What is the Logistic Map?

The logistic map is a recurrence relation that models population growth with limited resources:

x_{n+1} = r × x_n × (1 - x_n)

Where x_n is the population (as a fraction of the maximum, so between 0 and 1), and r is the growth rate (typically between 2 and 4).

Despite its simplicity, the logistic map exhibits an astonishing range of behaviors:

• r between 2 and 3: The population stabilizes at a fixed point
• r between 3 and 3.45: The population oscillates between 2 values (period-2 cycle)
• r between 3.45 and 3.55: Period-4 cycle
• r around 3.57: The onset of chaos — the behavior never repeats
• r between 3.57 and 4: Fully chaotic behavior, sensitive to tiny changes in initial conditions

In our puzzle, you're given r, x₀, and the number of steps N. Iterate N times and the final value × 100 mod 26 gives you a letter.

A Concrete Example

Let's work through an example with r = 3.83, x₀ = 0.35, and N = 10.

Step 1-10: Iterate the logistic map

x₀  = 0.350000
x₁  = 3.83 × 0.35 × 0.65      = 0.871325
x₂  = 3.83 × 0.871325 × 0.128675 = 0.429247
x₃  = 3.83 × 0.429247 × 0.570753 = 0.938374
x₄  = 3.83 × 0.938374 × 0.061626 = 0.221476
x₅  = 3.83 × 0.221476 × 0.778524 = 0.660189
x₆  = 3.83 × 0.660189 × 0.339811 = 0.859341
x₇  = 3.83 × 0.859341 × 0.140659 = 0.463134
x₈  = 3.83 × 0.463134 × 0.536866 = 0.952255
x₉  = 3.83 × 0.952255 × 0.047745 = 0.174136
x₁₀ = 3.83 × 0.174136 × 0.825864 = 0.550654

Step 2: Map the final value to a letter

x₁₀ × 100 = 55.0654 → rounded = 55
55 % 26 = 55 - 26 × 2 = 55 - 52 = 3
3 → 'd' (a=0, b=1, c=2, d=3)

The answer letter is 'd'.

Note: r = 3.83 is in the chaotic regime, so small changes in initial conditions produce completely different outcomes after 10 steps. This is the famous butterfly effect.

Why It Works: Chaos Theory in One Paragraph

Simple rules can produce complex behavior. The logistic map is a quadratic equation — as simple as math gets — yet it produces chaos. The key is that the parabola x(1-x) "stretches and folds" the interval [0,1] onto itself, just like a baker kneading dough. Points that start close together get separated (stretched) and then brought back together (folded). This stretching-and-folding creates sensitivity to initial conditions: the famous butterfly effect. The Feigenbaum constant δ ≈ 4.669, which describes the rate at which period-doubling occurs, is universal across all one-dimensional maps, not just the logistic map.

Solving Tips

• Start with x = x0, then loop: x = r × x × (1 - x)
• Repeat N times, then compute int(round(x × 100)) % 26
• For r between 2 and 3, the map converges to a fixed point regardless of x₀
• For r between 3 and 3.57, the map settles into a cycle — iterate extra steps for stability
• For r > 3.57, the map is chaotic — every decimal place matters
• Use double-precision floating point for accuracy

Difficulty Table

Real-World Applications

• Population Biology: The logistic map was originally developed to model population growth with limited resources. It explains why some animal populations fluctuate wildly while others stabilize.
• Epidemiology: The spread of diseases follows similar dynamics. The logistic map's chaotic regime helps explain why influenza outbreaks are hard to predict beyond a few weeks.
• Economics: Stock market dynamics show chaotic behavior. The logistic map is a toy model for how simple economic rules can produce apparently random market movements.
• Weather Prediction: Edward Lorenz discovered chaos while studying atmospheric convection equations. The logistic map is the simplest model that shows the same sensitivity to initial conditions that limits weather forecasts to ~10 days.
• Cryptography: Chaotic maps are used in some encryption schemes because their unpredictable behavior is similar to cryptographic pseudorandom number generators.

Want to Try It?

Head over to the puzzle browser. If a Logistic Map puzzle is active, you'll get r, x₀, and N. Iterate the map and compute the letter!