The Mathematics of Number Plate Spotting

There are (at least) two variations on the game of Number Plate Spotting: Consecutive Number Plate Spotting (CNPS) and Non-Consecutive Number Plate Spotting (NCNPS). CNPS, popularised by the comedian and broadcaster Richard Herring, involves spotting all numbers in consecutive order, i.e. you must spot a 1 before you can start looking for a 2, and so on. NCNPS involves simply ticking off each number once it has been spotted, in whichever order you happen to spot them.

With the introduction of the new system of number plates the game has become more complicated as some people allow the spotting of, e.g. BN02 FDC as a “2”, and OY59 DXG as a “59”, while purists argue that these numbers are in fact abbreviated or encoded versions of the year and month in which the number plate was issued, and so do not represent integers in the range 1 – 999. I am personally persuaded by the purist argument in this case.

Even the pre-Millennial version of the game had variants as there were still different formats to the number plate, specifically the antique ABC 123, followed by the
PQR 123 S format, and finally the W 123 XYZ. Again some might argue that only one format should be chosen at the outset and number plates of other formats discounted, though this does not affect the mathematics.

On Twitter the other day, and in response to a message read out on the Andrew Collins and Josie Long BBC 6Music radio programme, I stated that the non-consecutive version of the game was “approximately 1,000 times easier than true CNPS”. But as soon as I sent the message I began to question this figure. I have since done the maths.

Assume then that we are only counting number plates whose format contains an integer in the range 1 – 999.

For CNPS each number must be spotted before the next. Assuming that there is an even distribution of numbers then on average it will take 999 distinct number plates before the given number is spotted. As there are 999 numbers to be spotted this means that the average game of CNPS will take 999 x 999 number plates, which is 998,001 in total.

The maths of NCNPS is a little more complicated*. Each number has to be spotted but in any order, hence:

The first number plate will contain a number not previously spotted, so the probability of it containing a number we are looking for is 1.

The probability that the second number plate contains a number that we have not already spotted is 998/999, which means that, on average, we will have to view 999/998 number plates before we can spot a new number.

Once we have the second number, the probability that the next number plate contains a number that we haven’t already spotted is 997/999, which means that, on average, we will have to view 999/997 number plates before we can spot the third number.

And so on, until the 999th number, for which we will, on average, have to view 999/1 number plates before we spot the final number we are looking for.

This means that, on average, the total number of number plates we need to view in order to spot all 999 numbers is

1 + 999/998 + 999/997 + 999/996 + … 999/1

This is a harmonic series and there is a formula** for calculating the total:

n x [ln(n) + γ]

where γ is the Euler constant (here approximated to 0.57722) and n is the total number of items to be spotted, in this case 999.

Hence we have

999 x [ln(999) + 0.57722] = 7,476.49 (2 d.p.)

Finally we can divide 998,001 by 7,476.49 which gives

133.49 (2 d.p.)

Which means that, purely in terms of the average number of number plates needed to be viewed to complete each version of the game, and discounting the modern format of number plate entirely, NCNPS is about 133.5 times easier than CNPS.

* I am of course here indebted to Simon Whitehouse’s excellent article on the topic of Panini football stickers and the coupon collector problem.

** A formula which I initially got wrong, and remains an approximation, but it’s close enough to illustrate the point I think.

What to drop

Here is the only thing I have written in a year. It was, of course, produced for the fantastic Summer Special edition of All The Rage.

What to drop
I fell down the stairs yesterday. I forget which foot it was that glanced the cliff-edge of one stair and slid across it, causing me to shoot down the slope, and banging my right arm quite badly on the way. It was fortunate, perhaps, that I wasn’t carrying our one-year-old daughter at the time, as though I would like to think that hard-wired primordial coding would have kicked in and instructed my reflexes to do all I could to protect the child, even at risk of more serious injury to myself, who knows what might have happened?

Let’s assume that evolution in all its complexity would ensure that I saved the child at the expense of myself and any other items I happened to be carrying at the time. It is not uncommon for example for me to be making my way downstairs with the infant firmly tucked under one arm and my laptop under the other. The laptop is at least as delicate as a baby, but while it has been with me for longer, and contains countless priceless documents which I seem incapable of backing up, I think in a head-to-head with my actual biological progeny, I’m guessing – this is all conjecture: how could we know? – it is the computer that would be sacrificed.

Which leads us inevitably onto the question:

Not including babies, what is the hierarchy of objects to be dropped when falling down the stairs?

To investigate this I propose first a list of not quite randomly selected items, in alphabetical order: a balloon; a canary; a cat; a cup of tea; a fried egg sandwich; a laptop; the laundry; a piano; a plaster of Paris bagel and cream cheese paperweight; ten chocolate layer cakes; an unbound manuscript; a vase.

