If $a, b, c$ are three distinct natural numbers, all less than $100$, such that $|a - b| + |b - c| = |c - a|$, then the maximum possible value of $b$ is ______

To find the maximum value of $b$ where $a$, $b$, $c$ are distinct natural numbers less than 100 satisfying $|a - b| + |b - c| = |c - a|$.

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First, let's analyze the equation $|a - b| + |b - c| = |c - a|$

This equation is true if and only if $b$ lies between $a$ and $c$ (either $a < b < c$ or $c < b < a$).

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To maximize $b$, we need to make it as close to 100 as possible while ensuring it satisfies our constraint.

Let's try $b = 98$ with two possible arrangements:

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Case 1: If $a < b < c$

$b = 98$

$c = 99$ (largest possible value less than 100)

$a = 1$ (can be any number less than 98)

Verifying:

$|a - b| + |b - c| = |1 - 98| + |98 - 99| = 97 + 1 = 98$

$|c - a| = |99 - 1| = 98$

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Case 2: If $c < b < a$

$b = 98$

$a = 99$ (largest possible value less than 100)

$c = 1$ (can be any number less than 98)

Verifying:

$|a - b| + |b - c| = |99 - 98| + |98 - 1| = 1 + 97 = 98$

$|c - a| = |1 - 99| = 98$

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Since both cases work with $b = 98$ and we cannot choose a larger value for $b$ (as we need distinct natural numbers less than 100), the maximum possible value of $b$ is 98.