[It Got Away] Vermont License Plates

So I’m in Colchester, VT and I’m noticing the license plates.

Three letters followed by three numbers. Every time. The second you fix a format like that, you’re dealing scarcity. Just like in Costa Rica, I’m wondering, “How many license plates are available?” I’m also wondering, “Is it possible to determine the order in which those plates were issued? Can we guess it?”

Then one of the teachers at the school where I’m facilitating the workshop mentions that the first license plate issued was “AAB100” and that Vermont only recently issued “F-----.” Now we’re on our way to a timeline and we’re dealing with the much more interesting question, “When will they run out?”

I needed data. I needed to know when the first license plate in the series was issued and I needed to know what license plate they last issued. I got way more than that.

I called the Vermont Department of Motor Vehicles, which pinballed me around between various confused functionaries until I reached a person who e-mailed me a trove of data, including the one-, two-, and three-letter combinations they don’t allow under any circumstances on their plates. (NSFW, I suppose.) Now we have a scarcity problem with prurient appeal.

So here’s the question:

If AAB100 was issued in January 1990 and if EYT100 was issued in July 2009, when is Vermont going to run out of plates?

You’d have to calculate the total possible plates, excluding the disallowed combinations while respecting the DMV’s various constraints. (eg. you’ll never find a zero in the hundreds place; “FU----” is disallowed while “-FU---” is allowed.) You have to gather some data on yearly issuance of plates. The better the model’s fit (ie. if exactly 60,000 plates are issued every year) the better our mathematical investigation will focus on our targeted skills.

It Got Away

Alas, the model is all over the place.

[BTW: I changed the chart type per Tom’s recommendation.]

So maybe you reduce the data to a single average of 77,229 plates per year and proceed as planned. Maybe you leave the problem at “how many license plates are available?” and set aside the issue of time. Both options kind of bummed me out, though, so I scrapped this one.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Not all problems in the real world have exact solutions.

    Solving the problem to the point that you have a formula with a single variable for the rate of new and used car purchases in Vermont would be a useful demonstration.

    There are probably many organizations which have an interest in and therefore a prediction of what those future numbers might be. Plugging these numbers into the formula to see a range of possibilities is useful.

    Showing students the range of possibilities may in fact encourage some of them to pursue a ‘better’ answer on their own.

    ps: Your graph type is misleading as your datapoints are not continuous (they are yearly totals, descrete points every twelve months). A bar chart would be more visually truthful.

  2. The logic of running out of plates is flawed of course. I live in Victoria Australia and we have the same plate structure. When I was a lad my dads car was J…… My current car is U….. some 20odd years later, but they have been through the entire series in the interim. To run out of plates you would need to have that number of cars registered at the same time, no small feat I would suggest.

  3. Dan,

    Thanks for posting these. This new series is a great addition to your already-intriguing blog. The “it got away” problems are what allow the truly golden problems to exist. It reminds me of this, from Ira Glass (creator of This American Life on NPR): http://www.youtube.com/watch?v=KW6x7lOIsPE&feature=channel

    I agree with you on killing this project, BUT, just for the sake of discussion, you could ask kids how this variation came about. Interest rates spike, and people buy less cars? The economy tanks and people buy less cars? This information may not be accessible without a lot of digging on the kids’ parts, and it may get in the way of an easy solution to the question of when the numbers will run out, but isn’t this the real world? If this problem requires kids to pose a solution based on a lot of assumptions that they create and defend regarding why the data looks the way it does, and why they hypothesize it will look a certain way in the future, I say it can’t be all that bad. Again, it may not be the absolute best use of class time, particularly if you have better questions to ask. But it may not be totally junk.

    Just my two cents…

  4. What if you entered into a discussion on the shape of the graph? Why does the data behave this way? What forces caused a spike in the early 2000’s, and why has it dropped in the last couple of years? Based on that, can we speculate a possible realistic data point in the future, maybe 2015, or 2020? What kind of model do we expect best represents the long-term behavior: linear, quadratic, sinusoidal, other? Now let’s build a regression model and extrapolate. This, of course, directly leads to a discussion on the dangers of extrapolation, and a discussion on what the DMV can do to improve the estimated time until the plates run out.