This list instantly throws up some mouthwatering fixtures – canary vs cat is the stand-out tie – but also several potential “local derbies”, such as “balloon: helium or air? and “laundry: ascending (clean) or descending (dirty)?”.

So, and in no particular order:

We’re in strictly Laurel and Hardy / PG Tips territory here, the consequences of dropping a piano on the stairs being well documented. It is to be avoided, especially if you’re the pusher upper rather than the puller upper. But is it possible to drop a piano, if you’re pushing it upstairs? Doesn’t it drop you?

Ten Chocolate Layer Cakes There is a scene in Sesame Street, in one of the many songs dedicated to the number 10, in which a baker, holding a pyramid of ten chocolate layer cakes at the top of some stairs, sings “ten chocolate layer cakes”, and then falls down the stairs. It makes a right mess.

Fried Egg Sandwich We all understand the pain of losing a lovingly prepared fried egg sandwich before the first bite, and there’s the risk of some staining here too, especially if we consider the high probability of tomato sauce. However, the potential for serious damage is limited. Perhaps if that was the last egg, the safety of the sandwich should be prioritised, otherwise, regrettably, it might have to go.

Cup of Tea It’s the fried egg sandwich again, but with more valuable crockery – is it part of a set?

Cat and Canary The ability of a cat to right itself after release is legendary, and even on the uneven surface of a staircase, it should be able to escape the incident relatively unscathed. It is a cat. The canary is more contingent: a loose canary, a bird in the hand as it were, should, like the cat, be fairly self-reliant, unless it is being carried upstairs because it is has fallen asleep watching the snooker. Equally, a caged canary would only be able to mitigate slightly the effect of gravity, so an effort should be made to hold on to it.

Traditionally considered a “non-breakable” item, there is minimal risk involved in letting go of laundry in an emergency, the only slight concern might be that the clean laundry becomes dirty on falling, especially if dropped in conjunction with an egg sandwich. We shall assume for the sake of the argument that the laundry is not sufficiently dirty as to become a potential source of embarrassment if allowed to tumble out of the basket.

Balloon Dropping a standard domestic air/breath-filled balloon is relatively harmless. Dropping, or at least letting go of, a helium-filled balloon runs the risk of balloon loss, especially if, to add a consideration not hitherto introduced, the staircase is exposed to the open air. This in turn depends on the presence of a string attached to the balloon, and whether the string is tied to part of the body. Perhaps the balloon is being carried upstairs in 1994 after a Christmas party for employees of a leading insurance company, this particular employee having walked the four miles from the Botanical Gardens to their house in the early hours of the morning with the balloon tied to their wrist.

Plaster of Paris Bagel and Cream Cheese Paperweight
In all honesty, this item was serving merely as a pretext to acquire the girl in the coffee shop’s phone number, and as such can be released without guilt. Almost entirely without guilt.

Vase Are you just working in the museum in the summer holidays, or is this your career? You are about to find out.

Unbound Manuscript Inevitably, whilst you had finished the dissertation with plenty of time to cycle down to the department office before 4.45, you had seriously underestimated how long it would take to spell check and print it, and now you are cutting it very fine, and, gambling on the secretary having one of those really industrial staplers or something, just gather it into a bundle and set off down the stairs. It is at this point that you start to wish you had bothered to number the pages.

Laptop Hey, it’s a MacBook.

And here it is, the order in which you should drop various random items if you are about to fall down the stairs.

1. Balloon (air)
2. Cat
3. Canary (uncaged, awake)
4. Plaster of Paris Bagel and Cream Cheese Paperweight
5. Balloon (helium)
6. Laundry (descending)
7. Laundry (ascending)
8. Fried Egg Sandwich
9. Cup of Tea
10. Ten Chocolate Layer Cakes
11. Unbound Manuscript
12. Canary (caged/asleep)
13. Piano
14. Vase
15. Laptop

Remember, it’s very important to try to hold on to the baby.


Suitably minimal “flyer” below:

Enter ‘The Zone’ with
Stalker Night @ A E Harris.

On Friday 17th July at 7.30
Stan’s Cafe celebrate one of their greatest inspirations

Andrei Tarkovsky‘s extraordinary film Stalker

Alongside a DVD screening of this wonderful film there will be

a presentation of

Zone a photographic series by acclaimed photographer David Bate
inspired by the film, shot in an around its location.

two short performances devised by Stan’s Cafe
in homage to the film, one a ‘reworking’ of our first
major international hit It’s Your Film

of course there will also be a vodka bar

and, for those inspired by the tonsorial style of the film’s
eponymous ‘hero’, there will be a free hair clipping service on offer.

Tickets £5 on the door.

