Mathematics
Olympiad Mathematics
Junior
Water Pouring Puzzles

Water Pouring Puzzles

Theory

Water pouring puzzles are classic brain teasers where you have to measure an exact amount of liquid using only a few unmarked containers. You can fill them, empty them, or pour from one to another—but only until one is completely full or completely empty.

They look simple at first, but solving them takes careful thinking and a bit of creativity. That’s why they’re a favourite both for children who love puzzles and for teachers looking for engaging problem-solving activities.

The Story

Imagine stepping into an alchemist’s laboratory. The shelves are packed with jars, bottles, and flasks, but not a single one has markings to measure with. To finish the experiment, you must measure the ingredients exactly. One drop too many, and the potion fails!

This is the world of water pouring puzzles.

Example Puzzle

A river flows beside the lab. You have two barrels: one holds 3 gallons, the other 5 gallons.

How can you measure exactly 4 gallons of river water?

Water Pouring Puzzle

Measure exactly 4 gallons using a 3‑gallon and a 5‑gallon barrel.

3 gal
5 gal
Step-by-Step Solution
  1. Fill the 5-gallon barrel from the river.
  2. Pour into the 3-gallon barrel until it’s full (leaving 2 gallons in the big barrel).
  3. Empty the 3-gallon barrel into the river.
  4. Pour the 2 gallons from the big barrel into the small barrel.
  5. Refill the 5-gallon barrel.
  6. Pour into the 3-gallon barrel until it’s full (only 1 gallon needed).

Now the 5-gallon barrel has exactly 4 gallons.

Why It Works

Behind the puzzle is a neat bit of mathematics: whether a certain amount can be measured depends on the greatest common divisor (GCD) of the container sizes. With 3 and 5 gallons, the GCD is 1 - so you can measure any whole number of gallons from 1 to 8.

‍

Problems

  1. How to measure exactly 2 liters of water: a) If you have 4 and 5-liter bowls. b) If you have 4 and 3-liter bowls.
  2. Considering three bowls with the following capacities:In bowl A (8 liters capacity), there are 5 liters of water.In bowl B (5 liters capacity), there are 3 liters of water.In bowl C (3 liters capacity), there are 2 liters of water.Can you measure exactly 1 liter, using only 2 pours?
  3. If you need precisely 4 liters of water for your special formula for the award-winning roses during your gardening, but you only have a 5-liter and a 3-liter bowl, and have access to an infinite amount of water, how would you measure exactly 4 liters?
  4. A river flows beside the laboratory. Using two barrels of 3 gallons and 5 gallons, how can you measure exactly 4 gallons of river water?
  5. In a special flask there are 12 milliliters of sulfuric acid, and you also have two resistant measuring cups of 5 ml and 7 ml. How can you obtain exactly two equal portions of 6 ml each? (The acid will dissolve any other container in the lab.)
  6. An alchemist has collected 8 salamander tears (a precious substance) in one vessel. He also has two empty flasks of 2 tears and 3 tears capacity. How can he measure exactly 4 tears? Remember: the tears evaporate quickly! He has time for only three pourings before the substance disappears.
  7. Another ingredient of the elixir is cobra’s blood. A bowl contains 10 spoons of snake blood. The scientist has two ladles of 3 spoons and 4 spoons capacity. How can he measure exactly 5 spoons of blood? The solution must be done in no more than 5 pourings, otherwise the blood will coagulate and become useless.
  8. In the basement of the lab grow mandrakes, and there is an unlimited supply of mandrake extract. Using the 5 ml and 7 ml measuring cups from problem 2, how can you measure exactly 4 ml of extract? But beware! At no point can any cup contain exactly 3 ml—otherwise the mandrakes will scream and destroy the laboratory.
  9. In the furnace there is a cauldron with 9 liters of molten tin. During the experiment, it must be added to the elixir in three equal portions of 3 liters each, at fixed time intervals. You have only three fireproof goblets of 5 liters, 4 liters, and 2 liters. How can you prepare exactly three portions of 3 liters each?
Ages
Parity
Worst Case Scenario
Racing
Percentages
Sums and averages
Equations
Pigeonhole principle
Productivity
Divisibility
Strategies in Games
River Crossing Riddles
Mathematics in Chess
Cryptarithms
Olympiad BIDMAS
Weighing Puzzles
Water Pouring Puzzles
Knights and Liars
Magic Square
Reductio ad impossibile or Proof by contradiction
Base Numbers Problems
Patterns in Numbers and Digits
Matchsticks
Invariant Principle
Method of Colouring
Induction Principle
Patterns
Symmetry
Heads and Feet
Ratio
Number Theory
Combinatorics (Primary)
Gauss's Method and the Sum of Consecutive Numbers
Paths
How many shapes?
Maximum and Minimum
Exponents (Last Digit)
Rationalily
Triangle Inequality
GCD and LCM
Extreme Principle
Remainders of Squares and Cubes
Arithmetic Operations (Primary)
Algebraic Identities
AM–GM Inequalities

FAQ

Find quick answers to common questions about our lessons, pricing, scheduling, and how Exact Science can help your child excel.
Where do you hold your classes?
We hold our classes online or on-site on Saturdays at our branch in Pimlico Academy, London.
You can find our timetable here.
What do you need to start learning online?
For lessons you only need a computer or phone with a microphone, camera and Internet access. Wherever you are - in London, Nottingham, New York or Bali - online lessons will be at hand.
When can I take the trial lesson?
You can get acquainted with the school at any time convenient for you. To do this, just leave a request and sign up for a lesson.
What should I expect from the trial lesson?
The trial lesson is a 30-minute online session designed to get a sense of how your child approaches mathematical thinking and problem solving. (In practice, it often runs a bit longer if the student is engaged!)

We typically explore a range of fun and challenging problems drawn from competitions. We adapt the difficulty based on how the student responds, aiming to make it both accessible and stimulating.

After the session, we’ll have a quick conversation with the parent to share observations and suggest a personalised path forward.
I can't attend class, what should I do?
It is OK, it happens! Students have the opportunity to cancel a lesson up to 8 hours before the scheduled time without loss of payment. So you can reschedule it for a convenient time, and the teacher will have the opportunity to
I don't have much free time, will I have time to study?
Learning can take place at your own pace. We will select a convenient schedule and at any time we will help you change the schedule, take a break or adjust the program.
How long is one lesson?
All classes last 1 hour.

Meet our team

Our teachers will tell you how to prepare for exams, help you cope with difficult tasks and win the Olympiad
‍
They will tell you about the pitfalls of exams and the most common mistakes, and explain how to avoid them
George Ionitsa
Founder &
Maths and Coding Coach

What our students and parents say about learning with us

"Olympiad Maths Lessons helped me a lot to get the Gold medal in Junior Maths Challenge"
St. Paul's Student
"Thanks to the 'Data Science' and 'Coding in Python' lessons I got accepted to my dream university."
Michael
Data Science Student
Warwick University
"Great courses, which thoroughly explained topics beyond the capability of the GCSE answer sheet. Thanks so much."
Ivan
GCSE Student in Dubai
"Financial Mathematics! Best course to understand Python and Mathematics behind Finance!"
Gleb
VC Investor
"We got silver in PMC! Thanks George!"
Mum of St. Paul's Student
Prepare for the Primary Maths Challenge
"My daughter took a batch of 10 classes with George to understand Python with Turtle. I found George extremely knowledgeable and accessible."
Dad of Latymer School Student
Python with Turtle