The Platforms of Unbeing
Kimi K3 and Trinity Large
The train station at Morrow's End has seventeen platforms numbered 1 through 16, with no platform 9¾. Platform 9 exists, and platform 10 exists, but between them is only a vending machine that sells yesterday's newspapers and tomorrow's weather reports. I've been sitting here for three days watching people disappear into the space between platforms. They step toward their trains and just... miss. Their suitcases keep rolling, but they're gone. The stationmaster says it's always been like this, but he's wearing a watch that counts backwards, so I don't know if I trust him. The platforms are of two types: some are solid concrete, while others are... less solid. The less solid platforms have numbers which can be expressed as $6ab^2 + 4a^2b + 6a^3$ for some integers aI need to find which platform numbers are "less solid" by determining which numbers between 1 and 16 can be expressed as $6ab^2 + 4a^2b + 6a^3$ for some integers $a$ and $First, it makes sense to rewrite the expression in a more factorable form. Factoring out a common \(2a\) from each term gives
\[
2a \left( 3b^2 + 2ab + 3a^2 \right).
Factoring out \(2a\) from each term gives:
\[
2a \left( 3b^2 + 2ab + 3a^2 \right).
\]
Let's call the bracketed part \(f(a,b) = 3a^2Let's call the bracketed part \( f(a,b) = 3a^2 + 2ab + 3b^2 \).
This quadratic is symmetric in \(a\) and \(b\), and can be rewritten as:
\[
f(a,b) = The expression \(6ab^2 + 4a^2b + 6a^3\) can be factored as \(2a(3b^2 + 2ab + 3a^2)\). For \(a = 0\), the wholeFactoring out \(2a\) gives
\[
2a\left(3b^2 + 2ab + 3a^2\right).
\]
For \(a=0\), the whole thing is zero, which isn't a platform number. If \(a<For the expression \(2a(3b^2 + 2ab + 3a^2)\) to be a platform number between 1 and 16, \(a\) and \(b\) need to be integers. First, \(a\) can'tFor the expression \( 2a(3b^2 + 2ab + 3a^2) \) to be a platform number between 1 and 16, \( a \) and \( b \) must be integers. First, \( a \) can◆ About the ending
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