Cluedo - Crunching The Numbers

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BBP
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Cluedo - Crunching The Numbers

Post by BBP »

Cluedo and Math: Probabilities

A few days ago, for the second time in my life, I won a game of Cluedo on my first turn. A combination of awe, win, but also slight disappointment and playing another game to get your fix, BF and I got to think about the probability that this happens.

A standard game of Cluedo has 6 weapons, 6 suspects and 9 rooms. One of each category is selected at random (random often isn't random, but that's another story).

That gives us 6*6*9, or 324, possibilities.

However, as it turns out, the chance of guessing correctly on your first round, isn't 1/324. You have a hand of 6 cards in a 3 player game, a hand of 3 cards in a 6 player game and 3-5 cards in anything inbetween.

Taking a 3 player game, and using a Suspect-Weapon-Room format, your cards can be divided in the following ways:

5-0-1 and 0-5-1 - 5 weapons or suspects, and 1 room.In this case you know who or what, this increases your instant win probability dramatically. Now there are only 48 possibilities.

5-1-0 and 1-5-0 - 1 suspect and 5 weapons or the other way round, and no rooms.
Probability is 1*5*9 or 45 possibilities.

4-0-2 and 0-4-2 give 2*6*7 = 84 possibilities
4-1-1 and 1-4-1 give 5*2*8 = 80 possibilities

3-3-0 gives 3*3*9=81 possibilities
3-0-3 or 0-3-3 =108 possibilities

3-2-1 or 2-3-1 gives 3*4*8 = 96 possibilities
3-1-2 or 1-3-2 gives 3*5*7 = 105 possibilites

2-2-2 gives 4*4*7 = 112 possibilies
2-1-3 gives 4*5*6 = 120
2-0-4 gives 4*6*5 = 120

1-1-4 gives 5*5*5 = 125
1-0-5 gives 5*6*4 = 120

0-0-6 gives 6*6*3 = 108

I had often thought about having more rooms being an advantage due to the fact you can't ask them as freely as the others, but putting the numbers together does show that, when you have no rooms, it tends to cut the possible outcomes a lot more than when you have more of them.

With 6 players each has 3 cards. With 3 cards, the division is:
3-0-0 = 162
2-1-0 = 180
2-0-1 = 192
1-0-2 = 120
1-1-1 = 200
0-0-3 = 216

So it's looking like the more rooms you have, the worse off you are.

Stay tuned for what happens after your first turn!

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ZachBlack000
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Re: Cluedo - Crunching The Numbers

Post by ZachBlack000 »

This is exactly what I used to do on a daily-basis, when I had free time on my hands.

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Re: Cluedo - Crunching The Numbers

Post by BBP »

After 1 turn you may learn of either a suspect or a weapon, or a room. What does the realm of possibilities look like after 1 card has been shown on your first turn?

Suspects and weapons are interchangeable.

When you have 5 cards of either weapons or suspects, the amount of possibilities is as follows:
5-0-1 = 48 possibilities.
5-1-0 = 45 possibilities
The card of which you have 5 can't be shown. The balance can be:
5-2-0 = 1*4*9 = 36 possibilities
5-1-1 = 1*5*8 = 40 possibilities
5-0-2 = 1*6*8 = 48 possibilities

4 suspects (or weapons) give:
4-0-2 = 2*6*7 = 84 possibilities
4-1-1 = 2*5*8 = 80 possibilities
4-2-0 = 2*4*9 = 72 possibilities
Upon being shown a suspect, we get:
5-0-2 = 1*6*7 = 42 possibilities
5-1-1 = 1*5*8 = 40 possibilities
5-2-0 = 1*4*9 = 36 possibilities
Upon being shown a weapon:
4-1-2 = 2*5*7 = 700 possibilities
4-2-1 = 2*4*8 = 64 possibilities
4-3-0 = 2*3*9 = 54 possibilities
Upon being shown a room:
4-0-3 = 2*6*6 = 72 possibilities
4-1-2 = 2*5*7 = 70 possibilities
4-2-1 = 2*4*8 = 64 possibilities

With 3 suspects there are:
3-0-3 = 3*6*6 = 108 possibilities
3-1-2 = 3*5*7 = 105 possibilities
3-2-1 = 3*4*8 = 96 possibilities
3-3-0 = 3*3*9 = 81 possibilities
Shown a suspect, this becomes:
4-0-3 = 2*6*6 = 72 possibilities
4-1-2 = 2*5*7 = 70 possibilities
4-2-1 = 2*4*8 = 64 possibilities
4-3-0 = 2*3*9 = 54 possibilities
Shown a weapon, this becomes:
3-1-3 = 3*5*6 = 90 possibilities
3-2-2 = 3*4*7 = 84 possibilities
3-3-1 = 3*3*8 = 72 possibilities
3-4-0 = 3*2*9 = 54 possibilities
Shown a room, this becomes:
3-0-4 = 3*6*5 = 90 possibilities
3-1-3 = 3*5*6 = 90 possibilities
3-2-2 = 3*4*7 = 84 possibilities
3-3-1 = 3*3*8 = 72 possibilities

