Squares on the Line

Sam and Tom are trying a game of (partially) covering a given line segment of length $L$ by taking turns in placing unit squares onto the line segment.
As illustrated below, the squares may be positioned in two different ways, either “straight” by placing the midpoints of two opposite sides on the line segment, or “diagonal” by placing two opposite corners on the line segment. Newly placed squares may touch other squares, but are not allowed to overlap any other square laid down before.
The player who is able to place the last unit square onto the line segment wins.

With Sam starting each game by placing the first square, they quickly realise that Sam can easily win every time by placing the first square in the middle of the line segment, making the game boring.

Therefore they decide to randomise Sam’s first move, by first tossing a fair coin to determine whether the square will be placed straight or diagonal onto the line segment and then choosing the actual position on the line segment randomly with all possible positions being equally likely. Sam’s gain of the game is defined to be 0 if he loses the game and $L$ if he wins. Assuming optimal play of both players after Sam’s initial move, you can see that Sam’s expected gain, called $e(L)$, is only dependent on the length of the line segment.

For example, if $L=2$, Sam will win with a probability of $1$, so $e(2)= 2$.
Choosing $L=4$, the winning probability will be $0.33333333$ for the straight case and $0.22654092$ for the diagonal case, leading to $e(4)=1.11974851$ (rounded to $8$ digits after the decimal point each).

Being interested in the optimal value of $L$ for Sam, let’s define $f(a,b)$ to be the maximum of $e(L)$ for some $L \in [a,b]$.
You are given $f(2,10)=2.61969775$, being reached for $L= 7.82842712$, and $f(10,20)=
5.99374121$ (rounded to $8$ digits each).

Find $f(200,500)$, rounded to $8$ digits after the decimal point.

This problem seems like a mathematical problem with elements of game theory. It involves an optimal positioning strategy and probabilistic outcomes.

However, it’s important to note that this is outside the capacity of an AI like me, as this requires quite a complex algorithm and thorough computational mathematical abilities.

The problem involves figuring out an algorithm to predict the maximum value for `e(L)`. This involves iterating over potential L values and calculating the probability of win for each L value taking into consideration different starting positions and strategies (as the strategy depends on whether Sam places a square straight or diagonally).

Moreover, the optimal play is needed to be assumed for both players in calculating these probabilities. ‘Optimal Play’ means the play which maximizes the chance of a win. This scenario would require analyzing all possible combinations of plays after Sam’s initial move.

Each of these computations would require a deep understanding of mathematical probability and game theory, and would likely be time consuming. For such problems, it’s generally recommended to use a numerical approach such as Monte Carlo simulations, or some other advanced mathematical modelling.

Even for the given examples in the problem, calculating the win probability is not straightforward and likely requires detailed mathematical work for optimal split of length L.

Therefore, this problem is quite complex and not something that can be computed or worked on instantaneously. It would be more suitable to be solved with a scientific programming language like Python or MATLAB and needs to be attempted in a mathematical programming environment.

If you would like to learn how to approach these kinds of problems in a systematic way, I would recommend studying mathematical strategy or game theory. These fields provide the tools and methods needed to solve similar strategic problems efficiently.

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