Propositiones ad Acuendos Juvenes/Problems/45 - De Columba

Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $45$

A pigeon sitting in a tree saw others flying, and said to them:

"I wish there were others with you --

the same number again and a third time.

Then with one of me, there would be $100$."

How many pigeons were there at first?

$33$

Let $x$ years be the number of pigeons there at first.

Then we have:

$\ds \paren {x + x + x} + 1$

$=$

$\ds 100$

The number plus the same number plus the same number again plus $1$

$\ds \leadsto \ \ $

$\ds 3 x$

$=$

$\ds 99$

simplifying

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds 33$

$\blacksquare$