@ A E Harris, 110 Northwood Street, Birmingham, B3 1SZ

There is no why

One of my earliest memories is watching Blue Peter at what must have been the very end of 1973, making me five and a bit. The TV was filled with footage of Viking ships on fire, and one of the presenters was saying “and so we say goodbye to 1973…”. But what I was aware of at the time, and even more so now, was “what has 1973 got to do with Viking ships on fire?”

And so we say goodbye to 2008.

2008 has been a shit year for many people, including some personal friends, with bereavements and so on. It is not, I am sure, “the year in which everyone gets what they deserve”. I would have to be something more of an arrogant sod than I actually am to believe that it was the year in which I personally got what I deserved, as, overall it has been very good. I sipped champagne standing next to Ronnie Corbett. I visited Australia and New York for the first, and who is not to say only time. I became a forty-year-old man and got married on the same day. I made a return to theatrical performance after an eighteen year break. Lots of things.

Birmingham continues on its course of becoming a more interesting place  in which to live. Stan’s Cafe are instrumental in this, and this has been another in a series of triumphant years for them, culminating in bringing Of All The People In All The World to the city in September. In 2009 they take occupation of the A. E. Harris factory for who knows what fun. 7inch Cinema are another native cultural outfit that enhance the city and my life in it. I’ve been to a few Pilot Nights too, performing Reflections on the Counting of Sneezes in December. One of my favourite short theatrical performances was Edward Rapley‘s 10 Ways to Die on Stage, during which I failed to get wet, and I was also very taken by Where We Live and What We Live For by Kings of England, with its gentle eulogy to the passing of time.

The back bedroom office, as visitors to Sneezecount will know, has reverberated to the sound of the Collings and Herrin podcast, and the Adam and Joe radio programme. I made a video for the Adam and Joe Video War, which I was quite pleased with, even though it lost.

And finally, my favourite cultural event of the year was probably Man on Wire, in which Philippe Petit sums it all up. “There is no why”.

A Geek Misery Memoir

This month’s All The Rage theme is Games and Play, and it is brought to you alongside the London Games Festival Fringe. Here below are some of my thoughts on the topic.

I think it was for my eighth birthday that I received a pair of binoculars. They were not toy binoculars but proper grown-up binoculars, hefty, with a binoculary smell. Naturally I was very pleased with them, but I was also distracted by two items that I found in the box. The first was a small sachet of granules which ate moisture that would otherwise get inside the lenses and spoil them. I would of course go on to encounter many more of these granules over the years, in leather goods, in the packaging of gadgetry, even in drawers, but this was my first exposure to them. You never forget the first time. The idea of these moisture eating granules intrigued me, as did the thought that moisture might get inside the lenses and ruin the binoculars. Where did this moisture come from? But what fascinated me more than the granules, perhaps more than the binoculars themselves, was the other item, the five year guarantee. I was eight years old, and this felt to me like a ridiculously exuberant promise from the manufacturers, a Lifetime Guarantee in all but name, and it made me dizzy. Looking forward in time as an eight year old I was holding the binoculars the wrong way round, time stretching out impossibly in front of me. Reversing the perspective as I prepare to enter middle age, the telescopic lenses have a foreshortening effect, as, for example, I blinkingly realise that those episodes of Saturday Night Fry that I have just listened to on YouTube first entered my consciousness over half a life ago.

It is the fate of binoculars never quite to live up to the excitement that their acquisition generates. They are in this respect very much at the same end of the toy box as spy kits and invisible ink pens, or walkie-talkies, wherein their potential is never fully realisable in the eight-year-old universe. Who is to be spied on, what secrets are to be encrypted, who will hear the over-and-out? It doesn’t matter. The function of these playthings is not to be found in their use, but in the giving and the receiving, and in the first heady moments of possession.

Those childhood years of Action Man and espionage were rich and fun, but as a teenager I was deprived. Attending a school alongside fifteen hundred boys, my only exposure to girls were the tantalising glimpses through the bus or car window as the traffic inched its way up the Palatine Road, the whining school-boy, with his satchel and shining morning face, creeping like snail unwillingly to school. Shining with grease and spots, of course.

I was not even able to take solace in technology, as the other boys did. Our household budget somehow never stretched to one of the VCRs that were laid out, with their gaping cassette trays, in the Littlewoods catalogue. This, combined with my relative geographical isolation, meant that I was never invited to the weekend video club meetings that clustered in the outer lying suburbs, and so I was excluded from the informal education of Escape from New York, or The Life of Brian, or Porkies, and from the Monday conversations as the retelling lubricated and reinforced the social networks of the video-owning classes. I once forced my way into one of these sessions one afternoon by resolutely staying on the bus as the others talked excitedly of a trip to the video shop, and then more or less just following them home. They didn’t seem to mind, but the video that we shared was King Frat, and I always got off the bus at my own stop after that.

