Happy Numbers

Rule, exact count and rarity

To test a happy number, square each decimal digit, add the squares, and repeat the operation on the result. A starting number is happy if this process eventually reaches 1. The process is not ordinary digit summation: squaring matters. For example, 19 leads to 82, then 68, then 100, then 1. A path that repeats without reaching 1 is not happy. The engine detects these outcomes and includes 1 itself, while 0 stays at 0 and fails. Reordering the same digits produces the same first sum of squares and therefore the same happy status, although other patterns and the final rarity score may differ. The definition here always refers to decimal digits.

The exact count, 143,071, comes from evaluating all 1,000,001 integers from 0 through 1,000,000, including both endpoints. It is not a sample or an estimate. Each matching integer is counted once for this category, even if it contains multiple qualifying segments. Its frequency of 14.307086% describes the chance that a uniformly chosen integer passes this one test. It is not the global score percentile of every member, and overlapping pattern counts must not be added as if they represented disjoint sets.

The pattern supplies −log₂(143,071 / 1,000,001) bits before its within-family weight is applied. Other matched patterns, digit sum, distinct-digit count and short-length signals can change the complete score. Consequently, members of this category need not share a ranking. The highest-scoring representative is 1, with a rarity score of 1,504 and Top 0.0002%. This uses its complete score across all features, not just this pattern. Equal scores are displayed in ascending numerical order.

Example numbers

The highest-scoring match

The highest-scoring representative is 1, with a rarity score of 1,504 and Top 0.0002%. This uses its complete score across all features, not just this pattern. Equal scores are displayed in ascending numerical order.