Patterns /Number Theory
Triangular Numbers
A triangular number counts the dots in a triangle: 1, 3, 6, 10, 15, 21 — each one the running total 1 + 2 + … + k.
How long might triangular numbers take?
1 in 707.2 per roll · 707 rolls on average.
At one daily specimen a day, the mean is about 1.9 years. A 50% chance takes 490 rolls; 90% takes 1,628.
7 daily rolls
1 in 101
30 daily rolls
4.16%
365 daily rolls
40.34%
4.16% chance of at least one match in 30 independent rolls.
Waiting times are averages, not deadlines. Previous misses do not increase the next roll’s chance. Only Daily Specimens stamp your Atlas.
Because the k-th triangular number is k(k + 1) / 2, they grow quadratically, and their density falls like 1 / √(2n) — roughly the same shape as perfect squares, but with more members in range.
They have an unusually direct relationship with squares: eight times any triangular number, plus one, is always a perfect square. Small identities like this are why triangular numbers show up so often in puzzles.
A handful of numbers in range are both triangular and square. Those are genuinely rare objects, and they rank accordingly on the leaderboard.
Highest-scoring examples
The best-scoring numbers of four digits or more that carry this pattern. Shorter numbers are excluded here because their length alone dominates any score ranking.
Smallest examples
The first numbers in the range that match this pattern.
Related number theory patterns
Prime Numbers
A prime has no divisors other than 1 and itself. They are the building blocks every other number is assembled from.
Semiprime Numbers
A semiprime is a product of exactly two primes, counted with multiplicity — 15 = 3 × 5, and 49 = 7 × 7 counts too.
Perfect Squares
A perfect square is some whole number times itself: 144 = 12², 10000 = 100². The range ends neatly on 1,000,000 = 1000².
Perfect Cubes
A perfect cube is a whole number raised to the third power: 216 = 6³, 970299 = 99³. There are only about a hundred in the entire range.
Powers of Two
Each power of two is double the last: 1, 2, 4, 8, 16, and on up through 1024, 65536 and 524288. Doubling exhausts the range in only twenty steps.