I was further deprived of the natural recourse of the awkward and excluded teenage male, as our house contained neither computer nor computer games console. Actually strictly speaking it had contained a ZX81, but once I had deployed the customary two line program to fill the left-hand part of the TV with my name, the device lost its novelty, and was put aside. For some reason this dipping of the toe into the lukewarm waters of early ’80s home computer did not progress into the more fruitful areas of the Spectrum, or Commodore 64, or even Atari. Space Invaders, Pac-Man and Defender were therefore holiday amusement arcade treats only, but my lack of practice meant that these machines just ate my money, and I always felt more comforted by the capitalist allegories of the coin nudging cascades.

All Creatures Great and Small

It’s time for the first All The Rage of a new autumn, and so here is my bit.

There was an old woman who swallowed a fly.
I don’t know why she swallowed a fly.
Perhaps she’ll die.

I swallowed a fly once, but I survived. The key, I think, is not to panic.

But the old woman in the story overreacted to the initial incident, resulting in a catastrophic escalation of hostilities. This is how the biological arms race played out:

Fly v. Spider v. Bird v. Cat v. Dog v. Goat1 v. Cow v. Horse

Applying the “enemy’s enemy” principle, we can divide the animal combatants into two sides, or teams, as follows:

Team A

Team B

The sequential ordering of the teams in the original story puts us in mind of the tradition playground “first pick, second pick” system, though on this occasion we find a curious reversal, wherein the supposed weakest players are selected first, Team A effectively taking the first pick.

It is worth perhaps taking some time to examine in more detail how the two teams line up.

Sometimes a team is only as strong as its weakest player (butterfingered relay sprinters coming to mind) in which case Team A is in for a drubbing. Certain spiders could of course “have” any of the other animals, very much “punching above their weight”. (Equally, the bird that was swallowed to catch the spider would be in for something of a shock if the spider in question turned out to be the South American Goliath Bird Eating Spider.2) In all, it seems safe to conclude that the spider might well turn out to be Team B’s secret weapon, not least taking advantage of its opponent’s tendency to underestimate its power, a scenario endlessly exploited in David Carradine’s popular Kung Fu television series.

Assuming that the bird gets lucky, keeps its head, and overcomes the spider, it would then come up against the cat. Popular wisdom generally presents the cat as a wily and effective protagonist, the only possible problems being represented in the natural-order-overturning cartoons such as Tom and Jerry, or, most pertinently in this case, Sylvester and Tweety Pie. Did the old woman swallow Tweety Pie? It seems unlikely. Notch this one up to Team B.

The Cat versus Dog contest is as old as fictional animal conflict itself, and as hard to call as that between a monkey and robot, depending as it does on so many variables: age, breed of dog, “street” experience of the cat, previous martial arts training, etc. With the evidence presented to them, the Pools Panel award a score draw.

Sending in a goat to catch the dog is a curious strategy, given the dog’s more traditional role as a herder and “worrier” of domestic ruminants. Goats, however, are noted for their willingness to eat anything, and their stomachs are reported to be able to digest almost any organic substance – “ logically this would include dogs. Folklore surrounding the goat is also instructive. The three Billy Goats Gruff first outwit and then defeat in combat the apparently more powerful and aggressive troll, and research indicates that “a common superstition in the Middle Ages was that goats whispered lewd sentences in the ears of the saint”3. Sending in the goat, therefore, while at first sight quixotic, turns out to be a tactical masterstroke. Combining tried and tested butting and kicking skills with stamina, intelligence, the element of surprise and “PsyOps” (in the form of dirty whispering)4, that’s another round to Team B.

In desperation, Team A deploys the cow, an animal less suited to catching a goat it would be difficult to imagine. Unless the cow is planning on mooing the goat into submission, the situation is becoming hopelessly one-sided. If this were boxing, as commentators of all sports (except boxing) are prone to surmise, the referee would have stopped it a long time ago. Team B finishes the job itself though, with the straightforward deployment of the horse, since antiquity the experts’ choice when it comes to the rounding up of cattle.

1 I had no idea that there was a goat until I checked. Where would I be without the Internet? Nothing but a little heap of bones.
2 According to my favourite user-compiled Internet encyclopaedia, the Bird Eater, or Theraphosa blondi, while capable of seriously ruining a bird’s plans for the evening, is “fairly harmless” to humans, despite having a leg span of up to 30cm, and having fangs capable of biting off a human finger. Females have a life span of between 15 and 25 years, but males only 3 to 6, a discrepancy no doubt partly due to the idiosyncratic arachnid female’s attitude to one-night stands.
3 Internet encyclopaedia, ibid.
4 For more information about PsyOps, goats, and all manner of military crazybonk, see The Men Who Stare At Goats by Jon Ronson