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 35 cards in anything inbetween.
Taking a 3 player game, and using a SuspectWeaponRoom format, your cards can be divided in the following ways:
501 and 051  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.
510 and 150  1 suspect and 5 weapons or the other way round, and no rooms.
Probability is 1*5*9 or 45 possibilities.
402 and 042 give 2*6*7 = 84 possibilities
411 and 141 give 5*2*8 = 80 possibilities
330 gives 3*3*9=81 possibilities
303 or 033 =108 possibilities
321 or 231 gives 3*4*8 = 96 possibilities
312 or 132 gives 3*5*7 = 105 possibilites
222 gives 4*4*7 = 112 possibilies
213 gives 4*5*6 = 120
204 gives 4*6*5 = 120
114 gives 5*5*5 = 125
105 gives 5*6*4 = 120
006 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:
300 = 162
210 = 180
201 = 192
102 = 120
111 = 200
003 = 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!
Cluedo  Crunching The Numbers
 ZachBlack000
 Junior Detective
 Posts: 713
 Joined: Thu Sep 10, 2020 7:05 pm
 Location: Shooting billiards with a rope
Re: Cluedo  Crunching The Numbers
This is exactly what I used to do on a dailybasis, when I had free time on my hands.
Re: Cluedo  Crunching The Numbers
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:
501 = 48 possibilities.
510 = 45 possibilities
The card of which you have 5 can't be shown. The balance can be:
520 = 1*4*9 = 36 possibilities
511 = 1*5*8 = 40 possibilities
502 = 1*6*8 = 48 possibilities
4 suspects (or weapons) give:
402 = 2*6*7 = 84 possibilities
411 = 2*5*8 = 80 possibilities
420 = 2*4*9 = 72 possibilities
Upon being shown a suspect, we get:
502 = 1*6*7 = 42 possibilities
511 = 1*5*8 = 40 possibilities
520 = 1*4*9 = 36 possibilities
Upon being shown a weapon:
412 = 2*5*7 = 700 possibilities
421 = 2*4*8 = 64 possibilities
430 = 2*3*9 = 54 possibilities
Upon being shown a room:
403 = 2*6*6 = 72 possibilities
412 = 2*5*7 = 70 possibilities
421 = 2*4*8 = 64 possibilities
With 3 suspects there are:
303 = 3*6*6 = 108 possibilities
312 = 3*5*7 = 105 possibilities
321 = 3*4*8 = 96 possibilities
330 = 3*3*9 = 81 possibilities
Shown a suspect, this becomes:
403 = 2*6*6 = 72 possibilities
412 = 2*5*7 = 70 possibilities
421 = 2*4*8 = 64 possibilities
430 = 2*3*9 = 54 possibilities
Shown a weapon, this becomes:
313 = 3*5*6 = 90 possibilities
322 = 3*4*7 = 84 possibilities
331 = 3*3*8 = 72 possibilities
340 = 3*2*9 = 54 possibilities
Shown a room, this becomes:
304 = 3*6*5 = 90 possibilities
313 = 3*5*6 = 90 possibilities
322 = 3*4*7 = 84 possibilities
331 = 3*3*8 = 72 possibilities
With 2 suspects, the division can be:
204 = 4*6*5 = 120 possibilities
213 = 4*5*6 = 120 possibilities
222 = 4*4*7 = 112 possibilities
231 = 4*3*8 = 96 possibilities
240 = 4*2*9 = 72 possibilities
Shown a suspect, this becomes:
304 = 3*6*5 = 90 possibilities
313 = 3*5*6 = 90 possibilities
322 = 3*4*7 = 84 possibilities
331 = 3*3*8 = 73 possibilities
340 = 3*2*9 = 54 possibilities
Shown a weapon, this becomes:
214 = 4*5*5 = 100 possibilities
223 = 4*4*6 = 96 possibilities
232 = 4*3*7 = 84 possibilities
241 = 4*2*8 = 64 possibilities
