California License Plates Part 1

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License plates in the state of California follow an interesting pattern with 7 letters and numbers, and we're going to run out soon! Everyone panic!! However, this journal isn't about that just yet, but we're building up to it.

Simple Numeric Sequence

First things first, if we have let's say a simple 3-digit license plate of all numbers, going sequentially up, like this:

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1	`001, 002, 003, ... 010, 011, ... 100, 101, ... 999`

The formula to determine the sequence number \(N\) from the license plate NNN would be as follows:

\[N = 10^{2}N_{1} + 10^{1}N_{2} + 10^{0}N_{3}\\ N = 100N_{1} + 10N_{2} + N_{3}\]

For example, the license plate 123 is what place in the sequence?

\[N = 100*1 + 10*2 + 3 = 100 + 20 + 3 = 123\]

Awesome. If you have the sequence number, lets say \(123\), and want to know the corresponding plate, things are a little bit more complicated, but still easy for this example.

\[N_{1} = floor\left(\frac{N}{10^{2}}\right) = floor\left(\frac{123}{100}\right) = 1\]

\[N_{2} = floor\left(\frac{N - 10^{2}N_{1}}{10^{1}}\right) = floor\left(\frac{123 - 100*1}{10}\right) = 2\]

\[N_{3} = N - 10^{2}N_{1} - 10^{1}N_{2} = 123 - 100*1 - 10*2 = 3\]

Nice! We get right back to 123. Easy peasy.

We could also determine the total number of combinations like this:

\[C = 10*10*10 = 1000\]

There are 1,000 3-digit combinations for license plates with only numbers, which makes perfect sense. Count them all!

Letter Sequence

Assume you have a 2-letter license plate in the pattern LL, things get a little more complicated since we are no longer dealing with base-10.

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1	`AA, AB, ... AZ, BA, BB, ... ZZ`

How many possible plates are there?

\[C = 26*26 = 676\]

This is an easy one, just 26 squared, or 676. The formula to go from the license plate to the sequence number \(N\) would be as follows:

\[N = 26^{1}L_{1} + 26^{0}L_{2}\\ N = 26L_{1} + L_{2}\]

Each letter would correspond to a number with A starting at zero:

Letter	Value
`A`	\(0\)
`B`	\(1\)
`C`	\(2\)
`D`	\(3\)
`E`	\(4\)
`F`	\(5\)
`G`	\(6\)
`H`	\(7\)
`I`	\(8\)
`J`	\(9\)
`K`	\(10\)
`L`	\(11\)
`M`	\(12\)
`N`	\(13\)
`O`	\(14\)
`P`	\(15\)
`Q`	\(16\)
`R`	\(17\)
`S`	\(18\)
`T`	\(19\)
`U`	\(20\)
`V`	\(21\)
`W`	\(22\)
`X`	\(23\)
`Y`	\(24\)
`Z`	\(25\)

So, if you had license plate NV, that would be this number in the sequence:

\[N = 26*13 + 21 = 359\]

So going backwards from sequence number 359, you'd get this:

\[L_{1} = floor\left(\frac{N}{26}\right) = floor\left(\frac{359}{26}\right) = 13\]

\[L_{2} = N - 26L_{1} = 359 - 26*13 = 21\]

That translates back to NV!

Really, all we needed to do here was change our 10's to 26's, so this was actually pretty simple. Now what about letter-number combinations? That's where things get dicey. Tune in next time!