Counting to 21 game
probability - What is the best play strategy for a game called "21"?
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There is a game called 21, played orally by n players and the rules are following:
First player starts by randomly saying an integer form [1,3] for his turn, then the next player randomly adds to the previous number an integer also from [1,3] an so on by circle, until someone will be forced to say 21 and lose (the sum can't exceed 21, so if the previous number was 20, player is forced to say 21(his range is{1}). Likewise, if the previous number was 19, the his range is [1,2]). Then the next player starts a new round. The game continues until there is only 1 player left.
The question is somehow similar to the Josephus problem, but the "random" part is where I got lost. I found that if a player says "$X$", where X = 21-(n-1) then he definitely won't lose, so I was trying to think of an algorithm that leads players to "$X$", but I am stuck.
My question is: Let one player to be "strategic" and use a certain algorithm in order to win, and the other players will choose entirely randomly, can it be such an algorithm for n players?
- probability
- algorithms
- problem-solving
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I recommend looking at the game "Nim" like Sean Xie suggested. https://www.archimedes-lab.org/How_to_Solve/Win_at_Nim.html or https://boardgamestrategy.blog/2017/02/13/counting-to-21-strategy/
As the number of players increase, however, there less of a strategy (except in extreme cases) because the other players can cooperate against you; they have a wider range of options, collectively, than you do, so you can't use the sort of strategy that you can use in the two-player version.
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combinatorics - Counting to $P$ game - strategy?
I believe that there is no possibility of systematic winning strategy for a single player in such a multiplayer version of the game.
In a scenario with $N=2 \,$ players, the winning strategy for player A is to end his sequences on the numbers given by $P - 1 -k (m+1) \, \, \,$, since this allows to conclude his last sequence on $P-1\,$. For example, in the case $P=21 \, $ (as in the previous question) and $m=5\,$, since player A wins if he ends his last sequence on $21-1=20 \, \,$ , stopping his second to last sequence on $21-1-(5+1)=14 \, \, \,$ allows him to be sure about the possibility of ending his successive sequence on $20$, whatever the choice of player B. Similarly, stopping his third to last sequence on $21-1-2 \cdot (5+1)=8 \, \, $ allows him to be sure to end his successive sequence on $14$ whatever the choice of player $B $, and so on.
However, in the scenario with $\geq 3$ players, this strategy cannot be successful for a single player (let us call him again player A). As pointed out in the answer to the previous question for the original version with $m=3$ , even in this new modality of the game we can imagine that all remaining players could cooperate against him, with the advantage of having a considerably larger range of options.
Let us consider for example the case with three players. Players B and C could cooperate so that the last of them (i.e., that preceding player A) ends his last sequence on one of the numbers included in the range between $P - 1 - (2m+1) \, \, $ and $P - 1 - (m+2) \, \, $ . In this way, whatever the choice of player A, they can force him to lose: since the distance $d $ to their target $P-1$ ranges from $m+2$ to $2m+1$ , any successive choice of player A characterized by a sequence of $k $ terms (where $1 \leq k \leq m \, \, $ ) can be followed by two sequences of players B and C which in total include $d-k $ terms (where $2 \leq d-k \leq 2m \, \,$) , and the target is achieved. Note that the range between $P - 1 - (2m+1) \,$ and $P - 1 - (m+2) \,$ includes $m $ terms, so that there is no way for player A, if the other players collaborate against him, to avoid that the player who precedes him ends within this range during the game.
Generalizing, for the case with $N$ players where $N-1$ of them cooperate against player A, they can choose their sequences so that, at a given point of the game, the last of them ends his sequence on one of the numbers included in the range between $P - 1 - [(N-1)m+1] \,\, \, $ and $P - 1 - (m+N-1) \, \,$ .
In this way, whatever the successive choice of player A, they can force him to lose. The distance $d $ to their target $P-1$ ranges from $m+N-1 \,\,$ to $(N-1)m+1 \,\, $ . Therefore, any successive choice of player A characterized by a sequence of $k $ elements (where $1 \leq k \leq m \, \,$ ) can be followed by $N-1$ sequences of the other players including in total $d-k $ elements, where $N-1 \leq d-k \leq (N-1)m \,\, \, \, $ , and the target is again achieved.
Cheers to the Governor | Rules and how to play
📖 Content:
- 😄 What is the game Cheers to the Governor
- 🧐 How to play Cheers to the Governor
- 😎 Cheers to the Governor Rules
- 😜 cheers to the governor to rule the ideas
- More Rule Ideas Cheers to the Governor
What is Cheers to the Governor game?
Looking for something to play at a party that doesn't require playing cards, dice or other game materials? Cheers to the Governor is a fun turn based party game where you only need two things - alcohol and lots of imagination to have a great time with your friends!
Be sure to shout “Cheers to the Governor” as loudly as you can when the game starts! Also known as the 21 Drinking Game, Cheers to the Governor is perfect for playing with three or more people.
👉 Are you looking for other games that you can play with three people? Check out our list of the best drinking games for three!
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How to play Cheers to the Governor
One of the key things you need to remember when learning how to play Cheers to the Governor is that players need to count from 1 to 21. Everyone takes turns saying the number until one of the players reaches 21. Once the number 21 is called, all players must shout “Cheers to the Governor!” and drink.
Sounds easy, right? After each round it gets a little more difficult. Each time you and your friends get to the number 21 and complete the round, whoever says the number 21 gets the opportunity to formulate a rule.
As new rules keep being added, your game won't be as easy as you first thought.
The more you play, the more it becomes more confusing and interesting.!
Cheers to the Governor rules
Getting used to the rules of Cheers to the Governor is pretty easy. In short, it mostly involves counting and drinking. Easy peasy! The main rule is that players need to count to 21 while performing certain actions at certain numbers, and of course yell “Cheers to the Governor!” when the player reaches the number 21.
Here are the basic rules for playing Cheers to the Governor:
N e w !
Play
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All players sit in a circle counting from 1 to 21 in turn. until the number 21, he says “Cheers to the Governor!” and everyone drinks.
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The player who called the number 21 gets the opportunity to come up with a new rule for the next round.
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Each time the round ends, whoever said the number 21 can add a new rule.
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If players fail to follow the rules assigned to their number, they must drink.
The game starts again at number 1. -
The game ends as soon as you work out the 21 rules or the players are too tired or drunk to play!
New rules might be: “Clap five times on the number 5” or “Say the name of a celebrity on the number 8” . Feel free to make up rules as you play!
Cheers to the Governor Rule Ideas
What makes Cheers to the Governor especially fun is coming up with all sorts of crazy rules and trying to follow each rule, which gets even more difficult when the players start getting drunk.!
📖 Here's to you: Three Man Dice Drinking Game Rules & Guides
Check out some of these Cheers to the Governor rule ideas already assigned specific numbers. You can also tweak these rules to make your game more exciting!
1 - jump
When players say the number 1, they must jump once.
2 - Swap the number
When the players reach the number 2, they must say the number 3 instead.
The next player must then say the number 2.
3 - Make animal sounds
When the players reach the number 3, they must make an animal sound.
4 - Use another language
When players reach the number 4, they must say the number four in another language. For example, 4 is quatro!
5 - The Power of a Girl
When a player says the number 5, all female players must drink.
6 - Cursed number
Say a curse word after saying the number 6. If it's not a curse word, you should drink.
7 - On 7, say 11
When players reach the number 7, they should say the number 11 instead. 7-11… get?
8 - Boys World
When the players say the number 8, all male players must drink.
🤓 We offer you: Avalanche drinking game: Rules and how to play
9 - Finger counting
When players reach the number 9, they must raise 9 fingers. If they say the number 9 out loud, they must drink.
10 - Everyone drinks
When the player says the number 10, everyone must drink.
11 - Around the World
When players say the number 11, they must say the country as well.
12 - Skip the number
No one can say the number 12. Once the player reaches the number 12, he/she must say the number 13 instead.
13 - Dance away
When players reach the number 13, they should dance instead of saying the number 13 out loud.
14 - Don't say anything
When the players reach number 14, they should say nothing and just take a drink.
15 - Stand up
When players reach number 15, they must stand up and take a drink.
16 - Trouble
When the players reach the number 16, they must clap 16 times.
17 - Rhyme
When players reach number 17, they must call out "Clementine".
18 - Sing a song
When the players reach number 18, they must sing a song.
19 - Pose
When players reach number 19, they must do a selfie pose.
20 - Sharing is caring
When players reach the number 20, they can indicate who should take the drink.
We offer you: Circle of Death
21 - Swap places
When players say the number 21, they can force everyone to switch places to confuse things even more!
More Rule Ideas Cheers to the Governor
Still looking for more rules? Here are some more ideas that you can use at any time during the game.
1. Be healthy, boss!
Instead of yelling “Cheers to the Governor!”, players should yell “Cheers, Chief!”.
2. Do not laugh
Any player who laughs during the round must drink.
3. Be brave
Any player who hesitates to call a number should have a drink.
4. Go left
During one round, all players must use their left hand whenever they have to drink.
5. Count backwards
Play the round counting backwards. One player starts shouting the number 21. The player who reaches the number 1 must shout: “Cheers to the Governor!” and add a rule for the next round.
XXXI Summer Olympic Games
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The Summer Olympic Games are the largest international competitions in summer and all-season sports, held once every four years under the auspices of the International Olympic Committee.
Games have been held since 1896.
According to the Committee's decision, the 2016 Summer Olympic Games will be held in the most famous metropolis of Brazil - Rio de Janeiro. Competitions will be held from 5 to 21 August, that is, when the calendar winter is on that continent. The temperature during this period ranges from +18 to +25 degrees Celsius.
This Olympics is notable in that it will be the first Olympic Games to be hosted by South America. And now, among the unconquered continents, not counting Antarctica, only Africa remains, which has never hosted this large-scale event.
Compared to previous Games, the program includes two new sports - golf and rugby-7s. The number of disciplines is 33. During the Games, a record number of sets of medals will be played - 306, the number of participating countries will also be the maximum in the history of the Olympic movement - 206.
Dancing little men became the emblem of the Olympics, each has its own meaning: bright yellow - the sun, blue - the sea, rich green - the mountain ranges of Brazil.
The mascots of the Games are two animals representing the flora and fauna of Brazil. One talisman is yellow in color and symbolizes the brightest and most common representatives of the animal world of Brazil - a monkey and a parrot. The second beast has become a collective image of the Brazilian flora; in its features one can recognize both elements of wood and elements of a flower. Thanks to a vote that was held among the fans of the games, the names were chosen for them - Vinicius and Tom.
The flame of the Olympias was lit on April 21, 2016 in the temple of Hera in Olympia. The Greek stage of the relay race lasted for 8 days, until April 28, after which the Olympic flame arrived in Lausanne, and a day later in Geneva. On May 2, a one-day visit to Colombian Bogota took place. On May 3, the Olympic flame arrived in Brasilia, from where the Brazilian part of the relay began. Within its framework, the Fire will visit more than 300 cities, including the capitals of all Brazilian states.
The relay race will end on August 5, 2016 at the Opening Ceremony of the Olympic Games.
http://www.rio2016.com/en Official website of the Summer Games
Information about the 2016 Games is posted in Portuguese, English, Spanish, French.
http://olympteka.ru/olymp/rio2016 Olympteka
Encyclopedia of the Olympic Games, emblem, mascots, torch, coins, stamps, Games schedule, sports, sports facilities, news.
http://www.olympic.ru Russian Olympic Committee
http://valovs.ru All-Russian Association of Summer Olympic Sports
The task of the Association is to unite and protect the interests of federations of summer Olympic sports through joint efforts.
http://team-russia2016.ru Russian team
The composition of the Russian team for the Olympic Games will be approved by the executive committee of the Russian Olympic Committee no later than July 22.
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