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.

Matches in range
143,071
Share of all numbers
14.31%
Out of
1,000,001

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.

1,000,000 Top <0.01% Singular 1,000 Top <0.01% Singular 1,212 Top <0.01% Singular 2,121 Top <0.01% Singular 7,447 Top 0.01% Singular 95,959 Top 0.01% Singular 76,767 Top 0.01% Singular 4,040 Top 0.01% Singular

Smallest examples

The first numbers in the range that match this pattern.

1 7 10 13 19 23 28 31

Related digit arithmetic patterns

Harshad Numbers

A Harshad — or Niven — number is divisible by the sum of its own digits. 18 works because 1 + 8 = 9 and 18 ÷ 9 = 2.
Explore this pattern Analyze a number All patterns