# Stadium Paradox

## Paradox

Consider three rows of bodies:

- $\begin{array} {ccccc}

\text {(A)} & 0 & 0 & 0 & 0 \\ \text {(B)} & 0 & 0 & 0 & 0 \\ \text {(C)} & 0 & 0 & 0 & 0 \\ \end{array}$

Let row $\text {(A)}$ be at rest, while row $\text {(B)}$ and row $\text {(C)}$ are travelling at the same speed in opposite directions.

- $\begin{array} {ccccccc}

\text {(A)} & & 0 & 0 & 0 & 0 & \\ \text {(B)} & 0 & 0 & 0 & 0 & & \\ \text {(C)} & & & 0 & 0 & 0 & 0 \\ \end{array}$

By the time they are all in the same part of the course, $\text {(B)}$ will have passed twice as many of the bodies in $\text {(C)}$ as $\text {(A)}$ has.

Therefore the time it takes to pass $\text {(A)}$ is twice as long as it takes to pass $\text {(C)}$.

But the time which $\text {(B)}$ and $\text {(C)}$ take to reach the position of $\text {(A)}$ is the same.

Therefore double the time is equal to half the time.

## Resolution

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## Historical Note

The Stadium Paradox is one of Zeno's Paradoxes, as famously raised by Zeno of Elea.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Zeno's paradoxes** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**stadium paradox** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Zeno of Elea**(5th century bc) - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**Zeno of Elea**(5th century bc)