Puzzle
3 minute read

Ponder This Challenge - December 2018 - Two circles multiplication create a square

You get an integer number as a challenge and your goal is to make it a square of an integer number.
To do that, you use two circles with four integers each.
You can change the number you got in steps. In each step we multiply it by the number in the bottom of one of two circles and rotate this circle clock-wise to its next number.

For example, if the two circles are [3,14,15,92] and [6,5,3,5] (the first number is initially at the bottom) then we can solve the input number 42 (i.e. make it a square) by multiplying as follows:

1st circle [3,14,15,92]    42*3=126
2nd circle [6,5,3,5]    126*6=756
2nd circle [5,3,5,6]    756*5=3,780
2nd circle [3,5,6,5]    3780*3=11,340
2nd circle [5,6,5,3]    11340*5=56,700
1st circle [14,15,92,3]    56700*14=793,800
2nd circle [6,5,3,5]    793800*6=4,762,800
1st circle [15,92,3,14]    4762800*15=71,442,000
2nd circle [5,3,5,6]    71442000*5=357,210,000
and getting a square (357,210,000 = 18,900**2).

BTW, There is a much simpler solution, can you find it?

Your challenge, this month, is to find two circles (with four numbers each) such that the same initial state will allow you to solve at least 2,187 different integers between 1 and 65,536.
To make the problem more interesting, you must not use "boring" circles. A circle is called boring if the set of solvable numbers using only this circle is closed under multiplication (i.e. if you can solve X and Y using only this circle, then you can also solve X*Y with it). In the example above, one of the circles (which?) is boring.

Solution

  • Every natural number can be represented as a multiplication of powers of prime numbers.
    The exponents of this representation, modulo 2, can be viewed as a linear vector space.
    Each “boring” cycle is a linear sub-space, and therefore it is easy to combine two sub-spaces of dimension 3 to a sub-space of dimension 6.
    The interesting fact is that one can combine two sets that are *not* linear sub-space together into a linear sub-space.

    Many of you found a boring solution with 2,251 solvable numbers, like [[3, 7, 11, 13], [2, 5, 14, 65]].
    George Lasry, Deron Stewart, Jingran Lin, Andreas Stiller, Alper Halbutogullari and Oscar Volpatti found solutions with 2,556 solvable numbers such as: [[14, 231, 374, 34], [13, 1482, 95, 10]].
    Alper Halbutogullari also found a solution with exactly 2,187 solvable numbers: [[10, 5, 13, 39], [17, 51, 7, 66]].

Solvers

  • *Alper Halbutogullari (28/11/18 03:46 PM IDT)
  • Bert Dobbelaere (28/11/18 11:55 PM IDT)
  • *Arthur Vause (29/11/18 12:53 PM IDT)
  • *Eden Saig (29/11/18 11:35 PM IDT)
  • *Dominik Reichl (30/11/18 12:35 AM IDT)
  • *Deane Stewart (30/11/18 05:48 AM IDT)
  • *Deron Stewart (30/11/18 05:47 PM IDT)
  • *Lorenz Reichel (30/11/18 11:08 PM IDT)
  • JJ Rabeyrin (01/12/18 12:01 AM IDT)
  • George Lasry (01/12/18 06:04 AM IDT)
  • *Hugo Pfoertner (02/12/18 03:39 PM IDT)
  • *Justin Fletcher (03/12/18 02:59 AM IDT)
  • Harald Bögeholz (03/12/18 09:44 PM IDT)
  • *Graham Hemsley (04/12/18 03:16 PM IDT)
  • *David Greer (05/12/18 02:37 AM IDT)
  • *Thomas Egense (05/12/18 10:26 AM IDT)
  • *Serino Dario (05/12/18 12:38 PM IDT)
  • *Fabio Filatrella (05/12/18 05:40 PM IDT)
  • *Robert Lang (05/12/18 06:22 PM IDT)
  • *Motty Porat (05/12/18 11:57 PM IDT)
  • *Jingran Lin (06/12/18 11:08 PM IDT)
  • Almog Yair & NIr Yanowsky (07/12/18 02:19 AM IDT)
  • *Haye Chan (08/12/18 10:23 AM IDT)
  • *Dieter Dobbelaere (08/12/18 10:48 PM IDT)
  • *Armin Krauß (09/12/18 12:18 AM IDT)
  • *Florian Fischer (09/12/18 02:57 PM IDT)
  • Peter Gerritson (09/12/18 06:57 PM IDT)
  • Danny Antonetti (11/12/18 08:55 PM IDT)
  • Kang Jin Cho (12/12/18 10:52 AM IDT)
  • Muhyeddin Ercan (13/12/18 05:06 AM IDT)
  • *Ritesh Ranjan (13/12/18 02:44 PM IDT)
  • Alex Fleischer (14/12/18 03:21 PM IDT)
  • Daniel Bitin (16/12/18 08:45 PM IDT)
  • *Antonio García Astudillo (17/12/18 05:01 PM IDT)
  • Vladimir Dorofeev (18/12/18 03:59 PM IDT)
  • Thomas Huet (18/12/18 07:51 PM IDT)
  • *Reda Kebbaj (21/12/18 11:38 AM IDT)
  • Charles Joscelyne (21/12/18 05:22 PM IDT)
  • *Andreas Stiller (22/12/18 04:20 AM IDT)
  • Vamsi Krishna (22/12/18 08:16 AM IDT)
  • Dan Dima (22/12/18 06:22 PM IDT)
  • Andreas Stiller (23/12/18 01:09 PM IDT)
  • *Daniel Chong Jyh Tar (24/12/18 04:45 PM IDT)
  • *Andreas Knüpfer (29/12/18 11:06 PM IDT)
  • *Alexander Daryin (30/12/18 12:27 AM IDT)
  • Frank Mayer (30/12/18 08:37 AM IDT)
  • *Radu-Alexandru Todor (31/12/18 02:39 PM IDT)
  • *David F.H. Dunkley (31/12/18 07:25 PM IDT)
  • *Li Li (01/01/19 12:00 PM IDT)
  • Pieter Vandercammen (01/01/19 03:00 PM IDT)
  • Todd Will (01/01/19 07:26 PM IDT)
  • *Oscar Volpatti (02/01/19 01:35 AM IDT)

Related posts