250 = 4*1*9 = 36 possibilities
Shown a room, this becomes:
205 = 4*6*4 = 96 possibilities
214 = 4*5*5 = 100 possibilities
223 = 4*4*6 = 96 possibilities
232 = 4*3*7 = 84 possibilities
241 = 4*2*8 = 64 possibilities
With 1 suspect, the division can be:
105 = 5*6*4 = 120 possibilities
114 = 5*5*5 = 125 possibilities
123 = 5*4*6 = 120 possibilities
132 = 5*3*7 = 105 possibilities
141 = 5*2*8 = 80 possibilities
150 = 5*1*9 = 45 possibilities
Shown a suspect, this becomes:
205 = 4*6*4 = 96 possibilities
214 = 4*5*5 = 100 possibilities
223 = 4*4*6 = 96 possibilities
232 = 4*3*7 = 84 possibilities
241 = 4*2*8 = 64 possibilities
250 = 4*1*9 = 36 possibilities
Shown a weapon, this becomes:
115 = 5*5*4 = 100 possibilities
124 = 5*4*5 = 100 possibilities
133 = 5*3*6 = 90 possibilities
142 = 5*2*7 = 70 possibilities
151 = 5*1*8 = 40 possibilities
160 = 5*0*9 = reshuffle
Shown a room, this becomes:
106 = 5*6*3 = 90 possibilities
115 = 5*5*4 = 100 possibilities
124 = 5*4*5 = 100 possibilities
133 = 5*3*6 = 90 possibilities
142 = 5*2*7 = 70 possibilities
151 = 5*1*8 = 40 possibilities
With 0 suspects the division can be:
006 = 6*6*3 = 108 possibilities
015 = 6*5*4 = 120 possibilities
024 = 6*4*5 = 120 possibilities
033 = 6*3*6 = 108 possibilities
042 = 6*2*7 = 84 possibilities
051 = 6*1*8 = 48 possibilities
Shown a suspect, this becomes:
106 = 5*6*3 = 90 possibilities
115 = 5*5*4 = 100 possibilities
124 = 5*4*5 = 100 possibilities
133 = 5*3*6 = 90 possibilities
142 = 5*2*7 = 70 possibilities
151 = 5*1*8 = 40 possibilities
Shown a weapon, this becomes:
016 = 6*5*3 = 90 possibilities
025 = 6*4*4 = 96 possibilities
034 = 6*3*5 = 90 possibilities
043 = 6*2*6 = 72 possibilities
052 = 6*1*7 = 42 possibilities
Shown a room, this becomes:
007 = 6*6*2 = 72 possibilities
016 = 6*5*3 = 90 possibilities
025 = 6*4*4 = 96 possibilities
034 = 6*3*5 = 90 possibilities
043 = 6*2*6 = 72 possibilities
052 = 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 selfcontained. 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.
Suspects and weapons are interchangeable.
When you have 5 cards of either weapons or suspects, the amount of possibilities is as follows:
501 = 48 possibilities.
510 = 45 possibilities
The card of which you have 5 can't be shown. The balance can be:
520 = 1*4*9 = 36 possibilities
511 = 1*5*8 = 40 possibilities
502 = 1*6*8 = 48 possibilities
4 suspects (or weapons) give:
402 = 2*6*7 = 84 possibilities
411 = 2*5*8 = 80 possibilities
420 = 2*4*9 = 72 possibilities
Upon being shown a suspect, we get:
502 = 1*6*7 = 42 possibilities
511 = 1*5*8 = 40 possibilities
520 = 1*4*9 = 36 possibilities
Upon being shown a weapon:
412 = 2*5*7 = 700 possibilities
421 = 2*4*8 = 64 possibilities
430 = 2*3*9 = 54 possibilities
Upon being shown a room:
403 = 2*6*6 = 72 possibilities
412 = 2*5*7 = 70 possibilities
421 = 2*4*8 = 64 possibilities
With 3 suspects there are:
303 = 3*6*6 = 108 possibilities
312 = 3*5*7 = 105 possibilities
321 = 3*4*8 = 96 possibilities
330 = 3*3*9 = 81 possibilities
Shown a suspect, this becomes:
403 = 2*6*6 = 72 possibilities
