I celebrated a *Presidential Age *for the last half of 2008. By that, I mean I was 43 years old during the term in office of the USA’s 43rd President. And in fact, later this coming year I will turn 44, and have a *second* Presidential Age.

In this post, I want to explore some of the properties of this notion of “Presidential Age”. In particular:

- Everyone who lives long enough should expect to experience a Presidential Age of their own,
- only some people are likely to experience two, and
- experiencing three or more is rare but possible, and has happened in our nation’s history.

In what follows, I want to prove the first statement, explore the likelihood of the second, and give examples of the third.

### Everyone who lives long enough will experience a Presidential Age.

Assumptions: A) People live arbitrarily long lives, and B) over the course of your life, the mean length of a Presidential term in office will be greater than 1 year.

Roughly speaking, the reason why everyone gets a PA is the fact that your age increases by 1 each year, while the President Number increases more slowly than that on average, so eventually your age must become the larger of the two quantities. (Granted, in a few hundred years, the Presidential numbers will be larger than 100, and it will be harder for people to live long enough to experience their PA, but the general principle still holds.)

Making the previous paragraph more rigorous is at first glance non-trivial, as the two functions involved (your age and the Presidential number) are not continuous, so the Intermediate Value Theorem from the calculus doesn’t apply. And in fact you can have rational valued functions where f(a) < g(a) for all small enough a, and f(b) > g(b) for all large b, without f(x) = g(x) at any intervening point. (See the comments to a puzzle on winning percentages from jd2718′s blog for one such example.)

For the Age and Prez functions, though, it is significant that they are integer-valued, their values only increase, and they always increase by exactly 1. Thus the only way my Age can first become strictly larger than the Prez number is if I have a birthday (so my Age jumps by 1) on a day when there is no change in administration (so the Prez number stays unchanged), and hence just prior to my birthday my Age must have actually matched the Prez number.

### Getting more than one Presidential Age

Clearly lots of people will have more than one Presidential Age. Anyone who turned 43 between January 21, 2007 and January 19, 2009 will have been 43 years old during the GWB era, and will be 44 years old during the BHO era. But those who turned 43 earlier during the Bush presidency will have missed their chance, having turned 45 before President Obama took the oath of office.

The likelihood of getting a second Presidential Age depends, then, on the length of the term of the relevant President. If all Presidents served their full term in office, then in order to get more than one Presidential Age, your Age and Prez numbers must first coincide within two years of the end of that President’s term, and we’d expect this to occur between 1/2 and 1/4 of the time (depending on whether that president served one or two terms).

Reality is of course not so simple as all that, as some Presidents served less than their full term in office. Some examples:

### Nixon-Ford-Carter

Richard Nixon was our nation’s 37th President. He resigned the presidency on August 9, 1974, and was succeeded by Gerald Ford. Jimmy Carter followed as the 39th President on January 20, 1977.

An individual whose was born between 1/21/37 and 8/9/37 would have been 37 during the last months of the Nixon administration, 38 during Ford’s, and 39 at the beginning of Carter’s term in office.

### Van Buren – Harrison – Tyler

A more extreme example occurs in the 1840s, with the death of William Henry Harrison just one month after taking office. In that case, anyone who was nine years old between March 4, 1841 and April 4, 1841 would have three Presidential Ages (8, 9, 10).

### More than three?

In principle, it is possible to have more than three Presidential Ages, although for that to happen would require consecutive abbreviated terms in office, something our nation has never experienced, and hopefully never will.