QR codes?
Yummymath.com
Have you seen them? Do you know what they are? I had seen them but didn’t know what they were
until I read the newspaper yesterday. So, I started getting interested.
They are called QR codes (Quick Response codes) and are two-dimensional bar codes that were
created in 1994 by Denso Wave for Toyota to track car parts and the Toyota manufacturing process.
I’m used to regular barcodes. Grocery stores use them and almost all merchandise has barcodes. I
guess they make the cash register have the right price, a product name and an item description.
Barcodes also help stores keep their inventories up to date after each sale and speed up the check out
process. They were first used in 1973 to facilitate grocery store check out. Actually Rigley chewing gum
was the first product that carried a barcode on its package.
 In the U.S. bar codes are 12 ordered numbers.
 4 skinny, thin, medium or fat lines represent each digit.
 The first 6 digits in a barcode represent the manufacturer’s code and the second 6 numbers
represent the product code.
 Actually the first digit has to do with the category of the product like a pharmaceutical product or a
product that must be weighed to price it.
But since there are only 6 digits for each part of a product’s identification, there is a limit to how many
representations can be created using bar codes. I think it is a large number but maybe manufacturers
think it is not large enough. Let’s figure it out.
If the first 6 digits describe the manufacturer of the item, I wonder how many possible manufacturers
could be represented with those 6 digits?
I know that there are 10 possible digits to use since the values of the digits can be 0,1, 2, 3, 4, 5, 6,
7, 8, and 9.
1. To help you think about this calculation, first figure out how many different permutations
(combinations of numbers where order is important) are possible with two digits (remember
numbers 14 and 41 count as two different possible combinations).
2. Now try to determine how many different permutations are possible in a three-digit number.
How about a four-digit number? Are you seeing a pattern? If so, describe:
The number of possible orders to those 6 digits is called the number of permutations of those
numbers. To figure out the number of combinations that can be created, you have to count all of
the possible orders of the numbers. I don’t care if a digit is repeated in the sequence.
0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 1
0, 0, 0, 0, 0, 2
0, 0, 0, 0, 0, 3
…
This could take a while.
But, maybe this logic will make sense. There are 10 ways to choose that last number and 10 ways
to choose the 5th number and 10 ways to choose the 4th number, etc = 10 x 10 x 10 x 10 x 10 x 10
= 1,000,000 or 106 ways.
3. In the second 6 digits of a barcode, there are only 5 meaningful digits since that last digit is a way
of checking if the numbers have been read correctly (because barcodes get damaged all of the
time). How many different products can be defined by those last five meaningful digits?
4. So all together, how many products can be defined by a barcode with 11 significant digits?
In 1994, Denso Corporation supplied to Toyota with a way of communicating a lot more information
through a scan by making two-dimensional barcodes … QR codes.
The QR code at the top of this paper is the QR code for Yummymath.com. It can be read from any
rotation since those 3 squares, , in three of the corners define its orientation.
So QRs are two-dimensional bar codes. Instead of having 12 places for one of 10 digits they have a lot
more places for each digit or letter or symbol.
5. There are many versions of QR codes. Version 1 allows for 21 x 21 bits of information. That
means that in this version you can scan 21 x 21 numbers, letters, or symbols. About how many
characters of information is that?
a. 21 x 21 = _____ characters
b. How many different characters could you use?
# of digits + # of letters + your guess of how many symbols could be used = _______
c. Calculate the number of permutations possible with that many possible characters in a
space filled with 21 x 21 characters.
6. How does a 21 x 21 QR code compare with the amount of data in a US bar code, UPC? (UPC =
Universal Product Code)?
7. Version 40 allows for 177 x 177 bits of information. How many characters can you store in one of
those Version 40 QR codes?
8. Let’s assume that there are 100 possible symbols that you can use. I don’t think this is even
close to the actual number. How many permutations of QR symbols could you make? You
definitely should use scientific notation for this.
9. Are you getting the idea?
Where you see them
In selling real estate, in the realtor’s front window you might see a picture of a house and its QR code.
You can scan the code with your smart phone (you need to download an app that works with your smart
phone camera) and you will see how many rooms the house has, see a slide show of the home’s
features, watch a video with views of the home and the street where it is located, find out what school
your children would go to, and lots more useful information about the house.
1. Where have you seen QR codes used?
2. What are some other uses that you think that being able to have people scan the code with their
phone and receive urls, videos, slide shows, etc. would be good for?
Warning: QR codes can also be used to track your whereabouts by your phones GPS readout and
gather more information about you than you might want to share. Malicious QR codes combined with a
permissive reader can put a computer's contents and user's privacy at risk. QR codes intentionally
obscure and compress their contents and intent to humans. They are easily created and may be affixed
over legitimate QR codes.
Risks include linking to dangerous websites, enabling the microphone/camera/GPS and then streaming
those feeds to a remote server, extracting sensitive data (passwords, files, contacts, transactions), and
sending email/SMS/IM messages, corrupting privacy settings, stealing identity, and even containing
malicious logic themselves such or a virus. These actions may occur in the background while the user
only sees the reader opening a harmless webpage.
Sources: http://www.bostonglobe.com/metro/2011/10/07/quick-response-codes-provide-lots-information-you-know-what-theyare/gBirCbQ7ZPRL7kQizD5ldN/story.xml
http://qrcode.kaywa.com/
http://www.denso-wave.com/qrcode/aboutqr-e.html
http://www.cut-the-knot.org/Curriculum/Arithmetic/UPC.shtml
Brought to you by: Yummymath.com