All a mouse can answer is a single yes-or-no.
One of 1,000 wine bottles is poisoned. You have a single night. How many mice do you need to find it?
The number of mice is the number of questions, and each question folds the bottles in half.
Experiment
Hands-on experiment
🍷 Jump in! One of these 8 bottles is poisoned (which one is secret). You have 3 mice and one night, one test. Assign which mice sip from each bottle. In the morning, the pattern of dead mice alone must pin down the poison.
💡 Stuck? The secret of the bottle numbers
Write bottle number minus one as a 3-digit binary number. Mouse 1 owns the first digit, mouse 2 the second, mouse 3 the third. Have exactly the 1-digit mice drink, and all eight combinations come out different automatically.
| Bottle | Binary | Drinking mice |
|---|---|---|
| ① | 000 | none |
| ② | 001 | 🐭1 |
| ③ | 010 | 🐭2 |
| ④ | 011 | 🐭1 🐭2 |
| ⑤ | 100 | 🐭3 |
| ⑥ | 101 | 🐭1 🐭3 |
| ⑦ | 110 | 🐭2 🐭3 |
| ⑧ | 111 | 🐭1 🐭2 🐭3 |
📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
The night before a banquet, one of 1,000 wine bottles is poisoned. The poison takes a full night to act, so there is exactly one round of testing. A long-loved classic puzzle.
The first idea is 'one mouse per bottle' — 999 mice. Yet the answer is ten. The secret is numbering the bottles and reading the numbers in binary: each mouse takes charge of one binary digit.
Underneath, this puzzle is the mathematics of counting information. One yes/no answer is one bit. Why computers store everything in 0s and 1s, why twenty questions can pin down a million things — it all grows from this root.
Insight
Insights from the video
“One answer folds the world in half.”
A single yes/no question halves the candidates. Fold 1,000 bottles in half ten times and one bottle remains. Ten mice are ten questions asked in advance.
“Combinations grow faster than counts.”
Each extra mouse doubles the number of distinguishable bottles. Growth by multiplying, not adding — the power of exponents is this puzzle's engine.
Misconception
Common misconceptions
A thousand bottles should need about a thousand mice.
Tying one mouse to one bottle is the trap. Let each mouse sip from many bottles and the whole pattern of deaths becomes one answer. Ten mice give 2¹⁰ = 1,024 patterns — more than enough for 1,000 bottles.
You only get information when a mouse dies.
Survival is information of exactly the same size. 'It didn't die' erases half the bottles too. Even the all-alive pattern points at one specific bottle.
Formula
Writing it as math
Everything reduces to giving each bottle a different combination of mice. Write the bottle numbers in binary and the table completes itself. The next morning, the pattern of dead mice is the poisoned bottle's binary number.
🔬 Formula anatomy — 1,024 answers from ten mice
=
Binary assignment
Write bottle number minus one in binary; the mice on the 1-digits drink from it. Read the dead mice back as binary and the poisoned bottle appears.
The limit of n mice
n mice produce 2ⁿ life-death patterns. With more bottles than that, no assignment can tell them apart.
Flipping it with a log
'How many mice?' is a logarithm question: how many times must you double to pass 1,000 — the very question from the log lab.
In Real Life
Where you meet it in real life
Twenty questions
Twenty yes/no answers distinguish 2²⁰ ≈ 1.05 million things. That is why the game almost always wins.
Binary search
Looking up a word, you open the dictionary in the middle and discard half. A billion records take about 30 steps. The bread and butter of computer search.
Pooled testing
Mix samples from many people, test the batch, then narrow only the positive pools. A real technique that cut test counts during epidemics.
The computer's bit
Photos and songs are, in the end, strings of yes/no. The smallest unit of information, one bit — exactly what one mouse in this puzzle is.
Practice
Practice — conquer the types
✏️ 4 practice problems — solve to conquerTap to solve ▾
To find 1 poisoned bottle among 1,000 in one night, the minimum number of mice is:
What information can a single mouse deliver overnight?
What is the maximum number of bottles 4 mice can distinguish in one night?
If the cellar grew to one million bottles, how many mice would you need?
Watch
Related video
Connection
Concepts connect
Previous concept
The Bridge and Torch
You found the arrangement that saves time; now find the one that saves information.
← The Bridge and Torch labLeads to next
The Prisoners' Hats
Loading information onto a single word, perfected — one parity call saves everyone.
Go to the The Prisoners' Hats lab →Related
Labs worth exploring together
Related lab
Logarithm
'How many mice?' is log₂1000 — the mathematics of counting multiplications.
Go to the Logarithm lab →Related lab