George D. Logan(19/8/2023 8:02 PM IDT)
Anunay Kulshrestha(20/8/2023 4:32 AM IDT)
**Max Aehle(20/8/2023 11:49 AM IDT)
Kandala Srinivas(20/8/2023 8:24 PM IDT)
Soonho You(21/8/2023 5:32 AM IDT)
Quentin Higueret(21/8/2023 1:34 PM IDT)
Connor McCarthy(21/8/2023 5:17 PM IDT)
**Guglielmo Sanchini1(23/8/2023 6:15 PM IDT)
**Govind Jujare(23/8/2023 11:41 PM IDT)
**Dan Dima(24/8/2023 11:42 AM IDT)
Jose Coelho(24/8/2023 5:11 PM IDT)
Michael Tomicich(24/8/2023 8:30 PM IDT)
**Erik Wünstel(25/8/2023 12:47 PM IDT)
**Simone Foderà(25/8/2023 3:31 PM IDT)
**Motty Porat(27/8/2023 12:07 AM IDT)
**Carl Löndahl(27/8/2023 12:21 AM IDT)
*Julien Vincent Pradier(28/8/2023 8:38 PM IDT)
**Teawoo Kim(28/8/2023 10:40 PM IDT)
*Dario Filatrella(29/8/2023 7:34 PM IDT)
Rayan Swidan(29/8/2023 11:08 PM IDT)
Shiva Sud(30/8/2023 12:08 AM IDT)
**Daniel Copeland(30/8/2023 7:23 PM IDT)
*Matt Crawford(30/8/2023 10:07 PM IDT)
**Kay Lee(31/8/2023 5:22 AM IDT)
*Hengfeng Tian(31/8/2023 7:44 AM IDT)
**Evan Hur(31/8/2023 10:13 PM IDT)
**Hugo Pfoertner(31/8/2023 11:13 PM IDT)