Cicadas and Prime Numbers: Nature’s Mathematical Marvel

Every few years, an extraordinary natural phenomenon takes place: millions of cicadas emerge from the ground in a synchronized spectacle, filling the air with their unmistakable buzz. These insects may seem like just another summer nuisance, but their life cycles hide a fascinating secret—one deeply connected to the mysterious world of prime numbers.

The Cicada Countdown: 13 and 17-Year Cycles

But why? Scientists believe that evolving to emerge in prime-numbered intervals is an ingenious survival strategy. Here’s how:

Prime Numbers as a Survival Tactic

1. Avoiding Predators
   Many natural predators, such as birds or small mammals, have life cycles that repeat every 2, 3, 4, or 6 years. If cicadas emerged every 12 years (which is divisible by 2, 3, 4, and 6), they would often coincide with the peak populations of these predators. But by using 13 or 17-year cycles, their emergence rarely aligns with predator population booms, reducing the risk of being wiped out.
2. Minimizing Hybridization
   Different cicada broods (groups that emerge together) must avoid interbreeding, as mixed-cycle offspring might end up with irregular, non-prime emergence years—causing them to appear at the wrong time, possibly alone and vulnerable. Sticking to prime-numbered cycles ensures that broods remain distinct and synchronized.
3. Strength in Numbers
   When cicadas finally emerge, they overwhelm predators by sheer volume—a survival strategy called predator satiation. There are so many cicadas at once that predators can only eat a small percentage, leaving the rest to reproduce successfully.

A Mathematical Wonder in Nature

MathActivities inspired by the Cicadas

Using cicadas and their prime-numbered life cycles as a basis for math activities is a fantastic way to make learning engaging and relevant! Here are some ideas for different levels:

Elementary School (Basic Number Sense & Patterns)

1. Prime Number Discovery with Cicadas
   • List the cicada cycles (13 and 17 years) along with other numbers (10, 12, 15, etc.).
   • Have students determine which numbers are prime and discuss why prime numbers are special.
2. Cicada Skip Counting Game
   • Use a number line and have students “hop” forward by 13s or 17s to see how long it takes to reach 100.
   • Compare this with hopping by a composite number like 12—do they notice a pattern in how often they land on common multiples?
3. Cicada Art & Math
   • Have students draw or color cicadas with different cycle lengths and create a timeline showing when each brood would emerge together.

Prime Number Games

Station Prime – A Prime Factorization Game

The Great Escape

The Great Escape is a print-to-play game to practice multiples, factors, perfect squares and prime numbers.

Prime Numbers and “The Sieve of Eratosthenes

Learn About Eratosthenes and His Sieve

Middle School (Multiples, GCD, LCM, and Probability)

4. Least Common Multiple (LCM) of Cicada Emergences
   • Ask: If a 13-year brood and a 17-year brood emerge at the same time in 2024, when will they next appear together?
   • Solve using LCM(13,17) = 221 years!
5. Greatest Common Divisor (GCD) & Cicada Strategy
   • Discuss why cicadas use 13 and 17 rather than 12 or 16.
   • Have students calculate the GCD of cicada cycles and predator cycles (e.g., 13 & 6, 17 & 4).
   • Why would choosing a prime cycle make cicadas less predictable?
6. Probability of Predator-Cicada Overlap
   • If a predator has a 4-year life cycle and a cicada emerges every 13 years, what’s the probability they meet in a given cycle?
   • Explore different predator cycles (e.g., 2, 5, or 6 years) and see how often they align with cicada cycles.

LCM Race – Finding LCM (game)

High School (Algebra, Number Theory, and Modeling)

7. Cicada Prime Number Theorem
   • Have students investigate why prime numbers are relatively rare as numbers get larger.
   • Discuss why nature might have selected primes rather than just “big numbers” for cicada cycles.
8. Exponential Growth & Cicada Populations
   • Given that a cicada brood grows exponentially each cycle, have students model population growth using an exponential equation:
   • How large would the cicada population be after several cycles?
9. Computer Simulations of Cicada Life Cycles
   • Program a simple simulation to model the emergence of cicadas and when they overlap with different predator cycles.
   • Use coding (Python, Scratch, or spreadsheets) to visualize prime-numbered cycles.

Real-World Extensions

10. Cicada Math Scavenger Hunt
    • Hide “cicada eggs” (paper cutouts) around the room with math problems on them.
    • Each problem leads to a new clue about prime numbers and cicadas.
11. Cicada Prime Number Game

Cicada Prime Number Game: Survive & Emerge! 🎲

Objective:

Be the first to help your cicada emerge at the right time by landing on prime-numbered years! Although Cicadas emerge every 13 or 17 years for the sake of the game and for the sake of practicing prime numbers we will assume that they emerge whenever you land on a prime number.

Setup:

• 2 to 4 players
• One game board (100 chart)
• Each player gets a cicada token (use coins, buttons, or any small objects)
• A six-sided die
• A Prime Number Reference Card (optional for younger players)

How to Play:

1. Each player starts at year 1 (underground).
2. On your turn, roll the die and move forward by that many years.
3. If you land on a prime number (2, 3, 5, 7, 11, 13, etc.), your cicada emerges for that turn and you get a point.
   • If you land on 13 or 17, you get a bonus turn (because cicadas love those numbers)!
4. If you land on a non-prime number, your cicada stays underground. You must wait for your next turn to roll again. You can use your rolled number to go backwards in order to get to a prime number.
5. The first player to reach 10 points wins!

Challenge Mode (Optional Rules):

• Predator Attack: If you land on a multiple of 6 (e.g., 6, 12, 18), a predator finds your cicada! One point gets taken from you.
• Swarm Advantage: If two players land on the same prime number, both get a bonus turn!
• Swarm Disadvantage: If they meet each other on any other number the last to get there gives a point to the one that was there first.

I Hope that you find this game useful. Let us know what you think.