Definition
End at 1
Repeated plain digit sums finish at a single 1.
Check Magic Number in Python
Overview
What You’ll Learn
A magic number repeatedly reduces by summing its digits until one digit remains — and that digit is 1. This tutorial covers the definition, digital-root link, a live preview, worked Python examples, edge cases, and complexity.
Classic 19
Magic
19 → 10 → 1.
Digital Root
Root = 1
Magic here means digital root equals 1.
Not Happy
No squares
Happy uses squared digits; magic uses plain sums.
Live Preview
Show steps
Watch each digit-sum reduction in the browser.
O(1) shortcut
n % 9
Digital-root formula after you explain the loops.
Introduction
A magic number (in this interview sense) is a positive integer that reaches 1 when you repeatedly replace it by the sum of its decimal digits. Example: 19 → 10 → 1.
That final single digit is the digital root. Magic here simply means digital root equals 1 — not the same as happy numbers (which square each digit).
Why it matters?
It drills digit peeling and reduction loops — a clean warm-up before (or after) happy / Harshad style problems.
Key Highlights
Reduce to One Digit
Keep summing digits until n ≤ 9.
Magic = Root 1
Final digit must be exactly 1.
Two Styles
Nested loops or digital-root shortcut.
Positive Only
Reject 0 and negatives by convention here.
In short: keep replacing n with the sum of its digits; if you finish at 1, the number is magic.
Foundation
📝 Problem & Approach
Given a positive integer n, decide whether repeated digit-sum reduction ends at 1.
python
# 19 → 10 → 1 → magic
# 28 → 10 → 1 → magic
# 18 → 9 → not
# 1 → 1 → magicInputs & Outputs
| Item | Type | Description |
|---|---|---|
n | int | Positive integer (reject n ≤ 0). |
| Return / print | bool / text | True if digital root is 1. |
Minimal workflow
Pseudocode
function is_magic(n): // n > 0
while n > 9:
s = 0
while n > 0:
s += n mod 10
n = floor(n / 10)
n = s
return n == 1Method comparison
| Method | Idea | Notes |
|---|---|---|
| Nested loops | Digit sum until one digit | Interview default — clear and visual |
| Digital root | 1 + (n - 1) % 9 | O(1); mention after the loop story |
| Mod 9 check | n % 9 == 1 (n > 0) | Same shortcut; handle n%9==0 carefully for root 9 |
Cheat Sheet
⚡ Quick Reference
| Goal | Pattern |
|---|---|
| Last digit | n % 10 |
| Drop digit | n //= 10 |
| One digit-sum pass | total += n % 10 |
| Keep reducing | while n > 9 |
| Magic test | final == 1 |
| Shortcut | 1 + (n - 1) % 9 == 1 |
Compare
📋 Loops vs Shortcut vs Happy
Same digit idea — different stopping rules and formulas.
Nested loops
sum until 1 digitBeginner-friendly; easy to debug
Digital root
1+(n-1)%9Constant time after you know the math
Happy numbers
sum of squaresDifferent problem — do not mix them
Interview tip
loops firstOffer the % 9 shortcut as a follow-up
Context
When This Problem Shows Up
Reach for magic-number checks when digit reduction and digital roots appear.
Interview warm-ups
Digit loops with a crisp boolean stop condition.
Teaching digital roots
Same reduction used in divisibility by 9 stories.
Range listing tasks
Print magic numbers in 1…N for small N.
Beside happy / Harshad
Contrast plain digit sum vs squared digits vs divisibility.
Positive-only scope
State that 0 / negatives are out of scope here.
Key benefit: a short digit problem that still opens the door to digital-root theory and O(1) shortcuts.
Interactive
🔮 Live Preview
Enter a positive safe integer to see each digit-sum reduction step.
Hands-On
Examples Gallery
Three complete Python programs — single check for 19, range 1–50, and a digital-root shortcut. Click View Output to reveal sample console results.
📚 Getting Started
Nested loops that reduce until one digit remains.
Example 1 — Single Value: 19
Uses the classic nested-loop digit-sum reduction with a positive-input guard.
python
def is_magic_number(num: int) -> bool:
if num <= 0:
return False
while num > 9:
total = 0
while num > 0:
total += num % 10
num //= 10
num = total
return num == 1
number = 19
if is_magic_number(number):
print(f"{number} is a Magic Number.")
else:
print(f"{number} is not a Magic Number.")How It Works
The outer loop keeps reducing while the value is multi-digit. The inner loop performs one digit-sum pass. For 19: 1+9=10, then 1+0=1.