    As a side note, one kid will almost certainly ask why the DMV needs to know this! Can’t they just switch the plate design when they start to get close to the end of possible numbers? Another student may ask why they can’t recycle old plate numbers, both good questions that, depending on how they are answered, may push this question into the “pseudo-context” category.

  5. Virginia has the most personalized license plates per capita of any state. We are, it seems, quite vain.

    Some of the plate designs are hilarious, and pretty clever. We actually did a lesson around license plate permutations (and Nike iD, phone numbers, etc.) a while back. If you skip to slide 14 of the presentation, you might get a laugh. Row 3, column 2 is my personal favorite.

  6. I don’t know … this might be salvageable.

    Something that occurred to me is that the plates started with A– — and 19 years later are basically at the end of the Es so you don’t have to deal with the permutations at all. Then, account for the seven letters they exclude as starting letters and you get this:

    19 years = 5/19ths of the possible combinations.

    Again, rough numbers but I think the statistics will be pretty accurate at least for the purposes of the question – when will the numbers run out? The numbers work out to 72.2, Call it 70. After 2060, we can add a digit.

    Attempting to extrapolate more accurately would be akin to making much out of the airline cost vs miles, time vs miles data. Real life has too much fuzz to be all that accurate.

  7. I’ll have to fire up Excel sometime tomorrow.

    I think if you looked at the cumulative number of license places assigned, you would get a more interesting graph. It should be vaguely quadratic.

  8. I don’t really see enough through my math goggles, but from what I can tell I’m glad you stopped on this one. Almost as soon as you worded the thing you wondered about, it felt like a word problem from the old days. What I like about the ones that have worked is that they ask a compelling question that demands an answer. This one, not so much.

    And it’s as crucial to examine why *not* to do something as why *to* do something. We all need to think about both sides of the issue more in the quest to get it right.

  9. Can someone clarify this for me (from the list of restrictions):

    “I — O — Q — J — U — V — Z

    (No combinations starting with or including any of these letters.)”

    Does this preclude these letters entirely? Am I misreading, or is it worded incorrectly?

  10. After reading the comments so far, I’m left wondering about the value of this problem in the context of Todd’s “getting it right”. I believe that this problem is valuable to students because it can give them a better impression of how one goes about determining when *not* to continue to do something. So many students (myself included at times) do not know when to stop or even know the signs of completion within the problem solving process. The fact that this one “got away” speaks to the fundamental limitations that putting a functional framework on a problem can produce. The problem is no less important in my mind, and can be used to assist students in understanding that we can count the number of plates possible and give estimates for how long the count will last, but in the end there is no tidy little answer and we must accept a partial understanding of the situation.

    On a purely unrelated jam band note, the fact that Dan posts from Vermont and titled this entry “it got away” can’t be a coincidence as far as I’m concerned.

  11. Maybe you could get figures from a few other states and figure out how bad this system would be for California or some other larger state and then determine at what annual average of new plates it is unworkable if a state does not want to run out of numbers in less than 30 years? 50 years?

    Would this system work only in the smallest states or would it work in states with 2, 5, or 10 times more license plates than Vermont?

  12. New Jersey ran out of numbers and changed formats:

    From AAA-10A format to the reverse A10-AAA.
    (It appears from the article that NJ does not use 0 as the first of the two numbers.)

    VT & NJ both use 6 characters, but even without using I, O, & Q (to avoid confusion with 1 & 0), NJ has a lot more combinations using 4 letters & 2 numbers compared to VT’s 3 & 3 format. Even so, the article says NJ estimates this system will last only about 20 years.

  13. I think if you included data for the number of plates returned each year, or put back into circulation might give a fairly nice linear trend. At least that would be the hope.

    Also I noticed that WTF wasn’t disallowed.

  14. Here in Victoria (a state of Australia) we have the same licence plate system. Three letters, then three numbers. We’re currently up to the Uxx-nnn series I think, at least I haven’t seen any new plates starting with Vs yet. My car which was bought in 2000 has a numberplate of the form QIx-nnn.