# Order Theory

## Mr and Mrs Tan have five children: four boys and a girl. They compared their ages and this is what the children said:

Alfred: "I am the second place, counting either from the oldest or the youngest."

Brenda: "I have at least one younger brother."

Charles: "I am two places older than my sister, Brenda."

(i.e.. there is one sibling older than Brenda but younger than Charles)Darius: "Brenda is my younger sister."

Eric: "I am the...."

Unfortunately, Eric was interrupted. Can you complete his sentence?

There are three possibilities for the positions (in order of birth) of Charles and Brenda: \[\left(1^\text{st}, 3^\text{rd}\right), \left(2^\text{nd}, 4^\text{th}\right), \left(3^\text{rd}, 5^\text{th}\right).\] The second possibility is out as we see that Alfred is either \(2^\text{nd}\) or \(4^\text{th}\) so if the second possibility is the case, then there is no possible position for Alfred. The third possibility is also out as Brenda has at least one younger brother so she cannot be the youngest. This leaves us with the first possibility where Charles is the oldest and Brenda is the middle child.

We also see that Darius is older than Brenda, the \(3^\text{rd}\) child. The first place has already been taken by Charles so Darius must be the second oldest. Alfred is either \(2^\text{nd}\) or \(4^\text{th}\) but \(2^\text{nd}\) has been taken by Darius so Alfred is the second youngest. The only vacancy left for Eric is the \(5^\text{th}\) child. In other words, Eric is the

youngest. \(_\square\)