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
    • Two soldiers battling

      Spies Passing in the Night

      1 January, 2001
      January 2001

      The Persian and Greek armies march along a straight road at (different) constant speeds. They spy on each other by sending scouts back and forth on foot or on horseback. The scouts also travel at constant speeds (not necessarily at the same speed as each other). A traveller is walking at constant speed along the road between the two armies.

      The Greek army sends out two scouts simultaneously, one on horseback, the other on foot. The Persian army does the same, at a different time. The Greek horseman reaches the Persian army, and immediately turns around to return to the Greek army; the Persian horsemen behaves similarly when he reaches the Greek army. The Greek footscout arrives at the Persian army at the same time as the returning Persian horseman, and the Persian footscout arrives at the Greek army at the same time as the returning Greek horseman.

      On both their outward and return journeys, the two horsemen pass each other at the same time as they pass the traveller. Prove that, on their outward journey, the two footscouts also pass each other at the same time as they pass the traveller.

      You may want to use Pappus' Theorem.


      You can send your solution by e-mail to <plus@maths.cam.ac.uk>.

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

      We have not yet received a correct solution to the PLUS issue 12 puzzle. This one is proving to be difficult to crack!

      Solution link
      Spies Passing in the Night - Solution
      • 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