412 = 2*5*7 = 70 possibilities
421 = 2*4*8 = 64 possibilities
430 = 2*3*9 = 54 possibilities
Shown a weapon, this becomes:
313 = 3*5*6 = 90 possibilities
322 = 3*4*7 = 84 possibilities
331 = 3*3*8 = 72 possibilities
340 = 3*2*9 = 54 possibilities
Shown a room, this becomes:
304 = 3*6*5 = 90 possibilities
313 = 3*5*6 = 90 possibilities
322 = 3*4*7 = 84 possibilities
331 = 3*3*8 = 72 possibilities
With 2 suspects, the division can be:
204 = 4*6*5 = 120 possibilities
213 = 4*5*6 = 120 possibilities
222 = 4*4*7 = 112 possibilities
231 = 4*3*8 = 96 possibilities
240 = 4*2*9 = 72 possibilities
Shown a suspect, this becomes:
304 = 3*6*5 = 90 possibilities
313 = 3*5*6 = 90 possibilities
322 = 3*4*7 = 84 possibilities
331 = 3*3*8 = 73 possibilities
340 = 3*2*9 = 54 possibilities
Shown a weapon, this becomes:
214 = 4*5*5 = 100 possibilities
223 = 4*4*6 = 96 possibilities
232 = 4*3*7 = 84 possibilities
241 = 4*2*8 = 64 possibilities
250 = 4*1*9 = 36 possibilities
Shown a room, this becomes:
205 = 4*6*4 = 96 possibilities
214 = 4*5*5 = 100 possibilities
223 = 4*4*6 = 96 possibilities
232 = 4*3*7 = 84 possibilities
241 = 4*2*8 = 64 possibilities
With 1 suspect, the division can be:
105 = 5*6*4 = 120 possibilities
114 = 5*5*5 = 125 possibilities
123 = 5*4*6 = 120 possibilities
132 = 5*3*7 = 105 possibilities
141 = 5*2*8 = 80 possibilities
150 = 5*1*9 = 45 possibilities
Shown a suspect, this becomes:
205 = 4*6*4 = 96 possibilities
214 = 4*5*5 = 100 possibilities
223 = 4*4*6 = 96 possibilities
232 = 4*3*7 = 84 possibilities
241 = 4*2*8 = 64 possibilities
250 = 4*1*9 = 36 possibilities
Shown a weapon, this becomes:
115 = 5*5*4 = 100 possibilities
124 = 5*4*5 = 100 possibilities
133 = 5*3*6 = 90 possibilities
142 = 5*2*7 = 70 possibilities
151 = 5*1*8 = 40 possibilities
160 = 5*0*9 = reshuffle
Shown a room, this becomes:
106 = 5*6*3 = 90 possibilities
115 = 5*5*4 = 100 possibilities
124 = 5*4*5 = 100 possibilities
133 = 5*3*6 = 90 possibilities
142 = 5*2*7 = 70 possibilities
151 = 5*1*8 = 40 possibilities
With 0 suspects the division can be:
006 = 6*6*3 = 108 possibilities
015 = 6*5*4 = 120 possibilities
024 = 6*4*5 = 120 possibilities
033 = 6*3*6 = 108 possibilities
042 = 6*2*7 = 84 possibilities
051 = 6*1*8 = 48 possibilities
Shown a suspect, this becomes:
106 = 5*6*3 = 90 possibilities
115 = 5*5*4 = 100 possibilities
124 = 5*4*5 = 100 possibilities
133 = 5*3*6 = 90 possibilities
142 = 5*2*7 = 70 possibilities
151 = 5*1*8 = 40 possibilities
Shown a weapon, this becomes:
016 = 6*5*3 = 90 possibilities
025 = 6*4*4 = 96 possibilities
034 = 6*3*5 = 90 possibilities
043 = 6*2*6 = 72 possibilities
052 = 6*1*7 = 42 possibilities
Shown a room, this becomes:
007 = 6*6*2 = 72 possibilities
016 = 6*5*3 = 90 possibilities
025 = 6*4*4 = 96 possibilities
034 = 6*3*5 = 90 possibilities
043 = 6*2*6 = 72 possibilities
052 = 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 selfcontained. 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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Re: Cluedo  Crunching The Numbers
How does it work with 6 players (3 cards)?
