Propositiones ad Acuendos Juvenes/Problems/42 - De Scala Habente Gradus Centum

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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $42$

De Scala Habente Gradus Centum
One Hundred Steps
A stairway consists of $100$ steps.
On the first step stands a pigeon;
on the second, two pigeons;
on the third, three;
on the fourth, four;
on the fifth, five;
and so on every step up to the hundredth.
How many pigeons are there altogether?


Solution

We count them as follows.

Take the single one on the first step and add it to the $99$ on the $99$th step, making $100$.

Taking the $2$nd with the $98$th likewise gives $100$.

So for each step, one of the higher steps combined with one of the lower steps, in this manner, will always give $100$ for the two steps.

However the $50$th step is alone, not having a pair.

Similarly the $100$th step is alone.

Join all together and get $5050$ pigeons.

$\blacksquare$


Historical Note

David Wells calls to mind the apocryphal story (quite possibly untrue) about Carl Friedrich Gauss, where at the age of $8$ he solved the problem of adding the numbers from $1$ to $100$ in a few seconds.

David Singmaster remarks that the knowledge of how to sum an arithmetic series like this seems to have been known to the ancient Babylonians and ancient Egyptians.

It was also known to the ancient Greeks, but interestingly does not appear in Euclid's The Elements.

Alcuin's is the earliest text which dresses the problem up in fancy clothes, and many European texts follow his example.

In such renditions, the number of elements to add is almost always $100$.


Sources

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