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In game theory, the average branching factor refers to the average number of legal moves available to a player at each turn. In chess, the average branching factor is about .If Jules wants to write a chess simulation program that represents every possible sequence of moves in a game that lasts moves, the following function approximates the number of distinct sequences he would simulate.How many times greater does the number of sequences become each time the game lasts moves longer?
Full solution
Q. In game theory, the average branching factor refers to the average number of legal moves available to a player at each turn. In chess, the average branching factor is about .If Jules wants to write a chess simulation program that represents every possible sequence of moves in a game that lasts moves, the following function approximates the number of distinct sequences he would simulate.How many times greater does the number of sequences become each time the game lasts moves longer?
- Calculate Sequences for Moves: Let's first calculate the number of sequences for a game that lasts moves using the given function .
- Calculate Sequences for Moves: Now, let's calculate the number of sequences for a game that lasts moves, which would be .
- Find Ratio of Sequences Increase: To find out how many times greater the number of sequences becomes when the game lasts moves longer, we need to divide the number of sequences for moves by the number of sequences for moves. This gives us the ratio:
- Simplify Ratio Using Exponents: Using the properties of exponents, we can simplify the ratio by subtracting the exponents:
- Calculate Factor of Increase: Calculating gives us the exact factor by which the number of sequences increases:
