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 Socks on a Clothes Line

      Tick-tock...take socks

      12 July, 2010


      socks

      Unfortunately I am lazy and disorganised. One symptom of this is that I can never be bothered to fold up my socks in pairs when they come out of the wash. I just chuck them in the drawer. Another symptom is that I always wake up late for work and end up having to rush. Given that I only have white and black socks, how many socks do I have to grab out of my drawer at random to make sure the collection I've grabbed contains a matching pair?

      This puzzle is part of the Hands-on risk and probability show, an interactive event culminating in Who Wants to be a Mathionaire? workshop sessions, which you can book to perform at your school. The puzzle appeared in the book How any socks make a pair? by Rob Eastaway.


      For some challenging mathematical puzzles, see the NRICH puzzles from this month or last month.

      Solution link
      How many socks make a pair - solution
      • Log in or register to post comments

      Anonymous

      5 January 2011

      Permalink
      Comment

      2+2

      • Log in or register to post comments

      Anonymous

      19 January 2011

      In reply to ans by Anonymous

      Permalink
      Comment

      3
      first one is lets say black
      second is (worst case scenario) white
      third one will make a pair with either the first or second sock

      • Log in or register to post comments

      Anonymous

      20 May 2011

      In reply to 2+2?? by Anonymous

      Permalink
      Comment

      This is a simple application of the pigeonhole principle. If you have k pigeonholes and k+1 pigeons, then one pigeonhole will contain more than one pigeon.

      • Log in or register to post comments

      Anonymous

      27 September 2011

      In reply to 2+2?? by Anonymous

      Permalink
      Comment

      yep

      • Log in or register to post comments

      Anonymous

      29 March 2011

      Permalink
      Comment

      2

      • Log in or register to post comments

      Anonymous

      25 August 2011

      Permalink
      Comment

      4

      • Log in or register to post comments

      Anonymous

      12 November 2011

      Permalink
      Comment

      Likewise, an extension of the problem is 'how many socks do I have to pull out' if I have three colors? Four? n colors? When you see the solution for all of these, the answer, elegant (to me), is always 'colors + 1'. --That last one will always duplicate one of the previous colors for what is in this case defined as "pair". (Not that I would like to pull out six socks [for five colors] each morning, leaving the remains all over the sink, simply because I was too lazy to pair them in the first place.)

      A then more challenging question is 'how many pulls' for two colors, to make THREE identical socks? Or three colors, three socks, etc.

      • Log in or register to post comments

      Anonymous

      21 May 2014

      In reply to More colors by Anonymous

      Permalink
      Comment

      The answer to above problem is c*(n-1)+1. Because the answer is strictly greater than c*(n-1) as the combination c11,c12,...,c1n;c21,c22,...,c2n;...;c(n-1)1,c(n-1)2,...,c(n-1)n (where cij is color j) is a possible outcome and contains no n-group of any color. Now, let us assume that we took out c*(n-1) socks and still there is no n-group than I claim that the outcome is a permutation of above mentioned outcome.
      Proof of claim:
      let ni, for i=1,2,3,...c be number of socks there are which have color j. And assume that there is no n-group.
      Then:
      for all i, ni<=n-1
      summing up for all i:
      sum(ni)<=c(n-1)
      Now let us suppose that color I does not have n-1 number, then the sum(ni)

      • Log in or register to post comments

      Anonymous

      23 December 2011

      Permalink
      Comment

      3
      first lets say white
      second black
      third black or white either one it makes a pair

      • Log in or register to post comments

      Anonymous

      19 May 2012

      Permalink
      Comment

      Haha. Even after reading the question several times, I spent ages trying to figure out how to guarantee the guy gets one of each colour, instead of just an individual pair. Anyway my answer for that (wrong) question was 1/2 all socks + 1. ^0^

      Ho ho.

      • Log in or register to post comments

      Anonymous

      22 September 2012

      Permalink
      Comment

      3
      2 and a spare in cause you get a black and a white so you then will either have a pair of white or a pair of black socks

      • Log in or register to post comments

      Anonymous

      15 December 2012

      Permalink
      Comment

      3
      if sock a is white,
      and sock b is black,
      then sock c can be either black or white and would make a pair

      • Log in or register to post comments

      Anonymous

      23 August 2015

      Permalink
      Comment

      at first I was like 2 but then I thought this is supposed to be hard so then this puzzle got me thinking and I said it depends on the colour so for example you can have 1 black and 1 white you need another black or white sock to make a full pair so therefore the answer is ..... 3 ( in this case ) but logically you need 2 cause you only have 2 feet and to make a PAIR you need 2 like to apples make a pair 2 carrots make a pair etc. And that is my answer and I think most of you will agree !!!!

      By ... Zainab !!!

      • Log in or register to post comments

      Anonymous

      30 October 2023

      Permalink
      Comment

      If the first 2 socks don't match, then the third has to match with one of them as there are only 2 colours

      • 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