Python & Fifth Grade — A Math Challenge from Serbia 🌍🧠

In the world of mathematics competitions, young minds are often challenged with problems that test their reasoning, conceptual understanding, and problem-solving skills. A fascinating problem from a Serbian math competition for fifth graders involves set theory and Venn diagrams — fundamental tools for exploring mathematical concepts.🧮

Competition’s exercise for all grades can be found: here

The Riddle 🧐

Let T be the set of all even numbers less than 50. Represent with a Venn diagram the sets:

• A = {a ∈ T | a is divisible by 3},
• B = {b ∈ T | b is divisible by 4},
• C = {c ∈ T | c is divisible by 5}.

Determine A \ (B ∩ C). 📊

This exercise not only reinforces the arithmetic skills of divisibility but also encourages students to visualize mathematical relationships through the construction of a Venn diagram — a diagram that shows all possible logical relations between a finite collection of different sets.

Now, the challenge is to determine A \ (B ∩ C), which essentially means finding the elements in set A that are not in the intersection of sets B and C.

Let’s see how computational thinking can complement traditional math competition strategies using Python! 🐍

Step 1: Defining the Set T

We kick things off by defining set T, which contains all even numbers less than 50. With a dash of Python magic, we generate this set using the range() function:

# Define the set T as all even numbers less than 50
T = set(range(2, 50, 2))

# {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48}

Step 2: Defining the Sets A, B, and C

Next, we define sets A, B, and C, employing similar Python wizardry:

# Set A: Divisible by 3
A = {a for a in T if a % 3 == 0}  # {36, 6, 42, 12, 48, 18, 24, 30}

# Set B: Divisible by 4
B = {b for b in T if b % 4 == 0}  # {32, 4, 36, 8, 40, 12, 44, 16, 48, 20, 24, 28}

# Set C: Divisible by 5
C = {c for c in T if c % 5 == 0}  # {40, 10, 20, 30}

Here, we employ set comprehensions to create sets A, B, and C. These sets contain numbers from T that are divisible by 3, 4, and 5, respectively. The modulo operator `%` checks for divisibility.

Step 3: Calculating A \ (B ∩ C)

Now, let’s crunch some numbers. We calculate the intersection of sets B and C and then determine the elements in set A that are not in this intersection:

# Intersection of B and C
B_inter_C = B.intersection(C)  # {40, 20}

# A \ (B ∩ C)
result = A - B_inter_C         # {36, 6, 42, 12, 48, 18, 24, 30}      

And there we have it! A solution in just a few lines of Python code. 😎

Pay Attention to Brackets!

When working with set comprehensions, it’s crucial to remember the brackets. 🧐

• 👍 Correct: {x for x in some_list if condition} - This creates a set with elements that meet the condition.
• ❌ Incorrect: {} - This creates an empty dictionary, not a set.

Make sure to use the correct brackets to avoid unexpected Python surprises! 😄👌

Visualizing with Venn Diagrams

By using the Python library matplotlib-venn, we can visualize our sets with a Venn Diagram, showing the unique and shared elements.

Practical Example: TURF Analysis

Let’s take our understanding of sets to a real-world scenario with TURF (Total Unduplicated Reach and Frequency) analysis:

Imagine each set represents kids’ preferences for bubble gum flavors:

• 🍓 Set A: 3 kids prefer strawberry.
• 🍏 Set B: 6 kids are all about green apple.
• 🍇 Set C: 1 kid has a taste for grape.

When we analyze their overlaps:

• 🍓🍏 4 kids enjoy both strawberry and green apple (intersection of Sets A and B).
• 🍓🍇 1 kid likes strawberry and grape together (intersection of Sets A and C).
• 🍏🍇 2 kids delight in the mix of green apple and grape (intersection of Sets B and C).
• 🍓🍏🍇 No kid likes all three flavors (intersection of Sets A, B, and C).

To achieve the widest unique reach:

• Start with Set B (green apple), which has 6 unique enthusiasts.
• Add Set A (strawberry), bringing in 3 additional unique fans.
• Consider Set C (grape), which contributes just 1 more unique admirer.

Therefore, to reach the most kids, you’d offer green apple and strawberry flavors, reaching 9 unique kids in total (6 from green apple, 3 from strawberry). Adding grapes would only increase the unique reach by 1 additional kid, making it less impactful compared to the other two in terms of unduplicated reach. 📈

In summary, TURF analysis suggests focusing on the flavors that will attract the largest unique audience, which in this case would be green apple and strawberry bubble gums.

So, whether you’re a young math enthusiast or a seasoned data analyst, Python’s prowess in bridging the gap between abstract concepts and tangible solutions continues to be a valuable asset in your journey of exploration and discovery.

Happy coding and math solving! 🌟🧮🐍