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

by : 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.