The Folded Paper Tower

27 folds to exceed Everest (T(27) ≈ 13,422 m > 8,849 m). 42 folds to reach the Moon (T(42) ≈ 440,000 km > 384,400 km). Formula: n = ceil(log₂(10 × H_mm)) where H_mm is the target height in millimetres.

Exponential growth problems require solving 2^n ≥ target, which means taking log base 2 of the target. The results always feel impossibly small because our intuition underestimates doubling. Step 1: Thickness formula: T(n) = 0.1 × 2^n mm, where n is the number of folds. Step 2: Target 1 — Mount Everest height: 8,849 m = 8,849,000 mm. Solve: 0.1 × 2^n ≥ 8,849,000 → 2^n ≥ 88,490,000. Step 3: Take log₂: n ≥ log₂(88,490,000) = log₂(8.849 × 10^7). log₂(10^7) ≈ 23.25, log₂(8.849) ≈ 3.14. Total ≈ 26.39. So n = 27 folds. Step 4: Verify: 2^27 = 134,217,728. T(27) = 0.1 × 134,217,728 = 13,421,772.8 mm = 13,422 m > 8,849 m. T(26) = 6,711 m < 8,849 m. Confirmed: 27 folds. Step 5: Target 2 — Distance to Moon: 384,400 km = 3.844 × 10^8 m = 3.844 × 10^11 mm. Solve: 0.1 × 2^n ≥ 3.844 × 10^11 → 2^n ≥ 3.844 × 10^12. Step 6: log₂(3.844 × 10^12) = log₂(3.844) + 12×log₂(10) ≈ 1.94 + 12×3.3219 ≈ 1.94 + 39.86 ≈ 41.8. So n = 42 folds. Verify: 2^42 ≈ 4.4 × 10^12. T(42) ≈ 4.4 × 10^11 mm = 440,000 km > 384,400 km. Key Insight: 27 folds to reach Everest and 42 folds to reach the Moon — from a sheet of paper. This is the power of exponential growth. The gap between 27 and 42 (just 15 folds) covers the distance from Everest to the Moon — 375,000 km of extra reach from 15 doublings. Exponential functions dwarf linear intuition at every scale.