Puzzle
11 minute read

Ponder This Challenge - August 2023 - Square-triangular numbers

This puzzle was suggested by Michael Brand - thanks Michael!

A square number is a number of the form n2n^2 for some nn.

A triangular number is a number of the form 1+2+3+…+m1+2+3+\ldots+m for some m.

The number 1225 is both square and triangular: 1225=352=1+2+3+…+491225=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 101010^{10}.

Your goal: Find the number of positive square-triangular numbers below 1010010^{100} (googol).

A bonus "*" will be given for finding the number of positive square-triangular numbers below 101010010^{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 "*"

Solution

  • August 2023 Solution:

    The numerical solutions are 66 for googol and 653124034718489245484923772216924660058607596119243405888292084074025349410485980501955062384795 7752

    for googolplex.
    Since this is a very well-known problem, it can be solved by a quick journey to Wikipedia:

    Of course, it can be argued that part of the fun is finding a formula independently without resorting to known results.

    For googol, a formula is enough; for googolplex some manipluation of the formula is required, which results in a complex floating-point computation. A "*" was given for any good approximation resulting from this computation; a "**" was given to those who obtained an exact solution. The most common way to do this is by using an arbitrary precision floating point library such as mpmath:

    One possible way of solving the problem with general prior knowledge in number theory but not specific for square- triangular numbers is by using the method for solving **Pell's equation** after a suitable reduction of the problem to it (the exact derivation is given in Wikipedia).

    Another simple solution requiring only elementary tools in combinatorics is looking in the first few elements on the sequence:

    1,6,35,204,1189,6930,40391,235416,…1, 6, 35, 204, 1189, 6930, 40391, 235416, \dots

    And the first and second difference seqeunces:

    1,5,29,169,985,5741,33461,195025,…1, 5, 29, 169, 985, 5741, 33461, 195025,\dots

    4,24,140,816,4756,27720,161564,…4, 24, 140, 816, 4756, 27720, 161564, \dots

    It can be seen that the second difference sequence is the original sequence multiplied by 4; with little algebra this gives the recurrence relation

    an=6an−1−an−2a_n = 6a_{n-1} - a_{n-2}

    which can be solved to obtain

    an=αn−βn42a_n = \frac{\alpha^n-\beta^n}{4\sqrt{2}}

    α=3+22\alpha = 3+2\sqrt{2}

    β=3−22\beta = 3-2\sqrt{2}

    For googolplex it gives us

    n=⌊10100+log⁡1042α⌋n = \left\lfloor \frac{10^{100}+\log_{10}{4\sqrt{2}}}{\alpha}\right\rfloor

    We end up with a problem in high-precision floating point arithmetic. A common solution many solvers used was to turn to a suitable library, e.g. Python's mpmath.

Solvers

  • **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)
  • **Vladimir Volevich (1/8/2023 5:17 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)
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  • **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)
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  • **Dan Ismailescu (1/8/2023 6:21 AM IDT)
  • **Blaine Hill (1/8/2023 8:14 AM IDT)
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  • 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)
  • **Vladimir Volevich (1/8/2023 5:17 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)
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  • **Grant Boudreaux (2/8/2023 4:47 AM IDT)
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  • **Xiao Liu (2/8/2023 6:38 AM IDT)
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  • **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)
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  • **Martin Thorne (2/8/2023 7:51 PM IDT)
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  • 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)
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  • *Evert van Dijken (5/8/2023 3:36 PM IDT)
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  • **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)
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  • **A. Ramanujan (7/8/2023 5:37 PM IDT)
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