⚡ Range Output
Reuse the same helper to filter a beginner interval.
Example 2 — Magic Numbers in [1, 50]
Prints all magic numbers from 1 to 50.
python
def is_magic_number(num: int) -> bool:
if num <= 0:
return False
while num > 9:
total = 0
while num > 0:
total += num % 10
num //= 10
num = total
return num == 1
print("Magic Numbers in the range 1 to 50:")
for i in range(1, 51):
if is_magic_number(i):
print(i, end=" ")
print()How It Works
Numbers congruent to 1 mod 9 (in this positive range) end at digital root 1. The reusable checker keeps the range loop clean.
⚙️ Digital-Root Shortcut
Same verdict in O(1) once you know the formula.
Example 3 — Constant-Time Check
Uses 1 + (n - 1) % 9 and compares to 1.
python
def digital_root(n: int) -> int:
if n <= 0:
raise ValueError("n must be positive")
return 1 + (n - 1) % 9
def is_magic_shortcut(n: int) -> bool:
if n <= 0:
return False
return digital_root(n) == 1
for value in (19, 18, 1, 28):
label = "magic" if is_magic_shortcut(value) else "not magic"
print(f"{value}: {label} (root={digital_root(value)})")How It Works
Digital root collapses all digit-sum iterations into one modulus. In interviews, explain the loop first, then mention this O(1) shortcut.
🧠 How the Algorithm Decides
1
Validate
Reject n ≤ 0 under this page’s convention.
Guard
2
Sum digits
While n > 9, replace n with its digit sum.
Reduce
3
Compare to 1
Magic iff the final single digit is 1.
Verdict
=
Magic or not
Root 1 → yes; otherwise no.
Trace
🔎 Worked Walkthrough — n = 19
Trace the digit-sum reductions for the classic magic example.
| Step | n | Digit sum | Next |
|---|---|---|---|
| 1 | 19 | 1 + 9 | 10 |
| 2 | 10 | 1 + 0 | 1 |
| 3 | 1 | (single digit) | stop — magic |
Contrast: 18 → 9 stops at 9 → not magic.
Use Cases
Where magic-number checks show up beyond the interview prompt.
1. Interview Warm-Ups
Digit peeling with a clear stop condition.
Example: write is_magic_number(n).
2. Digital-Root Intro
Bridge loops to the % 9 formula.
Example: root of 19 is 1.
3. Range Filters
List magic numbers in a classroom interval.
Example: 1 to 50 list above.
4. Contrast Problems
Separate magic from happy and Harshad.
Example: plain sum vs squares.
5. Divisibility Stories
Digital roots relate to rules for 9.
Example: root 9 ↔ multiple of 9.
6. Debugging Traces
Print each reduction like the live preview.
Example: 19 → 10 → 1.
Pro Tip: say “magic means digital root is 1” before coding the loops.
Advantages
Why this pattern works well in interviews and classwork.
1. Clear Rule
One sentence: digit-sum reduction ends at 1.
2. Easy to Trace
19 → 10 → 1 is whiteboard-friendly.
3. Shortcut Available
Digital-root formula upgrades the solution later.
4. Famous Contrasts
19 vs 18 and magic vs happy catch misconceptions.
Pro Tip: lead with nested loops; offer the digital-root shortcut if asked about optimization.
Usage Tips
Small habits that keep magic-number solutions interview-ready.
1. Guard Positive n
Reject n ≤ 0 under this tutorial’s definition.
2. Stop at One Digit
Use
while n > 9, not an arbitrary iteration count.3. Spot-Check 19 and 18
Magic and not-magic classics catch bugs fast.
4. Do Not Square Digits
That is the happy-number map — different problem.
5. Mention the Shortcut Last
Explain loops first; then cite digital root.
Pro Tip: 1 is magic because it is already the target single digit — say that when asked about the base case.
Common Pitfalls
Mistakes that commonly break magic-number solutions.
1. Squaring Digits
Confusing magic with happy numbers.
→ Sum digits plain — no squares.
2. Stopping After One Pass
Checking only the first digit sum (e.g. 19 → 10) and quitting.
→ Keep going until n ≤ 9.
