Did you know you can find out where a UK car was first registered or how old it is just by looking at its number plate?

In this Python challenge, we will create a program that decodes a UK plate number. The program will ask the user to enter a UK car registration plate and the program will then:

• Identify the regional registration area of the car
• Calculate the approximate age of the vehicle

Decoding a UK Plate Number

Modern UK registration plates follow this format:

Regional Area Codes

The first letter of the area code tells us where the vehicle was registered sing the following letters/codes

Age Identifier Rules

The two numbers in the registration plate indicate when the vehicle was first registered. It covers a 6-month period, which is why new plates are issued twice a year and the age identifier is determined based on the following rules:

• For vehicles registered between March and August hold, we use the last 2 digits of the year it was put on the road. e.g for a car registered in March 2026, the age identifier code used is 26.
• For cars registered from September to February, we use the same approach but add 50 to the code. So for a car registered in September 2026, the age identifier will be 26 + 50 = 76

So to determine the approximate age of a car based on the age identifier we will use the following rule:

Age of a Car?

Once we have established the year of registration of a car, we can estimate its age (in years) using the following formula:

What about the last 3 letters?

Unlike the first 4 characters of a plate which have a meaning, the final 3 letters don’t mean anything. Instead, these random 3-letter combinations are used to help keep each plate number unique compared to other plate numbers for cars registered at the same time.

Python Challenge

Let’s write a Python program that will:

Ask the user to enter a valid UK registration plate number
Validate this number ensuring that the plate number is exactly 8 characters long (including the space before the last 3 letters)
Extract the first digit of the plate number (first digit of the area code)
Extract the 2 number digits (age Identifier: the 3^rd and 4^th digits of the plate number)
Lookup the area code within the list of area codes and output the area where the vehicle was registered.
Calculate and output the estimated registration year of the vehicle.
Calculate and output the estimated age (in years) of the vehicle.

We have started the code for you but this code is incomplete. Can you complete the code to calculate both the registration year and the estimated age of the car.

Testing:

Use you code to work out the area code and estimated age of the following vehicles:

Solution...

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Did you know you can estimate the value of π (pi) using probability and number theory?

This challenge explores a surprising mathematical relationship:
The probability that two randomly chosen integers are coprime, P(\text{coprime}) is equal to \frac{6}{\pi^2}

Coprime numbers?

Two numbers are coprime if their only common factor is 1.

Using this idea, you can estimate π by randomly generating pairs of integers and checking how often they are coprime.
​
If:
P(\text{coprime}) = \frac{6}{\pi^2} ​
Then we can rearrange to estimate π:
\pi \approx \sqrt{\frac{6}{P(\text{coprime})}} ​

So the aim of this challenge will be to write a program to:

• Generate many random pairs of integers
• Count how many of these pairs are coprime
• Use the ratio to estimate π

Not that to work out if two numbers are coprime, we will need to create a function to calculate their Greatest Common Denominator.

Step 1: Generating N pairs of random integers

For this step we will use the random library to generate random integer values (between 1 and 10,000), and a for loop, to generate N pairs.

import random, math

N = 1000
for i in range(0,N):
   number1 = random.randint(1,10000)
   number2 = random.randint(1,10000)

Step 2: Finding out if our pair of numbers are coprime

To check if two numbers are coprime, we first need to calculate their Greatest Common Denominator. We can do so using the following function, based on Euclid’s Division Algorithm:

def gcd(a, b):
    # Repeat until b becomes 0
    while b != 0:
        remainder = a % b   # Find the remainder when a is divided by b
        a = b               # Move b into a
        b = remainder       # Move remainder into b

    # When b is 0, a contains the GCD
    return a

We can use this function in our previous for loop to count how many pairs of numbers are coprime:

import random, math

N = 1000
coprime_count = 0

for i in range(0,N):
   number1 = random.randint(1,10000)
   number2 = random.randint(1,10000)
   if gcd(number1, number2) == 1:
      coprime_count = coprime_count + 1

Step 3: Outputting the ratio of coprime pairs:

We can now calculate and display the ratio of coprime pairs (Probability that two randomly generated numbers are coprime):

probability = coprime_count / N

Estimating Pi

We can now estimate Pi using the following formula:
\pi \approx \sqrt{\frac{6}{P(\text{coprime})}}

pi_estimate = math.sqrt(6 / probability)

And finally we can display our estimate on screen as well as calculate the percentage of accuracy of our estimate:

print("Number of pairs", N)
print("Number of coprime", coprime_count)
print("Estimated value of Pi:", pi_estimate)
print("Actual value of Pi:", math.pi)
print("Percentage of accuracy:", 100 - 100 * abs(math.pi - pi_estimate)/math.pi)

Python Code

Your turn! You can now complete the code below:

Solution...

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