Definition
Reach 1
Repeat sum of squared digits until 1 or a cycle.
Check Happy Number in Python
Overview
What You’ll Learn
A happy number reaches 1 by repeatedly summing the squares of its digits. This tutorial covers the digit map, Floyd tortoise-hare detection, a visited-set alternative, a live preview, worked Python examples, edge cases, and complexity.
Classic 19
Happy
19 → 82 → 68 → 100 → 1.
Floyd
O(1) memory
Slow/fast pointers detect the cycle.
Unhappy Cycle
Starts at 4
4 → 16 → … → 20 → 4.
Live Preview
Try any n
Classify positive integers instantly.
O(μ+λ)
Floyd steps
Tail plus cycle length in digit-sum steps.
Introduction
A happy number is a positive integer that reaches 1 when you repeatedly replace it by the sum of the squares of its digits. Example: 19 → 82 → 68 → 100 → 1.
If the process never hits 1, it enters a fixed unhappy cycle (starting at 4). Floyd tortoise-hare detection finds that loop with O(1) extra memory.
Why it matters?
It combines digit peeling with cycle detection — a clean bridge from number warm-ups to linked-list Floyd problems.
Key Highlights
Digit Map f(n)
Sum of squared decimal digits.
Happy = Hit 1
1 is a fixed point: f(1) = 1.
Floyd Detects Loops
Slow/fast pointers meet in a cycle.
Positive Only
Reject 0 and negatives by definition.
In short: keep replacing n with the sum of squared digits; if you reach 1 it is happy, otherwise you loop.
Foundation
📝 Problem & Approach
Given a positive integer n, decide whether repeated digit-square sums reach 1.
python
# 19 → 82 → 68 → 100 → 1 → happy
# 2 → 4 → 16 → ... cycle → unhappy
# 1 → 1 → happyInputs & Outputs
| Item | Type | Description |
|---|---|---|
n | int | Positive integer (reject n < 1). |
| Return / print | bool / text | True if n is happy. |
Minimal workflow
Pseudocode
function sum_square_digits(n):
s = 0
while n > 0:
d = n mod 10
s += d * d
n = floor(n / 10)
return s
function is_happy(n):
slow = n
fast = n
repeat:
slow = sum_square_digits(slow)
fast = sum_square_digits(sum_square_digits(fast))
until slow == fast
return slow == 1Method comparison
| Method | Idea | Notes |
|---|---|---|
| Floyd | slow = f(slow), fast = f(f(fast)) | O(1) extra space — interview favorite |
| Visited set | Stop when value repeats | Clearer; uses O(k) memory |
| Known cycle | False if hit any of 4,16,… | Fast shortcut after learning the cycle |
Cheat Sheet
⚡ Quick Reference
| Goal | Pattern |
|---|---|
| Last digit | n % 10 |
| Drop digit | n //= 10 |
| Square sum | total += digit * digit |
| Floyd step | slow = f(slow); fast = f(f(fast)) |
| Happy classics | 1, 7, 19, 23 |
| Unhappy classic | 2 (enters cycle at 4) |
Compare
📋 Floyd vs Visited Set vs Known Cycle
Three ways to decide happy vs unhappy — pick by memory and clarity.
Floyd
slow / fastO(1) space; interview default
Visited set
seen.add(n)Easy to explain; uses extra memory
Known cycle
hit 4 → falseShortcut once you know the cycle
Interview tip
Floyd firstThen mention the set alternative
Context
When This Problem Shows Up
Reach for happy-number checks when digit maps and cycles appear.
Interview warm-ups
Digit loops plus cycle detection in one prompt.
Teaching Floyd
Same tortoise-hare idea as linked-list cycle detection.
Range listing tasks
Print all happy numbers in 1…N for small N.
After GCD
Next classic number-theory style warm-up in this chain.
Positive-only scope
State that 0 / negatives are out of scope.
Key benefit: one short boolean check that forces clear thinking about functional graphs and cycles.
Interactive
🔮 Live Preview
Positive integers only, within JavaScript safe range.
Hands-On
Examples Gallery
Three complete Python programs — Floyd single check, range 1–50, and visited-set style. Click View Output to reveal sample console results.
📚 Getting Started
Floyd tortoise-hare with O(1) extra memory.
