Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • white x on blackboard background

      Mystery number

      6 December, 2012
      Hindu-Arabic numerals: much loved and indispensable.

      Find the number x that satisfies these two properties:

      • The digits of x add up to a number y where x equals y times the number you get when you reverse the digits of y.
      • Reverse the digits of x and find the prime factors of the number you get. Then take the sum of the squares of these prime factors and halve it. Removing the digit 0 from the new number yields back x.

      Here's a hint.



      This puzzle was contributed by Aziz Inan, Professor of Electrical Engineering at the University of Portland. He chose it in honour of a famous mathematician who would have turned 125 this year.

      If you have a puzzle you think might interest Plus readers, please email us!

      Solution link
      Mystery number: Solution
      • Log in or register to post comments

      Anonymous

      12 December 2012

      Permalink
      Comment

      There are two four digit numbers that satisfy condition 1.
      1458: sum digits is 18 and 18 ∙ 81 = 1458
      1729: sum digits is 19 and 19 ∙ 91 = 1729.

      But 1458 does not satisfy condition 2, so we delete it.
      1729 does satisfy condition 2: 9271 = 73 ∙ 127 and
      (73^2 + 127^2) / 2 = 10729.

      • Log in or register to post comments

      Anonymous

      26 December 2012

      In reply to mysterie number by Anonymous

      Permalink
      Comment

      Agreed. However, I wonder whether this solution is unique. It certainly is the only four digit solution but might there be solutions for numbers larger than 9999?

      Kevin

      • Log in or register to post comments

      Anonymous

      3 January 2013

      In reply to Mysterie Number by Anonymous

      Permalink
      Comment

      It is unique. For example, consider a 5-digit number x. The sum of the digits of x represented by y will be maximum 45 (if the number x happens to be 99999). Any 2-digit number y times its reverse (which also consists of maximum 2 digits) will not yield a 5-digit number. In fact, y will consist of maximum 2 digits as long as x has maximum 11 digits. Does this make sense?

      Aziz

      • Log in or register to post comments

      Anonymous

      18 December 2012

      Permalink
      Comment

      Without taking the hint into account, there are two more numbers satifying condition 1:

        1 : sum of digits is 1 and 1 * 1 = 1
      81 : sum of digits is 9 and 9 * 9 = 81

      Neigther of them satifies the second condition.

      • Log in or register to post comments

      Anonymous

      29 January 2015

      In reply to mystery number by Anonymous

      Permalink
      Comment

      Submitted by Anonymous on December 18, 2012.
      Without taking the hint into account, there are two more numbers satifying condition 1:

      1 : sum of digits is 1 and 1 * 1 = 1
      81 : sum of digits is 9 and 9 * 9 = 81

      Neigther of them satifies the second condition.

      • Log in or register to post comments

      Anonymous

      31 August 2013

      Permalink
      Comment

      between 1000 to 9999 only a single number(1729) satisfies both two properties. here is a 'C' Program to find the number.

      #include
      #include

      main()
      {
      long x,y,rem,sum,sum1,t,rev,div,flag;

      for (x=100;x<=99999;x++)
      {
      t=x;sum=0;
      while(t>0)
      {
      rem=t%10;
      sum=sum+rem;
      t=t/10;
      }
      y=sum;rev=0;
      while(sum>0)
      {
      rev=rev*10+(sum%10);
      sum=sum/10;
      }
      if(x==y*rev)
      {
      rev=0;sum=x;
      while(sum>0)
      {
      rev=rev*10+(sum%10);
      sum=sum/10;
      }
      div=2;sum1=1;flag=0;
      while(rev>1)
      {
      if(rev%div==0)
      {
      rev=rev/div;
      if(flag==0){
      sum1=sum1+(div*div);flag++;}
      }
      else
      {
      div=div+1;
      flag=0;
      }
      }
      sum1=sum1/2;
      printf("x=%d lastno=%d\n",x,sum1);
      }
      }
      getch();

      • Log in or register to post comments

      Anonymous

      27 December 2014

      Permalink
      Comment

      Nice puzzle, and a nice homage to Ramanujan and the taxi number that brought GH Hardy to see him in hospital. Fujiwara (1729) showed 1729 is one of four positive integers (with the others being 81, 1458, and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number.

      • Log in or register to post comments

      Anonymous

      25 August 2015

      Permalink
      Comment

      x = 1729
      1+7+2+9 = 19
      y=19
      19x91= 1729

      • Log in or register to post comments

      Alcides Facal

      3 August 2017

      In reply to x in the first part by Anonymous

      Permalink
      Comment

      Number 1729 = 1000 ➕ 729 = 10^3 ➕ 9^3 = 12^3 ➕ 1^3 .

      • Log in or register to post comments

      anymnous

      19 February 2019

      Permalink
      Comment

      Royston designed a game 'The Mystery Number' to help his friends in his study group learn Maths. In this game, 6 times of a mystery number is 2601 more than
      1
      3
      of the same mystery number. Find the mystery number.

      • Log in or register to post comments
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms