Chutes and Ladders has been a hit with the kindergarten set since it was introduced in the United States in , by Milton Bradley. The colorful 10 x 10 game board contains squares numbered 1 to The first player to spin his or her way to square at the top of the board is declared the winner.
How do u play Snakes and Ladders? Who invented Snakes and Ladders? Which country invented chess? What is the snake world record? Is it possible to win snake? How do you finish Snakes and Ladders?
What age is appropriate for Snakes and Ladders? How much does Chutes and Ladders cost? It's not exactly known when or who invented it, though it's believed the game was played at a time as early as 2nd century BC.
According to some historians, the game was invented by Saint Gyandev in the 13th century AD. Originally, the game was used as a part of moral instruction to children. The squares in which ladders start were each supposed to stand for a virtue, and those housing the head of a snake were supposed to stand for an evil.
Whilst the chances of a game running, and running are essentially negligible constantly landing on snakes and going around in circles , there is a theoretical chance that the main game loop could be executing forever.
This is bad, and will lock-up your code. Any smart developer implementing an algorithm like this would program a counter that is incremented on each dice roll.
This counter would be checked, and if it exceeds a defined threshold, the loop should be exited. Looking at any coded implementation that does not employ a fail-safe like this should ring loud warning bells in your ears. Even if, like in the current configuration, you are more likely to win the lottery whilst being struck by lightning for the third time that day than lock-up, who is to say that:.
Your estimate that the odds are trivial is correct. How good are you at estimating those Black Swan events? At a later date, someone tweaks the parameters of the game adding different dice, new rules, extra features … , massively changing the dynamics of the system. Maybe your random number generator is not as random as you thought and you get into a harmonic resonance that oscillates you between states so that you never progress.
There are various variants of the 10x10 game in circulation, here is description of the board used for my experiments. There are nine ladders and ten snakes. In the above graph, the x-axis shows the number of rolls, and the y-axis shows the percentage of games that were completed in that number of rolls. The shortest possible game takes just seven rolls. There are mulitple ways this can be achieved, it happens approximately twice in every thousand games played.
One possible solution is the rolls: 4 , 6 , 6 , 2 , 6 , 6 , 4 This takes the user up ladder 5 which boosts the player over half the board in one go! The Modal number of rolls required to complete a game is 20 represented by the peak on the top graph. The mode is the most frequent occurence — more games will be completed in 20 rolls than any other number of rolls. The chart below is a plot of the cummulative probability of finishing the game by turn- n.
The cummulative total, is the total of all probabilities up to and including roll n. The Median number of rolls required to complete a game is With a median of 29, this means that there are the same number of games that are completed in less moves than 29 as there are games that take more than 29 moves to complete. To complete the average trifecta, we'll calculate the Mean.
The arithmetic mean is simply the sum of all the rolls of the die divided by the number of games played. During the billion simulated games played, the die was rolled a total of 36,,, times which results gives an average of the number of die rolls per game of approx There is a subtle difference.
If the same snake was encounterd three times in a single game, the count for this snake would be increased by 3, not just 1 "Used or not in this game". Think of it like a "Toll" charge for using the ladder or snake. The least frequently used ladder, not surprisingly, is ladder 1 which can only be used if a player rolls a 1 on their initial roll. If a 1 is not rolled on the first roll, it is impossible to come back to this square.
Refreshingly, looking at the count of the number of times this ladder was used in the entire simulation run results in a percentage of The next least used ladder is ladder 2. Again, not a surprise since, like ladder 1, once passed, there is no way to return to take it again. Unlike ladder 1, however, there is more than one way of getting onto this ladder over a series of early rolls, so the percentage this ladder is used is higher than ladder 1.
Objective approaches often work extremely well, but there are some limitations. Sometimes, for instance, it is simply not possible to repeat an experiment multiple times. Sometimes you only get one shot. The work around for this leads us neatly to an alternative mechanism for calculating the distribution of expected game lengths.
Games like Chutes and Ladders are ideal candidates for Markov Chain analysis because, at any time, the probability of events that will happen in the future are agnostic about what happened in the past.
If a player is on grid square 18 of the board, the probability of what will happen on the next roll is independent on how the player got to square It is this memorylessness that enables Markov chain analysis to work. At the heart of Markov Chain analysis is the concept of a Stochastic Process. This is just a fancy word to say that, from a given state, there are a series of possibilities that could happen next, defined by a probability distribution.
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