Example 1 — Single Value: 19
Floyd cycle detection for one value with a positive-input guard.
python
def sum_of_squares(n: int) -> int:
total = 0
while n > 0:
digit = n % 10
total += digit * digit
n //= 10
return total
def is_happy(n: int) -> bool:
slow = n
fast = n
while True:
slow = sum_of_squares(slow)
fast = sum_of_squares(sum_of_squares(fast))
if slow == fast:
break
return slow == 1
number = 19
if number < 1:
print("Use a positive integer.")
elif is_happy(number):
print(f"{number} is a Happy Number.")
else:
print(f"{number} is not a Happy Number.")How It Works
The two pointers eventually meet. If they meet at 1, the number is happy; otherwise they met inside the unhappy cycle.
⚡ Range Output
Reuse the same helper to filter a beginner interval.
Example 2 — Happy Numbers in [1, 50]
Checks each number independently and prints only happy ones.
python
def sum_of_squares(num: int) -> int:
total = 0
while num > 0:
digit = num % 10
total += digit * digit
num //= 10
return total
def is_happy(num: int) -> bool:
slow = num
fast = num
while True:
slow = sum_of_squares(slow)
fast = sum_of_squares(sum_of_squares(fast))
if slow == fast:
break
return slow == 1
print("Happy numbers in the range 1 to 50:")
for i in range(1, 51):
if is_happy(i):
print(i, end=" ")
print()How It Works
Each number uses the same happy check. Floyd keeps memory constant per check.
⚙️ Visited-Set Style
Easier to explain — trade O(1) space for clarity.
Example 3 — Track Seen Values
Stop when you hit 1 (happy) or see a repeated value (cycle).
python
def sum_of_squares(n: int) -> int:
total = 0
while n > 0:
digit = n % 10
total += digit * digit
n //= 10
return total
def is_happy_set(n: int) -> bool:
if n < 1:
return False
seen = set()
while n != 1 and n not in seen:
seen.add(n)
n = sum_of_squares(n)
return n == 1
for value in (19, 2, 1, 7):
label = "happy" if is_happy_set(value) else "unhappy"
print(f"{value}: {label}")How It Works
A set records every intermediate value. Repeating a value means you entered a cycle; hitting 1 means happy. Prefer Floyd when the interviewer asks for O(1) space.
🧠 How the Algorithm Decides
1
Digit-square map
Define f(n) as the sum of squared digits.
f(n)
2
Advance pointers
slow = f(slow), fast = f(f(fast)).
Floyd
3
Meet and decide
If meeting value is 1 → happy; else cycle.
Verdict
=
Happy or not
Reach 1 → yes; otherwise no.
Trace
🔎 Worked Walkthrough — n = 19
Trace the digit-square path for the classic happy example.
| Step | n | Digit squares | Next |
|---|---|---|---|
| 1 | 19 | 12 + 92 | 82 |
| 2 | 82 | 82 + 22 | 68 |
| 3 | 68 | 62 + 82 | 100 |
| 4 | 100 | 12 + 0 + 0 | 1 |
Reached 1 → 19 is happy.
Use Cases
Where happy-number checks show up beyond the interview prompt.
1. Interview Warm-Ups
Digit peeling plus cycle detection.
Example: write is_happy(n).
2. Teaching Floyd
Same tortoise-hare idea as list cycles.
Example: slow/fast on f(n).
3. Range Filters
List happy numbers in a classroom interval.
Example: 1 to 50 list above.
4. Functional Graphs
Each n maps to exactly one next value.
Example: talk about μ and λ.
5. Digit Practice
% 10 / // 10 drills before harder digit problems.
Example: before Harshad next.
6. Unhappy Cycle Trivia
All unhappy positives share one 8-cycle.
Example: start at 4.
Pro Tip: say “happy means reach 1 under digit-square iteration” before coding Floyd.
Advantages
Why this pattern works well in interviews and classwork.
1. Clear Rule
One sentence: reach 1 via digit-square sums.
2. O(1) Detection
Floyd needs no hash set for the cycle.
3. Famous Tests
19 vs 2 makes verification quick.
4. Rich Follow-Ups
Visited set, known cycle, and linked-list Floyd.
Pro Tip: lead with Floyd; offer a visited set if asked for the simplest version.
Usage Tips
Small habits that keep happy-number solutions interview-ready.
1. Extract f(n) First
Write a clean sum-of-squares helper before Floyd.
2. Guard Positive n
Reject n < 1 under the standard definition.
3. Spot-Check 19 and 2
Happy and unhappy classics catch bugs fast.
4. Meet at 1 Means Happy
Floyd meeting value is the decision signal.
5. Mention the Unhappy Cycle
Shows you understand why Floyd terminates.
Pro Tip: 1 is happy because f(1) = 1 — say that when asked about the fixed point.
Common Pitfalls
Mistakes that commonly break happy-number solutions.
1. Summing Digits Without Squaring
Using digit sum instead of digit-square sum.
