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Since the first vehicle was officially registered in Spain in 1900, there have been three different registration systems. Until 1971, the system was to use one or two letters that identified the province (initially there were up to three letters in some cases) in which the vehicle was registered, followed by several digits.
Each province started with the number 1, so if it had continued with this system, there would have been no problem registering an infinite number of cars. However, the number of digits would grow, as would the cost of registration plates, and it would end up being very cumbersome to identify each vehicle.
So that the October 7, 1971 the system was modified, keeping the letters of the province, but using only four digits (which started with 0000), adding a letter at the end, from A to Z, not including Ñ, Q, or R, to avoid confusion with N, O, P, respectively.
In short, 24 letters, if I have not counted wrong. When the combination 9999 – Z arrived, a new letter would be added, continuing with 0000 – AA, and so on.
In addition to having to label fewer characters on the plates compared to the initial system, what was improved? Let’s count how many vehicles we would service. From 0000 to 9999 it is obvious that we have 10,000 figures. By combining them with a letter, we get up to 240,000 units, not too many. However, adding a second letter makes matters much better. How many possibilities do we get from AA to ZZ?
Still in shock when remembering the responses of some readers to the column from a fortnight ago, it has occurred to me to incorporate the acronym C4OP, whose meaning I explain below, into each review that I write in the future. Although they are not very fond of cinema, or of a certain galactic saga (‘Star Wars’), the name of one of the main androids will surely ring a bell. Well, not long ago, in a galaxy very close by, there was a series of units called C4OP, who thought that it was possible to do anything With 4 Operations (addition, subtraction, multiplication and division) only. The idea is none other than to show that around us we need more mathematical content than that, trying to make it playful and fun. In the elementary case proposed, a C4OP would simply write all the possible license plates, and start counting them. A couple of times at least, of course, it’s easy to lose count and/or get it wrong.
There were also the TEDI gentlemen (Initially Difficult Enunciated Nonsense), whose ‘strength’ was based on exhaustive scientific knowledge (physics, astronomy, chemistry, medicine, all known sciences along with mathematics, of course) together with a vast philosophical background. and humanistic that allowed them to discern and fully respond to all those GEDI questions raised wherever they were. That meant discarding wasting time on matters of their ancient ancestors such as telekinesis, clairvoyance, telepathy and other pseudoscientific nonsense (it was their respectable opinion), although they maintained mind control (to put up with certain barbaric assertions heard with some frequency) and abilities such as fencing, wrestling, martial arts, etc., because it was always possible to find some little evolved specimen to give the occasional lesson with their own tools. Obviously for a GEDI, the answer was immediate: 24 x 24 were all the possible variations with a pair of letters (variations with repetition they called them) that together with the 10^4 possible digits made a total of 5,760,000 possible license plates. Much more than the initial system.
However, the number of vehicles continued to skyrocket, and before reaching 9999 ZZ (in fact, it ended up with license plate M – 6814 – ZX), a new system was implemented (on September 18, 2000). The identifications of the provinces disappeared (thus avoiding unspeakable acts of vandalism), using four digits and three letters, excluding the Ñ, the Q and the five vowels (to avoid the formation of well-known three-letter words, such as ANA, FEO and others) . How many different license plates can be formed in this case? (If the C4OPs have finished with the previous account, let them start with this one).
In a similar way to that described above, we again have 10,000 different figures, and now 20 letters, so the total number will be 20^3 x 10^4, that is, 80,000,000, 80 million different vehicles, which gives us This leaves room for some years (according to data from the DGT, the last registration was 21** LYH; the complete registration is not provided for security reasons). New test to find out if you are a C4OP or a GEDI: assuming that there are no vehicles circulating with the previous systems (since by law they are obliged to request a license plate with the new system), how many cars are currently circulating in Spain assuming that the last license plate was 2199 LYH? If in the year 2020, the total registration was from the letters LDP to LLC (let’s put from 0000 of the first combination to 9999 of the second), at that rate, try to estimate for how many years we will have enough registrations with the current system .
Other countries
Far from having a common universal system, each country has its own. For example, the UK introduced its current format on September 1, 2001, and it’s a bit more cumbersome to describe. Each license plate consists of seven characters with a format defined as follows:
Apart from the flag and the abbreviation of the country, there are two letters, the area code, indicating the local vehicle registration office. As the first letter they only use one of these twenty: A, B, C, D, E, F, G, H, K, L, M, N, O, P, R, S, V, W, X, Y ( that is, from the usual alphabet without Ñ, they also eliminate I, J, Q, T, U and Z, for different reasons that I will not detail). The second letter is an identifier of the Driver and Vehicle Licensing Agency (DVLA identifier), and each area uses specific ones for each office, and also reserves some for specific cases. But most use 23 letters. Thus, for example, the license plate in the image was issued in Birmingham (B), at a fourth sector licensing agency (D). Next, there are two digits, the year identifier, which change twice a year, on March 1 and September 1. For example, if the registration was issued between March and August of the year 2022, a 22 appears, the last two figures of the current year. If it is done between September and February 2022, 50 is added to 22, that is, a 72 would appear (C4OPs should read this slowly so as not to explode their heads when trying to deduce the year of issue of the license plate from the image).
Next, there is a sequence of three random letters that uniquely distinguish all vehicles with the same initial area of ​​the previous four characters (remember that these identify the area and the year in which the vehicle was registered). In this case, the letters I, Q are excluded from the 26 letters of the alphabet, as well as combinations that may be offensive in English or any foreign language (take it now!), such as, for example, ASS, SOB, etc. . It is common for ‘neighboring’ letter sequences to correspond to the same vehicle manufacturer.
This scheme has obvious advantages:
1.- A buyer of a second-hand vehicle can determine the year of the first registration of the vehicle without having to look for it. However, a vehicle is permitted to display a license plate where the age identifier is older (but not newer) than the vehicle. The wide knowledge of the population of how the ‘age identifier’ works (the two figures that accompany the area identifier) ​​has led second-hand car dealers to use the license plate as reliable proof of the vehicle’s age and not having to accept the year displayed on a poster, which can be misleading.
2.- In the case of a police investigation of an accident or a crime related to a vehicle, the witnesses usually easily remember the letters of the initial area code (or better still the two letters and the two numbers), being easier to reduce suspicious vehicles to a smaller number of cases, consulting the authority’s database without having to know or memorize the complete number.
3.- The designers of the system anticipated that the scheme will have enough random sequences of 3 letters for each combination of area code and age identifier to be calm until February 28, 2051.
As is evident, the final questions are obvious: does the reader (indistinctly C4OP or GEDI) dare to determine how many vehicles the British can register with the aforementioned system? Is it better or worse than the Spanish system? And if you liked the matter, would you be able to design a different system to the previous ones that with no more than seven characters (note that this is the number for both Spain and the United Kingdom) provides a greater number of possible vehicle identifications ? Just think of the total number; the very precise British sophistications of identification we will leave them aside, although if they dare…
Alfonso Jesús Población Sáez is a professor at the University of Valladolid and a member of the Dissemination Commission of the Royal Spanish Mathematical Society (RSME).
The ABCdario of Mathematics is a section that arises from the collaboration with the RSME Dissemination Commission.
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