Mathematics in Action: Vehicle Identification Numbers

Motor vehicles have a 17-character Vehicle Identification Number or VIN on a metal plate like the one below, usually on the driver’s side dashboard, or on the driver’s side door jamb, or in front of the engine block:

A Vehicle Identification Number (VIN) plate (Photo: Michiel1972)

VINs offer an interesting example of check digit calculation. The central digit (or an X representing 10) is a check digit (calculated modulo 11) used to detect errors. Any letters in the rest of the VIN are decoded like this:

A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z
1	2	3	4	5	6	7	8	–	1	2	3	4	5	–	7	–	9	2	3	4	5	6	7	8	9

The check digit calculation involves decoding the VIN, and multiplying the resulting numbers by the weights shown in blue, giving the products in purple:

VIN	L	J	C	P	C	B	L	C	X	1	1	0	0	0	2	3	7
Decoded	3	1	3	7	3	2	3	3	10	1	1	0	0	0	2	3	7
Weights	8	7	6	5	4	3	2	10	0	9	8	7	6	5	4	3	2
Product	24	7	18	35	12	6	6	30	0	9	8	0	0	0	8	9	14

These products are added up modulo 11 (meaning the sum is divided by 11 and the remainder taken). In this case, the sum is 186 = 10 = X (mod 11), which makes the VIN valid, because it matches the original central X. What about the VIN on your vehicle?

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Solar Car Racing Status Check #2

Tokai, who came 4^th at WSC 2017, will race against Nuon at SASOL this year (photo: Anthony Dekker)

In further solar car racing news, preparations are continuing for the SASOL Solar Challenge in South Africa (September 22 to 30). Defending champions Nuon and Japanese team Tokai will attend this event, along with local teams, such as North-West University.

The Belgian car, Punch 2, which came 3^rd at WSC 2017, will be challenging Twente at iESC this year (photo: Anthony Dekker)

Thirteen teams have registered so far for the 24 hour iLumen European Solar Challenge (iESC) in Belgium (September 19 to 23), and Twente will be defending their title there. I am maintaining an information page and teams list for this race. See also the official iESC social media at  

Western Sydney’s car, Unlimited 2.0, which came 6^th at WSC 2017, will race at ASC this year (photo: Anthony Dekker)

Thirty-two teams have registered for the American Solar Challenge (ASC) in July. Scrutineering for this race begins on July 6, track racing on July 10, and the road race runs from July 14 to July 22. I am maintaining a detailed information page and teams list for this race too.

Missouri’s new car, Independence, was unveiled on 18 April (picture credit)

Five teams are attending with cars that raced at WSC 2017 (including one Australian team), although these cars will require adjustment to satisfy ASC rules. Seven other teams had existing cars (including one Russian team). The remaining teams have been building new cars.

Poly Montreal’s new car, Esteban 9, was unveiled on 23 April (picture credit)

Recently unveiled cars for the ASC include Missouri S&T (18 April), Poly Montreal / Esteban (23 April), and Georgia Tech (24 April). There are 17 cars still to be unveiled.

Georgia Tech’s new car, SR-2, was unveiled on 24 April (picture credit)

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Pencil charts for visualising colours

As a result of a discussion with a photographer friend of mine, I’ve been thinking (not for the first time) about visualising the colour palette of images. Consider this sunset, for example (a picture I took in Adelaide 8 years ago):

The photograph is rich in yellow and orange. However, the apparent blue in the sky is actually grey, and the apparent grey of the sea is actually brown. If we postulate a standard set of 35 plausible pencil colours, and map each pixel to the closest-matching pencil colour, we get this (I have done the comparison in RGB space):

Then we can visualise the colour palette of the image by showing the wear on the virtual pencils, if each virtual pencil has been used to colour the corresponding pixels. It can be seen that a lot of orange, brown, and grey was used (click to zoom):

Conversely, this beach scene (photographed in Vanuatu in 2016) is rich in blues:

The warm light greys of the beach don’t quite find an exact match among the pencils, but the other colours match fairly well:

And here is the pencil visualisation (click to zoom):

If, rather than using a standard set of colours, we extract the pencil colours from the image itself (image quantisation), fewer pencils will, of course, be required:

The fit to the original image will be much closer as well:

So this is a trick to remember for another day – pencil visualisations!

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Calendar for May

Here is the calendar for May (click for hi-res image). See more calendars here.

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Solar Car Racing Status Check

In solar car racing news, preparations are beginning for the SASOL Solar Challenge in South Africa (September 22 to 30). It seems that both Nuon and Tokai will attend this event, along with local teams.

Nuon at WSC 2017 (photo: Anthony Dekker)

Thirteen teams have registered so far for the 24 hour iLumen European Solar Challenge in Belgium (September 19 to 23), and Twente will be defending their title there. I am maintaining an information page and teams list for this race. See also the official iESC social media at  

Twente at WSC 2017 (photo: Anthony Dekker)

The American Solar Challenge is a lot closer than those two races, with scrutineering beginning on July 6, track racing on July 10, and the road race running from July 14 to July 22. I am maintaining a detailed information page and teams list for this race. At last count, 34 teams were registered, with Anderson, UCSD, Principia, UC Irvine, Phoenix, and UT Austin having, sadly, dropped out.

Six teams are attending with cars that raced at WSC 2017, although these cars will require adjustment to satisfy ASC rules (Michigan, Western Sydney, Principia, and Illini, plus the Cruisers PrISUm and Minnesota). Six other teams are attending with cars that previously raced at ASC.

PrISUm at WSC 2017 (photo: Anthony Dekker)

Twenty-two other teams are frantically building cars for ASC. Car unveils that have been announced include team 42 (Missouri) on 18 April, team 55 (Esteban) on 23 April, team 101 (Eclipse) in mid May, team 828 (AppState) in mid June, and team 65 (Calgary) on 16 June.

Missouri’s unfinished car (picture credit)

See my detailed information page and teams list for this race for more information and for social media links. I will continue to update that page as news comes in.

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Elements in the Human Body

The most common elements in the human body (logarithmic scale, data from Wikipedia, chart produced using R). Yes, we are mostly made of air!

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The Sand Reckoner

In his short work The Sand Reckoner, Archimedes (c. 287 BC – c. 212 BC) identifies a number larger than what he believed was the number of grains of sand which would fit into the Universe. He was hampered by the fact that the largest number-word he knew was myriad (10,000), so that he had to invent his own notation for large numbers (I will use modern scientific notation instead).

Archimedes’ began with poppyseeds, which he estimated were at least 0.5 mm in diameter (using modern terminology), and which would contain at most 10,000 grains of sand. This makes the volume of a sand-grain at least 6.5×10⁻¹⁵ cubic metres (in fact, even fine sand-grains have a volume at least 10 times that).

Archimedes estimated the diameter of the sphere containing the fixed stars (yellow in the diagram below) as about 2 light-years or 2×10¹⁶ metres (we now know that even the closest star is about 4 light-years away). This makes the volume of the sphere 4×10⁴⁸ cubic metres which means, as Archimedes shows, that less than 10⁶³ grains of sand will fit.

A more modern figure for the diameter of the observable universe is 93 billion light-years, which means that less than 10⁹⁵ grains of sand would fit. For atoms packed closely together (as in ordinary matter), less than 10¹¹⁰ atoms would fit. For neutrons packed closely together (as in a neutron star), less than 10¹²⁶ neutrons would fit. But these are still puny numbers compared to, say, 2^77,232,917 − 1, the largest known prime!

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