Discrete mathematics · final presentation
Collisions aren't bugs.
They're arithmetic.
Simulating 100,000 bank account registrations inside a 100 millisecond window, and measuring the resulting ID collisions against what the Pigeonhole Principle and the Birthday Paradox say must happen. No amount of better code changes the answer.
1. The Pigeonhole Principle
Guaranteed, not probable
If you have N pigeons and M holes and N > M, at least one hole holds more than one pigeon.
| Parameter | Value | What it is |
|---|---|---|
| N | 100,000 | registration requests |
| M | ~100 | distinct milliseconds in the window |
| N − M | 99,900 | collisions that must occur |
Minimum collisions = N − M = 100,000 − 100 = 99,900
2. The Birthday Paradox
The 50% point arrives far earlier than intuition allows
P(collision) ≈ 1 − e^(−n² / 2M) for M = 900,000
| Registrations | P(at least one collision) | Intuition |
|---|---|---|
| 100 | 0.55% | seems safe |
| 500 | 12.9% | getting risky |
| 1,000 | 42.6% | very likely |
| 1,178 | 50.0% | the coin flip |
| 5,000 | 99.99% | essentially certain |
| 100,000 | 100.0% | guaranteed |
With 900,000 available numbers, intuition says 1,178 users is nothing. The maths disagrees.
3. Measured against theory
SHA-256 tracks the prediction to within a tenth of a percent
| Input size | Naive mapper | Hash mapper | Theoretical |
|---|---|---|---|
| 1,000 | ~90% | 0.05% | 0.06% |
| 5,000 | ~95% | 1.4% | 1.4% |
| 10,000 | ~97% | 5.4% | 5.4% |
| 25,000 | ~98% | 29% | 29% |
| 50,000 | ~99% | 63% | 63% |
| 100,000 | ~99%+ | 89% | 89% |
The cryptographic hash is enormously better than the naive scheme and still fails at scale — because the output space, not the hash quality, is the binding constraint.
4. The plots
Generated by the simulation, not drawn by hand
