## Tuesday, January 3, 2017

### Age puzzles

Some quick puzzles created and discussed with J2 while at lunch today. For all, we made the simplifying assumptions that everyone has their birthday on the same day.

Basics/assumptions
Our friends were discussing their current ages: Jin 9, Jate 7, Panelia 6, Sophia 5, and Jane 4. The first two are boys, the latter three girls. All have their birthday on the same day of the year and are having their birthday today!

The boys wondered: will it or has it ever happened that the sum of our ages is the same as the sum of the ages of the girls?

Doubles
A little twist on the previous question: when will the sum of the girls ages be twice that of the boys?

For those that know algebra: can you make sense of the solution?

Halves
The previous puzzle was a bit strange. What if we go the other way: when will (or were) the sum of the ages of the boys twice that of the girls?

Murky waters with products
What is the current product of the boys ages? The product of the girls ages?

Have those products ever been equal? We argued this based on the intermediate value theorem.
When were those products equal? We numerically approximated to the closest half year.

## Some other families

J2 asked me to include these, which we'd discussed during a dinner last week.

Bill is five years younger than his sister. In seven years, Bill will be 2/3 his sister's age. How old are they now?

John is ten years older than his brother Joe. In six years, John will be twice as old as Joe. How old is Joe now?