→ Always square each digit.
2. Infinite Loop Without Cycle Check
Iterating forever on unhappy numbers.
→ Use Floyd or a visited set.
3. Accepting Zero / Negatives
Standard definition is positive integers only.
→ Reject n < 1.
4. Wrong Floyd Decision
Returning true whenever pointers meet.
→ Check that the meeting value is 1.
5. Mutating n Inside Helpers Carelessly
Reusing a destroyed working variable later.
→ Keep sum_of_squares pure on a local copy.
Edge Cases
Keep input positive and ensure the digit function is pure and deterministic.
n = 1
Immediate happy
It stays at 1.
n = 0
Not positive
Treat as invalid for the standard definition.
Negative
Input validation
Reject negatives instead of guessing behavior.
Base
Decimal assumption
This page uses base-10 digits only.
Unhappy
Cycle at 4
All unhappy positives enter the same 8-cycle.
Range
1 to 50
Expect 1 7 10 13 19 23 28 31 32 44 49.
Math
⚖️ Facts Worth Knowing
Handy follow-ups interviewers sometimes ask.
- Map. f(n) = sum of squared decimal digits; happy means some iterate equals 1.
- Fixed point. f(1) = 1, so 1 is happy.
- Unhappy cycle. 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.
- Bound. Digit-square sums shrink large n into a small range quickly.
Drill
🎯 Practice Problems
Try these variations to lock in the pattern.
1. Classify classics
- 1, 7, 19 → happy
- 2, 4 → unhappy
2. Match both styles
- Floyd vs visited set
- Assert identical booleans
3. Range 1 to 100
- List all happy numbers
- Compare with known lists
4. Print the path
- Show 19 → 82 → … → 1
- Stop at 1 or first repeat
Important
Notes
- Happy: eventually reaches 1.
- Detection: Floyd uses two-speed pointers; no hash set needed.
- Watch-outs: validate positive input and remember the unhappy cycle.
- Per check: O(μ + λ) digit-sum steps with O(1) extra space (Floyd).
Quick Takeaway: sum squared digits repeatedly; if you reach 1 the number is happy, otherwise you loop.
Analysis
⏱️ Time and Space Complexity
| Method | Time (per check) | Extra space |
|---|---|---|
| Floyd (this page) | O(μ + λ) digit-sum steps | O(1) |
| Visited set | same step class | O(k) |
Scan [1, N] | O(N) checks | O(1) beyond each check |
μ is the tail length before the cycle; λ is the cycle length. Each digit-sum step costs O(number of digits).
🎉 Conclusion
Happy numbers reach 1 under repeated digit-square sums; unhappy ones enter a fixed cycle. Prefer Floyd for O(1) space, or a visited set for clarity — and always restrict to positive integers.
Practice the three examples above, then continue to Harshad numbers for another digit-sum divisibility check.
Meeting at 1 means happy; meeting elsewhere means the unhappy cycle.
Pro Tips
💡 Best Practices
✅ Do
- Write a pure sum-of-squares helper
- Use Floyd for O(1) space
- Test 1, 19, and 2
- Reject nonpositive inputs
- Mention the unhappy cycle
❌ Don’t
- Forget to square digits
- Return true on any pointer meeting
- Loop forever without cycle detection
- Accept 0 or negatives silently
- Confuse digit sum with digit-square sum
Summary
Key Takeaways
Knowledge Unlocked
Five things to remember about happy numbers
Decide happiness the interview-friendly way.
5
Core concepts 01
Goal
Reach 1
Definition 02
Map
Σ d²
Digits 03
Floyd
O(1) space
Detect 04
Cycle
Unhappy 8-loop
Trivia 05
Cost
O(μ+λ)
Analysis❓ Frequently Asked Questions
Starting from a positive integer n, repeatedly replace n by the sum of squares of its digits. If this process reaches 1, the number is happy.
Yes. It immediately maps to itself.
It detects loops with O(1) extra memory by moving two pointers at different speeds.
Happy numbers are defined for positive integers. Handle nonpositive inputs separately.
Standard definition uses positive integers only. Usually reject negatives at input.
Each digit-square step is O(log n) in decimal digits, and Floyd takes O(mu + lambda) such steps.
Yes. Track seen values and stop when you repeat or hit 1. It is clearer but uses extra memory.
All unhappy positives eventually enter 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.
Did you Know? 🔊
If a positive integer is not happy, repeated digit-square sums enter the same unhappy cycle: 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.
Continue to Harshad Number
Learn how Harshad (Niven) numbers are divisible by the sum of their digits.
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