Math Problem Statement
can you solve
<p>Given two positive integers $a,b$, Alex and Bianca play a game in $ab$ rounds. They begin with a square piece of paper of side length $1$.</p> <p>In each round Alex divides the current rectangular piece of paper into $a \times b$ pieces using $a-1$ horizontal cuts and $b-1$ vertical ones. The cuts do not need to be evenly spaced. Moreover, a piece can have zero width/height when a cut coincides with another cut or the edge of the paper. The pieces are then numbered $1, 2, ..., ab$ starting from the left top corner, moving from left to right and starting from the left of the next row when a row is finished.</p> <p>Then Bianca chooses one of the pieces for the game to continue on. However, Bianca must not choose a piece with a number she has already chosen during the game.</p> <p>Bianca wants to minimize the area of the final piece of paper while Alex wants to maximize it. Let $S(a,b)$ be the area of the final piece assuming optimal play.</p> <p>Find $S(2,2)$. Give your answer in scientific notation rounded to ten significant digits after the decimal point. Use a lowercase e to separate the mantissa and the exponent.</p>Solution
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Math Problem Analysis
Mathematical Concepts
Game Theory
Optimization
Geometric Shapes
Formulas
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Theorems
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Suitable Grade Level
Advanced High School
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