3. Accepting Zero / Negatives
This page treats them as not magic.
→ Return False for n ≤ 0.
4. Wrong Shortcut for Multiples of 9
Using
n % 9without the digital-root adjustment.→ Prefer
1 + (n - 1) % 9for root values.5. Mutating Input Carelessly
Destroying n before you can print the original later.
→ Work on a local copy inside the helper.
Edge Cases
Keep input positive and remember magic is not the same as happy.
Zero
n = 0
This tutorial does not treat 0 as magic.
Negative
Sign convention
Reject negatives unless your task defines otherwise.
Definition
Magic vs happy
Happy uses squared digits; this page uses plain digit sum.
n = 1
Already one digit
Immediately magic — no loop needed.
18
Classic no
18 → 9, final digit is not 1.
Range
1 to 50
Expect 1 10 19 28 37 46.
Math
⚖️ Facts Worth Knowing
Handy follow-ups interviewers sometimes ask.
- Digital root. Magic on this page means digital root equals 1.
- Formula. For n > 0, root = 1 + (n - 1) % 9.
- Pattern. In small ranges, magic numbers often look like 1, 10, 19, 28, …
- Not happy. Happy numbers iterate sum of squared digits until 1 or a cycle.
Drill
🎯 Practice Problems
Try these variations to lock in the pattern.
1. Classify classics
- 1, 19, 28 → magic
- 18, 20 → not
2. Match both styles
- Loops vs digital-root shortcut
- Assert identical booleans
3. Range 1 to 100
- List all magic numbers
- Compare with n % 9 == 1 pattern
4. Print the path
- Show 19 → 10 → 1
- Stop when n ≤ 9
Important
Notes
- Definition: repeated digit sum ends at 1.
- Code: nested loop approach is beginner-friendly.
- Optional: digital-root formula provides a constant-time shortcut.
- Do not confuse with happy numbers (squared digits) or Harshad (divisible by digit sum).
Quick Takeaway: keep summing digits until one remains; magic means that digit is 1.
Analysis
⏱️ Time and Space Complexity
| Approach | Time (single n) | Extra space |
|---|---|---|
| Repeated digit-sum loops | O((log n)²) practical small | O(1) |
| Digital-root shortcut | O(1) | O(1) |
Scan [1, N] | O(N) checks | O(1) |
Each digit-sum pass costs O(number of digits); very few passes are needed in practice.
🎉 Conclusion
Magic numbers (here) are positive integers whose digital root is 1. Implement the nested digit-sum loops first, test 19 and 18, then mention the O(1) digital-root shortcut.
Practice the three examples above, then continue to matrix addition for a 2D array warm-up.
Final digit 1 means magic; 18 → 9 means not — and never square the digits on this page.
Pro Tips
💡 Best Practices
✅ Do
- Reduce with plain digit sums until one digit
- Test 1, 19, and 18
- Reject nonpositive inputs
- Explain loops before the % 9 shortcut
- Contrast with happy numbers when asked
❌ Don’t
- Square digits (happy-number habit)
- Stop after a single digit-sum pass
- Accept 0 or negatives silently
- Lead with the shortcut before the definition
- Confuse digital root 9 with root 0
Summary
Key Takeaways
Knowledge Unlocked
Five things to remember about magic numbers
Decide magic the interview-friendly way.
5
Core concepts 01
Goal
End at 1
Definition 02
Map
Digit sum
Digits 03
Root
Digital root = 1
Math 04
Shortcut
1+(n-1)%9
O(1) 05
Not happy
No squares
Contrast❓ Frequently Asked Questions
Repeatedly sum decimal digits until one digit remains; if that digit is 1, the number is magic.
Yes. It is already one digit and equals 1.
No. Happy numbers use sum of squared digits. Magic numbers here use plain digit sums.
This page follows interview convention: use positive integers only.
The final one-digit value is digital root. Magic here means digital root = 1.
Each pass sums digits in O(log n); repeated passes are tiny in practice for standard input sizes.
Yes. For n > 0, digital root is 1 + (n - 1) % 9. Magic iff that equals 1 (equivalently n % 9 == 1).
18 → 9, and the final digit is 9, not 1.
Did you Know? 🔊
Magic-number checks in this page are exactly digital-root checks for root = 1.
Continue to Matrix Addition
Learn how to add two matrices element by element in Python.
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