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Tyson Cung
Tyson Cung

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I Found Hidden Solutions to a Viral Math Puzzle — Here's How

#math #programming #python #beginners

A video went viral recently showing a neat math trick:

Take a number, find its nth root, and check if that equals the digit sum minus n.

The original showed just 4 examples:

Expression Digit Sum − n Result
√25 = 5 (2+5) − 2 = 5 ✅
√64 = 8 (6+4) − 2 = 8 ✅
³√125 = 5 (1+2+5) − 3 = 5 ✅
³√4096 = 16 (4+0+9+6) − 3 = 16 ✅

Cool party trick. But as a developer, my first thought was: are there more?

Brute-Forcing the Answer

I wrote a simple script to check every combination:

def digit_sum(n):
    return sum(int(d) for d in str(n))

solutions = []

for n in range(2, 20):        # root degree (2=square, 3=cube...)
    for number in range(1, 1_000_000):  # check up to 1 million
        root = round(number ** (1/n))

        # Verify it is a perfect nth root
        if root ** n != number:
            continue

        # Check the pattern: nth_root == digit_sum - n
        if root == digit_sum(number) - n:
            solutions.append((n, number, root))
            print(f"{n}root({number}) = {root}  "
                  f"(digit sum {digit_sum(number)} - {n} = {root})")
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The Results: 8 Total Solutions

The original video found 4. We found 8:

2-digit numbers (square roots) 

√25  = 5    →  (2+5) − 2 = 5  ✅
√64  = 8    →  (6+4) − 2 = 8  ✅
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3-digit numbers (square + cube roots) 

³√125 = 5   →  (1+2+5) − 3 = 5  ✅
√196  = 14  →  (1+9+6) − 2 = 14 ✅   ← NEW!
√289  = 17  →  (2+8+9) − 2 = 17 ✅   ← NEW!
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4-digit numbers (cube roots only) 

³√2744 = 14  →  (2+7+4+4) − 3 = 14  ✅  ← NEW!
³√3375 = 15  →  (3+3+7+5) − 3 = 15  ✅  ← NEW!
³√4096 = 16  →  (4+0+9+6) − 3 = 16  ✅
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That is it. No 5-digit solutions. No 4th roots. No 5th roots. The pattern completely dies.

Why Does It Die?

Here is the intuition:

  • An n-digit number has a maximum digit sum of 9n (all 9s)
  • A k-digit number is at least 10^(k-1)
  • For the pattern to work: nth_root ≈ digit_sum − n

As numbers grow, the nth root grows polynomially but the digit sum grows only logarithmically (roughly proportional to the number of digits).

The gap between them widens fast:

Number      | Max Digit Sum | Square Root
-----------:|:-------------:|:---------:
99          |      18       |    ~10
999         |      27       |    ~32
9,999       |      36       |    ~100
99,999      |      45       |    ~316
999,999     |      54       |    ~1000
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By 5 digits, the square root is already 7× larger than the maximum possible digit sum. The pattern mathematically cannot produce new solutions.

The Takeaway

This is why I love code. A viral video shows a cute trick with 4 examples and implies it is a deep pattern. Ten lines of Python proves there are exactly 8 solutions and the pattern is finite.

Not everything that looks like a pattern is a pattern. But finding the boundaries is where the real fun starts.

▶️ Watch the YouTube Short: https://youtube.com/shorts/ocnOgIYSDdE

I also made a YouTube Short about this — check it out above! 👆

Like mathematical puzzles + code? Follow me for more — I post daily about AI, development, and the occasional number theory rabbit hole.

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