With 2 suspects, the division can be:
2-0-4 = 4*6*5 = 120 possibilities
2-1-3 = 4*5*6 = 120 possibilities
2-2-2 = 4*4*7 = 112 possibilities
2-3-1 = 4*3*8 = 96 possibilities
2-4-0 = 4*2*9 = 72 possibilities
Shown a suspect, this becomes:
3-0-4 = 3*6*5 = 90 possibilities
3-1-3 = 3*5*6 = 90 possibilities
3-2-2 = 3*4*7 = 84 possibilities
3-3-1 = 3*3*8 = 73 possibilities
3-4-0 = 3*2*9 = 54 possibilities
Shown a weapon, this becomes:
2-1-4 = 4*5*5 = 100 possibilities
2-2-3 = 4*4*6 = 96 possibilities
2-3-2 = 4*3*7 = 84 possibilities
2-4-1 = 4*2*8 = 64 possibilities
2-5-0 = 4*1*9 = 36 possibilities
Shown a room, this becomes:
2-0-5 = 4*6*4 = 96 possibilities
2-1-4 = 4*5*5 = 100 possibilities
2-2-3 = 4*4*6 = 96 possibilities
2-3-2 = 4*3*7 = 84 possibilities
2-4-1 = 4*2*8 = 64 possibilities

With 1 suspect, the division can be:
1-0-5 = 5*6*4 = 120 possibilities
1-1-4 = 5*5*5 = 125 possibilities
1-2-3 = 5*4*6 = 120 possibilities
1-3-2 = 5*3*7 = 105 possibilities
1-4-1 = 5*2*8 = 80 possibilities
1-5-0 = 5*1*9 = 45 possibilities
Shown a suspect, this becomes:
2-0-5 = 4*6*4 = 96 possibilities
2-1-4 = 4*5*5 = 100 possibilities
2-2-3 = 4*4*6 = 96 possibilities
2-3-2 = 4*3*7 = 84 possibilities
2-4-1 = 4*2*8 = 64 possibilities
2-5-0 = 4*1*9 = 36 possibilities
Shown a weapon, this becomes:
1-1-5 = 5*5*4 = 100 possibilities
1-2-4 = 5*4*5 = 100 possibilities
1-3-3 = 5*3*6 = 90 possibilities
1-4-2 = 5*2*7 = 70 possibilities
1-5-1 = 5*1*8 = 40 possibilities
1-6-0 = 5*0*9 = reshuffle
Shown a room, this becomes:
1-0-6 = 5*6*3 = 90 possibilities
1-1-5 = 5*5*4 = 100 possibilities
1-2-4 = 5*4*5 = 100 possibilities
1-3-3 = 5*3*6 = 90 possibilities
1-4-2 = 5*2*7 = 70 possibilities
1-5-1 = 5*1*8 = 40 possibilities

With 0 suspects the division can be:
0-0-6 = 6*6*3 = 108 possibilities
0-1-5 = 6*5*4 = 120 possibilities
0-2-4 = 6*4*5 = 120 possibilities
0-3-3 = 6*3*6 = 108 possibilities
0-4-2 = 6*2*7 = 84 possibilities
0-5-1 = 6*1*8 = 48 possibilities
Shown a suspect, this becomes:
1-0-6 = 5*6*3 = 90 possibilities
1-1-5 = 5*5*4 = 100 possibilities
1-2-4 = 5*4*5 = 100 possibilities
1-3-3 = 5*3*6 = 90 possibilities
1-4-2 = 5*2*7 = 70 possibilities
1-5-1 = 5*1*8 = 40 possibilities
Shown a weapon, this becomes:
0-1-6 = 6*5*3 = 90 possibilities
0-2-5 = 6*4*4 = 96 possibilities
0-3-4 = 6*3*5 = 90 possibilities
0-4-3 = 6*2*6 = 72 possibilities
0-5-2 = 6*1*7 = 42 possibilities
Shown a room, this becomes:
0-0-7 = 6*6*2 = 72 possibilities
0-1-6 = 6*5*3 = 90 possibilities
0-2-5 = 6*4*4 = 96 possibilities
0-3-4 = 6*3*5 = 90 possibilities
0-4-3 = 6*2*6 = 72 possibilities
0-5-2 = 6*1*7 = 42 possibilities






We can see a connection between Cluedo, MasterMind and Guess Who?.

In Guess Who? we pick a character with certain traits (hair colour, shirt colour, gender, facial hair, eye colour). This is "find the hidden code in a number of codes by asking if a certain digit is in it."

