Skip over navigation
Cambridge University Faculty of Mathematics NRich logo
menu search
  • Teachers expand_more
    • Early years
    • Primary
    • Secondary
    • Post-16
    • Events
    • Professional development
  • Students expand_more
    • Primary
    • Secondary
    • Post-16
  • Parents expand_more
    • Early Years
    • Primary
    • Secondary
    • Post-16
  • Problem-Solving Schools
  • About NRICH expand_more
    • About us
    • Impact stories
    • Support us
    • Our funders
    • Contact us
  • search

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Developing positive attitudes
  • Cross-curricular contexts

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics
  • Mechanics
  • Calculus

For younger learners

  • Early Years Foundation Stage

Water Pistols

Age 16 to 18
Challenge Level Yellow starYellow starYellow star
  • Problem
  • Getting Started
  • Student Solutions
  • Teachers' Resources

There are n people in a field and no two pairs of people are at the same distance apart. Everyone has a water pistol and shoots at and hits the nearest person to them. Show that if n is even everyone may get wet but if n is odd we can be sure that there will be someone who does not get wet.

Here are the solutions to the different parts of the problem from three different people.

Roy M. proved that for an even number of people there must be some circumstances where they all get wet. "For the first problem it is possible that everyone is placed in pairs so that around the field each person shoots, and gets shot by, their pairs. Since there is an even number of people this will work and is therefore a possibility."

Gary from Winchester College proved that when there are three people one stays dry: "Let them be X, Y and Z with XY the shortest distance, thus X and Y each get shot by the other, we are left with 1 shot and 1 person Z, as he cannot shoot himself he has to shoot the one who is nearest to him...whoever that is, and Z is left unshot!"

Now suppose n is an odd number greater than 3. Chen of The Chinese High School uses an argument that reduces the number under consideration from n to n-2 which is the basis for a proof of this result by mathematical induction. "Since the distance between any two people is unique, we can consider the shortest length between 2 people. Then, these two people necessarily spray water to each other, i.e. they are both wet. Now, consider the remaining (n-2) people. If anyone of the (n-2) people spray water at the first 2 people, then there would be at least one person who is not wet. Hence, we consider the second shortest length among these n people (and this line does not connect to the 2 people who are previously chosen) and the two people which the length connects also spray water at each other. By continuing this sequence of actions till there are 3 people remaining, we see that it is impossible for everyone to be wet when n is odd, as the case for n=3 has already been proved to be impossible."


You may also like

Fixing It

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

Be Reasonable

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

  • Tech help
  • Accessibility Statement
  • Sign up to our newsletter
  • Twitter X logo

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

University of Cambridge logo NRICH logo