A one-in-a-quintillion bottle cap?

Coke has been running is My Coke Rewards program for more than a year now, offering soda drinkers the chance to earn "rewards" for just being loyal soda drinkers. Since I'm on a 2-3 Diet Coke-a-day habit, I've been saving my bottle caps and earning reward points (usually 3 points per bottle cap).

To track points, you go to a Coke Web site and type in the 12- or 15-digit code found on the bottle cap, and the system awards you the points. If you have enough points, you can then redeem them for various prizes or other rewards. If you mistype the code, the software politely tells you that the code isn't valid, and asks you to double-check the code.

Earlier today I was typing in the code, and the system kept telling me the code wasn't valid. After the third time typing it in, I realized the cap was upside down. Yet every number and letter in this particular code could be read upside down or right-side up. The incorrect code was "HW6N66WW00W6" – when I turned it the correct way it was "9M00MM99N9MH".

My computer technology brain started ticking. First, shouldn't the program that assigns the codes try to include at least one letter or number that can't be misinterpreted when upside down? Since you already have a program that generates a random code and then has to track it in a Web database (to confirm real codes vs. fake ones), it shouldn't have been too much of a problem to guarantee that each code has a non-upside-down number in it.

Then my math brain started ticking. What are the chances of getting a code that could be read both ways? With a choice of 26 letters, 10 numbers (zero through 9) and a 12-digit code (I'll ignore the 15-digit codes for a second), that gives the possibility of 4,738,381,338,321,616,896 different codes (36x36x36, etc., for 12 digits). For you math majors, that's more than 4.7 quintillion possibilities.

So the odds of getting that exact code are one in 4.7 quintillion, but the odds are lower that I could get another cap code that could be read upside down correctly. Considering that the letters H, I, M (confused as W), N, O, S, X and Z, as well as the numbers 0, 6 (read as 9) and 8 could be read upside down (I'm also assuming the number 1 has the base on it and isn't just a straight line), that's 13 different characters that could be combined into a 12-digit upside-down code. Following this logic, 13 to the 12th power would give you 23,298,085,122,481 or about 23 trillion upside-down codes.

At this point my head started to hurt with all of the math, so I'll let the readers take this on further (and correct my math or logic). Not knowing the number of valid codes that Coke has distributed (let's assume they picked 1 billion valid codes), what are the chances that there's another upside-down code available?

• math
• timekiller