In my travel edition of Guess Who?, all of the obvious traits are present in 5 characters, not in the other 25. Except, for some reason, facial hair, but we'll leave that out for the sake of argument now. If you're lucky you guess the right trait straignt away and flip 25 characters, but mostly, you'll flip 5 and look for a better guess.
The rule is you can only ask yes/no questions. To win the game as efficiently as possible, you need 50/50 questions - questions where you can always eliminate half of the characters. "Is your character's hair blonde or ginger?" "Is there an I in the name of your character?"
It beats sneakingly putting Bill back in the stack as tactic.

In Master Mind, a random code needs to be found. The code is made up of 4 variables that can have 8 different states (4 pegs randomly chosen out of 8 colours).
Master Mind is self-contained. There is only 1 person guessing the code. This means that, what is the best question, is always dependent on your current knowledge. Upon starting, the best combination to ask is 1122 since it cuts the most combinations. What becomes the best question after that, is dependent on the outcome.

Cluedo however is quite different in that there are multiple people chasing the same secret code. Their guesses provide information that you may only learn at a later turn. This means that the information above is hardly relevant for use within game - especially considering finding out which room it is, is so limited with all the walking around, and considering there is a good chance you will find out the secret combination by what you learn from your opponent's guesses, rather than your own.

So you can basically use this for your first move. Maaaaybe the 2nd. Global rule in this - only ask for a room if you have more than 5 of them at an early stage (or however is possible). Otherwise, if you have more weapons, aim for a weapon, if you have more suspects, aim for a suspect.
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BBP
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Re: Cluedo - Crunching The Numbers

Post by BBP »

How does it work with 6 players (3 cards)?

Possible:

3-0-0: 3*6*9 = 162 possibilities
If shown:
Sus - 4-0-0: 2*6*9 = 108 possibilities
Wea - 3-1-0: 3*5*9 = 135 possibilities
Roo - 3-0-1: 3*6*8 = 144 possibilities
Aim for a suspect


2-1-0: 4*5*9 = 180 possibilities
If shown:
Sus - 3-1-0 = 3*5*9 = 135 possibilities
Wea - 2-2-0 = 4*4*9 = 144 possibilities
Roo - 2-1-1 = 4*5*8 = 160 possibilities
Aim for a suspect

2-0-1: 4*6*8 = 192 possibilities
If shown:
Sus - 3-0-1 = 3*6*8 = 144 possibilities
Wea - 2-1-1 = 4*5*8 = 160 possibilities
Roo - 2-0-2 = 4*6*7 = 168 possibilities
Aim for a suspect

1-2-0: 5*4*9 = 180 possibilities
If shown:
Sus - 2-2-0 = 4*4*9 = 144 possibilities
Wea - 1-3-0 = 3*5*9 = 135 possibilities
Roo - 1-2-1 = 4*5*8 = 160 possibilities
Aim for a weapon

1-1-1: 5*5*8 = 200 possibilities
If shown:
Sus - 2-1-1 = 4*5*8 = 160 possibilities
Wea - 1-2-1 = 5*4*8 = 160 possibilities
Roo - 1-1-2 = 5*5*7 = 175 possibilities
Aim for suspect or weapon

1-0-2: 5*6*7 = 210 possibilities
If shown:
Sus - 2-0-2 = 4*6*7 = 168 possibilities
Wea - 1-1-2 = 5*5*7 = 175 possibilities
Roo - 1-0-3 = 5*6*6 = 180 possibilities
Aim for suspect


0-3-0: 6*3*9 = 162 possibilities
If shown:
Sus - 1-3-0: 5*3*9 = 135 possibilities
Wea - 0-4-0:
Sus - 4-0-0: 2*6*9 = 108 possibilities
Wea - 3-1-0: 3*5*9 = 135 possibilities
Roo - 3-0-1: 3*6*8 = 144 possibilities
Aim for a weapon


0-2-1: 6*4*8 = 192 possibilities
If shown:
Sus - 1-2-1 = 5*4*8 = 160 possibilities
Wea - 0-3-1 = 6*3*8 = 144 possibilities
Roo - 0-2-2 = 6*4*7 = 168 possibilities
Aim for a weapon

0-1-2: 6*5*7 = 210 possibilities
If shown:
Sus - 1-1-2 = 5*5*7 = 175 possibilities
Wea - 0-2-2 = 6*4*7 = 168 possibilities
Roo - 0-1-3 = 6*5*6 = 180 possibilities
Aim for a weapon

0-0-3: 6*6*6 = 216 possibilities
If shown:
Sus - 1-0-3 = 5*6*6 = 180 possibilities
Wea - 0-1-3 = 5*6*6 = 180 possibilities
Roo - 0-0-4 = 6*6*5 = 180 possibilities
Aim for anything


Global rule - aim for weapon or suspect, whatever you have more of.

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