Patterns /Digit Arithmetic
Happy Numbers
Square each digit, add them up, repeat. If you eventually reach 1 the number is happy; if not, you fall into a cycle. 19 → 82 → 68 → 100 → 1.
How long might happy numbers take?
1 in 7 per roll · 7 rolls on average.
At one daily specimen a day, the mean is about 0 years. A 50% chance takes 5 rolls; 90% takes 15.
7 daily rolls
66.07%
30 daily rolls
99.03%
365 daily rolls
100.00%
99.03% 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.
Every starting number ends up either at 1 or in a single fixed eight-step loop, so happiness is genuinely binary — there is no third outcome to worry about.
Roughly one number in seven is happy, and that proportion is remarkably stable as the range grows. That makes happiness one of the few traits here whose density barely changes with magnitude.
Because the test depends on digits rather than magnitude, happy numbers are scattered unpredictably through the range. There is no way to spot one at a glance, which is what makes them a fair Pattern Hunt target.
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.