Brain Teasers - Lethal Duel

Solved 4.7k times
4.6 5 54
Difficulty: Intermediate | Style: Brain Teaser | Topics: Brain teaser

Problem Definition



Three men, Adam, Bertie and Charlie, agree to take part in a lethal three-way duel. They each have a pistol and several rounds of ammo, and they draw lots to see who will shoot first and who second. Then they stand an equal distance apart, and they take turns to fire. When it is a player's turn, he can take one shot at any target of his choice. Then, the turn passes to the next player. The duel continues until only one player is left alive.

For simplicity, we can ignore the possibility of a player becoming injured and unable to shoot: a player is either alive or dead. We also assume that each player chooses to play in such a way as to give himself the best chance of winning.

Now Adam and Bertie are crack shots and will always get a kill, but Charlie is only a moderate shot and will only get a kill two thirds of the time. All three men know this.


Of the three men, who has the best chance of winning the duel?



A man was found murdered on Sunday morning. His wife immediately called the police. The police questioned the wife and staff and got these alibis:

  • The Wife said she was sleeping.
  • The Cook was cooking breakfast.
  • The Gardener was picking vegetables.
  • The Maid was getting the mail.
  • The Butler was cleaning the closet.


The police instantly arrested the murderer. Who did it and how did they know?



Three closed boxes have either white marbles, black marbles or both, and they are labeled white, black and both. However, you're told that each of the labels is wrong. You may reach into one of the boxes and pull out only one marble.


Which box should you remove a marble from to determine the contents of all three boxes?


Brain Teasers are short questions with difficult logical or creative answers. They are a good way to test your analytical skills and capacity to think out of the box. These short questions also show the interviewer your ability to elaborate on difficult tasks from the real world. Each Brain Teaser is supposed to be solved by the interviewee in about 5-10 minutes.

If the interviewee gets stuck in a brain teaser (i.e. spends more than 10 minutes), propose to go on to the next one.

Detailed Solution


Paradoxically, although Charlie is the worst shot, he has the best chance of winning the duel, and this holds true whatever the playing order.

Suppose Adam is drawn to fire first. He doesn't want to shoot Charlie because if he does he'll be inviting certain death from Bertie on the next turn. So he opts to shoot Bertie and take his chances that Charlie will miss him. With Bertie now dead, Charlie shoots at Adam and has a two thirds chance of killing him and winning. If Charlie misses then it's Adam's turn again and he will kill Charlie. So Charlie has a two thirds chance of winning and Adam only one third.

If Bertie is drawn to fire first then the same reasoning applies. Bertie will shoot Adam first and will have a one third chance of winning, while Charlie will have a two thirds chance.

What if Charlie is drawn to fire first? Charlie actually doesn't want to kill either of the others because if he does then it means certain death for him. So he elects to fire his pistol into the air, ensuring that he misses. The situation then reduces to one of the previous cases.

So overall, Charlie has a two thirds chance of winning the duel, while Adam and Bertie share a one third chance between them.


It was the Maid. She said she was getting the mail. There is no mail on Sunday!


Pick a marble from the box labeled “both”. Because “both” is the surely a wrong label, you know the real label of the box by the color you picked. Suppose it is black.

Additionally the white label has to be wrong, too. As the old “both” box is now “black”, the only box left than can be white is the old “black” box.

That leaves the old “white box” to be the new “both” box.

Related consulting question(s)

Hey, I am new to this too, but I think I have some ideas for your problem. Your approach looks good, but I think you have a mistake in how you time the trains. You say that there is one train e... (more)

How can I improve my speed of solving case studies/brainteasers?
Francesco gave the best answer on Oct 12, 2017 - 1 answer
#1 Expert for Coaching Sessions (2.100+) | 1.000+ Reviews with 100% Recommendation Rate | Ex BCG | 6 Years Coaching Experience
Book a coaching with Francesco

100% Recommendation Rate

2,173 Meetings

2,281 Q&A Upvotes

USD 319 / Coaching

Hi Anonymous, there may be several reasons for that; some of them, in order of frequency, are: Your structure is not complete enough – you are just using not MECE/ not complete structure... (more)

Brain Teaser: Solar panels
Sidi gave the best answer on May 14, 2018 - 1 answer
McKinsey Engagement Manager & BCG Consultant | Interviewer at McK & BCG for 7 years | Coached 80+ candidates secure MBB offers
Book a coaching with Sidi

100% Recommendation Rate

274 Meetings

1,654 Q&A Upvotes

USD 319 / Coaching

Hi Anonymous, first of all - this is not a brain teaaser, but an ordinary estimation question! Here is a high-level approach: I would start with the second question, because it is most straight forw... (more)

How Many Hair Salons In New York?
An (Jack)
gave the best answer on Jun 06, 2018 - 3 answers
Google Strategy Manager | Ex- BCG Consultant | References Available
Book a coaching with An (Jack)

100% Recommendation Rate

23 Meetings

96 Q&A Upvotes

USD 219 / Coaching

Hey there Here is one approach: # Hair Salons = (# hours required for men's haircuts per year + # hours required for women's haircuts per year) / average service capacity of 1 hair salon (which... (more)

Market size of an online-transportation service such as Uber/grab bike in a day
gave the best answer on Feb 27, 2017 - 2 answers
Specific coaching: company you are applying for; 1st or 2nd round; your professional background; ...
Book a coaching with Marco

100% Recommendation Rate

26 Meetings

55 Q&A Upvotes

USD 109 / Coaching

Hi, here's how I would approach it - Define the geographic area to be covered [the whole world? the U.S.? NYC? Manhattan?...for this example I'll use Manhattan] - Population living there [~ 1... (more)

Related BootCamp article(s)

Brain Teasers

Learn how to tackle brain teasers and what purpose they serve in consulting case interviews

2 Comment(s)

Typical Question Types in Written Tests

Questions among all tests can be categorized in math, GMAT-style, chart reading and brain teaser questions. Each type needs to be approached differently.

4 Comment(s)
4.6 (54 ratings)
4.6 5 54
Case exhibits

No case exhibits yet