> The Basel Problem: Euler's Masterpiece
How the sum of inverse squares converges to π²/6 — and how its digits hide letters.
What is the Basel Problem?
The Basel problem asks: what is the sum of 1/n² as n goes from 1 to infinity?
S = 1/1² + 1/2² + 1/3² + 1/4² + ... = ?
For decades, the best mathematicians couldn't find the exact value. Then in 1734, the 28-year-old Leonhard Euler shocked the mathematical world by proving that
S = π²/6 ≈ 1.644934...
The number π appearing in a sum of rational numbers was completely unexpected. It was one of Euler's greatest discoveries.
In our puzzle, you're given an N (how many terms to sum). You compute the partial sum, extract the fractional part (the digits after the decimal point), multiply by 10000, and mod by 26 to get a letter.
A Concrete Example
Let's work through an example where N = 10.
Step 1: Compute the partial sum
S = 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 1/49 + 1/64 + 1/81 + 1/100 Term by term: n=1: 1/1 = 1.000000 n=2: 1/4 = 0.250000 → cumulative: 1.250000 n=3: 1/9 ≈ 0.111111 → cumulative: 1.361111 n=4: 1/16 = 0.062500 → cumulative: 1.423611 n=5: 1/25 = 0.040000 → cumulative: 1.463611 n=6: 1/36 ≈ 0.027778 → cumulative: 1.491389 n=7: 1/49 ≈ 0.020408 → cumulative: 1.511797 n=8: 1/64 = 0.015625 → cumulative: 1.527422 n=9: 1/81 ≈ 0.012346 → cumulative: 1.539768 n=10: 1/100= 0.010000 → cumulative: 1.549768
Step 2: Extract the fractional part
S = 1.549768 Fractional part = 0.549768
Step 3: Multiply by 10000 and mod 26
0.549768 × 10000 = 5497.68 → int = 5497 5497 % 26 = 5497 - 26 × 211 = 5497 - 5486 = 11 11 → 'l' (a=0, b=1, ..., l=11)
The answer letter is 'l'.
Why It Works: Infinite Series in One Paragraph
The sum of 1/n² converges because the terms shrink fast enough. Each term 1/n² is smaller than the corresponding integral ∫ 1/x² dx from n to n+1, and that integral converges to 1. By comparing the sum to an integral, you can prove the sum is finite — but finding the exact value π²/6 required Euler's extraordinary insight. He recognized that the infinite product expansion of sin(x)/x, when expanded as a power series, must match the coefficients of the Taylor series, and that matching produced π²/6.
Solving Tips
- Compute S = sum(1/n² for n in 1..N) — use floating point arithmetic
- Extract the fractional part:
frac = S - int(S) - Compute
int(frac × 10000) % 26to get a number 0-25 - Convert to letter:
chr(ord('a') + number) - For N=5..15 (diff 1-2), the sum is around 1.46 to 1.61
- For larger N (diff 3+), the sum approaches π²/6 ≈ 1.644934
Difficulty Table
| Difficulty | N | Notes |
|---|---|---|
| 1-2 | 5-15 | Few terms, quick computation |
| 3-4 | 10-100 | More terms, closer to π²/6 |
| 5-7 | 50-1000 | Many terms, precision matters |
Real-World Applications
- Quantum Electrodynamics: Feynman diagrams for particle interactions involve infinite sums of the same form. The Basel sum appears in calculations of the electron's magnetic moment.
- Signal Processing: The total power of a signal is sometimes the sum of squared components. The Basel sum is the total power of Fourier series of certain waveforms.
- Riemann Zeta Function: ζ(2) = π²/6 is the first non-trivial value of the zeta function, which is central to the Riemann Hypothesis — one of the seven Millennium Prize Problems.
- Probability Theory: The Basel sum appears in the normalization constant of the Cauchy distribution and other probability distributions.
- String Theory: The sum 1+2+3+... = -1/12 (in zeta regularization) is related to the Casimir effect, a real physical force measurable between conducting plates.
Want to Try It?
Head over to the active puzzles page. If a Basel Problem puzzle is active, it will give you N. Compute the partial sum, extract the fractional digits, and find the letter!