Possible:
300: 3*6*9 = 162 possibilities
If shown:
Sus  400: 2*6*9 = 108 possibilities
Wea  310: 3*5*9 = 135 possibilities
Roo  301: 3*6*8 = 144 possibilities
Aim for a suspect
210: 4*5*9 = 180 possibilities
If shown:
Sus  310 = 3*5*9 = 135 possibilities
Wea  220 = 4*4*9 = 144 possibilities
Roo  211 = 4*5*8 = 160 possibilities
Aim for a suspect
201: 4*6*8 = 192 possibilities
If shown:
Sus  301 = 3*6*8 = 144 possibilities
Wea  211 = 4*5*8 = 160 possibilities
Roo  202 = 4*6*7 = 168 possibilities
Aim for a suspect
120: 5*4*9 = 180 possibilities
If shown:
Sus  220 = 4*4*9 = 144 possibilities
Wea  130 = 3*5*9 = 135 possibilities
Roo  121 = 4*5*8 = 160 possibilities
Aim for a weapon
111: 5*5*8 = 200 possibilities
If shown:
Sus  211 = 4*5*8 = 160 possibilities
Wea  121 = 5*4*8 = 160 possibilities
Roo  112 = 5*5*7 = 175 possibilities
Aim for suspect or weapon
102: 5*6*7 = 210 possibilities
If shown:
Sus  202 = 4*6*7 = 168 possibilities
Wea  112 = 5*5*7 = 175 possibilities
Roo  103 = 5*6*6 = 180 possibilities
Aim for suspect
030: 6*3*9 = 162 possibilities
If shown:
Sus  130: 5*3*9 = 135 possibilities
Wea  040:
Sus  400: 2*6*9 = 108 possibilities
Wea  310: 3*5*9 = 135 possibilities
Roo  301: 3*6*8 = 144 possibilities
Aim for a weapon
021: 6*4*8 = 192 possibilities
If shown:
Sus  121 = 5*4*8 = 160 possibilities
Wea  031 = 6*3*8 = 144 possibilities
Roo  022 = 6*4*7 = 168 possibilities
Aim for a weapon
012: 6*5*7 = 210 possibilities
If shown:
Sus  112 = 5*5*7 = 175 possibilities
Wea  022 = 6*4*7 = 168 possibilities
Roo  013 = 6*5*6 = 180 possibilities
Aim for a weapon
003: 6*6*6 = 216 possibilities
If shown:
Sus  103 = 5*6*6 = 180 possibilities
Wea  013 = 5*6*6 = 180 possibilities
Roo  004 = 6*6*5 = 180 possibilities
Aim for anything
Global rule  aim for weapon or suspect, whatever you have more of.
Possible:
300: 3*6*9 = 162 possibilities
If shown:
Sus  400: 2*6*9 = 108 possibilities
Wea  310: 3*5*9 = 135 possibilities
Roo  301: 3*6*8 = 144 possibilities
Aim for a suspect
210: 4*5*9 = 180 possibilities
If shown:
Sus  310 = 3*5*9 = 135 possibilities
Wea  220 = 4*4*9 = 144 possibilities
Roo  211 = 4*5*8 = 160 possibilities
Aim for a suspect
201: 4*6*8 = 192 possibilities
If shown:
Sus  301 = 3*6*8 = 144 possibilities
Wea  211 = 4*5*8 = 160 possibilities
Roo  202 = 4*6*7 = 168 possibilities
Aim for a suspect
120: 5*4*9 = 180 possibilities
If shown:
Sus  220 = 4*4*9 = 144 possibilities
Wea  130 = 3*5*9 = 135 possibilities
Roo  121 = 4*5*8 = 160 possibilities
Aim for a weapon
111: 5*5*8 = 200 possibilities
If shown:
Sus  211 = 4*5*8 = 160 possibilities
Wea  121 = 5*4*8 = 160 possibilities
Roo  112 = 5*5*7 = 175 possibilities
Aim for suspect or weapon
102: 5*6*7 = 210 possibilities
If shown:
Sus  202 = 4*6*7 = 168 possibilities
Wea  112 = 5*5*7 = 175 possibilities
Roo  103 = 5*6*6 = 180 possibilities
Aim for suspect
030: 6*3*9 = 162 possibilities
If shown:
Sus  130: 5*3*9 = 135 possibilities
Wea  040:
Sus  400: 2*6*9 = 108 possibilities
Wea  310: 3*5*9 = 135 possibilities
Roo  301: 3*6*8 = 144 possibilities
Aim for a weapon
021: 6*4*8 = 192 possibilities
If shown:
Sus  121 = 5*4*8 = 160 possibilities
Wea  031 = 6*3*8 = 144 possibilities
Roo  022 = 6*4*7 = 168 possibilities
Aim for a weapon
012: 6*5*7 = 210 possibilities
If shown:
Sus  112 = 5*5*7 = 175 possibilities
Wea  022 = 6*4*7 = 168 possibilities
Roo  013 = 6*5*6 = 180 possibilities
Aim for a weapon
003: 6*6*6 = 216 possibilities
If shown:
Sus  103 = 5*6*6 = 180 possibilities
Wea  013 = 5*6*6 = 180 possibilities
Roo  004 = 6*6*5 = 180 possibilities
Aim for anything
Global rule  aim for weapon or suspect, whatever you have more of.