🫗Measuring Puzzles

Some puzzles hand you almost nothing — a few unmarked jugs, some sticks, a single chess piece — and still demand real problem-solving muscle. Measuring puzzles are exactly that: the rules are obvious, the way out hides in the sequence of moves. This lesson's classic has circulated for decades, because what it teaches is not water but method.

Warm up first

Good solvers do not rely on flashes of inspiration; they rely on strategy. Loosen up with a classic:

Place 1–9 so every row, column and diagonal sums to 15

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Remaining

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💡 Show a sample solution↺

Placing 1 to 9 so every row and column sums to 15 (full walkthrough in Magic Square) trains two weapons: planning — which piece goes first, which goes later — and record-keeping — which paths you have already tried. You will need both shortly.

The classic: measure exactly 4 liters

You have a 5-liter jug and a 3-liter jug, neither with markings, and an endless water supply. How do you measure exactly 4 liters?

Pouring at random rarely lands on the answer. Do it honestly, with a state table — after every move, write down how much each jug holds:

Step	Action	5 L jug	3 L jug
0	Start	0	0
1	Fill the 5 L jug	5	0
2	Top up the 3 L jug from the 5 L jug	2	3
3	Empty the 3 L jug	2	0
4	Pour the remaining 2 L into the 3 L jug	0	2
5	Fill the 5 L jug again	5	2
6	Top up the 3 L jug from the 5 L jug	4	3 ✓

The last step is the trick: the 3 L jug is exactly 1 liter short, so just 1 liter leaves the 5 L jug — leaving precisely 4 liters behind.

Work backwards and the path appears

How would anyone find that table? By reasoning backwards from the goal: the target is 4 = 5 − 1, so first you need a 1-liter gap in the 3-liter jug; and 1 = 3 − 2, while 2 is just 5 − 3. Once the chain is linked, running it forward gives the six moves above.

Why trying beats guessing

Measuring puzzles drill a set of habits that transfer everywhere:

1. Write the state down — two numbers (or three) describe the whole situation; writing them keeps you from walking in circles;
2. Never repeat — skip every state you have seen, and you will exhaust all possibilities eventually, so the answer must appear;
3. Reason backwards — from "what I want" back to "what I would need first".

The same thinking clears levels in Logic Puzzles and moves disks in Tower of Hanoi. Puzzles are not about luck — they are about discipline.

Check yourself

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Quick quiz

↺ Try again

1. 1. At the end of step 2 (having topped up the 3 L jug), how much water is in the 5 L jug?

   1 L2 L3 L

2. 2. The 3 L jug already holds 2 L. Pouring from the 5 L jug, how much more can it take?

   1 L2 L3 L

3. 3. What is the most reliable habit for solving a measuring puzzle?

   Trying fast on instinctRecording both jugs after every moveUsing only the biggest jug