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Jun 21, 2026
Analyzed by GPT OSS 120B

Cracking the 4‑to‑Tail Quarter Puzzle: The Smallest Number Revealed

AI Summary
A classic 1983 Moscow Mathematical Olympiad puzzle asks for the smallest number beginning with 4 that becomes a quarter of itself when the leading 4 is moved to the end. The solution, 410 256, emerges after a systematic digit‑by‑digit search, illustrating elegant number‑theory reasoning.

Lead: A Century‑Old Brain‑Teaser Finds Its Minimal Solution

A decades‑old Olympiad challenge—find the smallest integer N that starts with 4 and turns into a quarter of itself when the leading digit is shifted to the end—has been solved with a tidy six‑digit answer: 410256. The step‑by‑step deduction showcases how simple arithmetic constraints can drive a precise search through the number line.

The 4‑Leading Digit Puzzle Unveiled

The problem can be expressed algebraically as:

  • N = 4 × 10^k + x, where x is the remaining k‑digit suffix.
  • Moving the leading 4 yields the number 10 × x + 4.
  • The condition N ÷ 4 = 10 × x + 4 must hold.

Solvers start with the smallest possible digit count and incrementally test feasibility, eliminating two‑, three‑, four‑ and five‑digit candidates before arriving at a six‑digit solution.

Numbers Behind the Solution: 410 256

Key numeric observations during the search:

  • Two‑digit trial forces the second digit to be 1, but 14 ≠ ¼ × 41.
  • Three‑digit trial yields a forced final digit 6 (since 4 × 4 = 16), yet 416 ≠ ¼ × 164.
  • Four‑digit trial introduces a penultimate digit 5 (4 × 64 = 256), but 4156 fails the quarter test.
  • Five‑digit trial requires an ante‑penultimate digit 2 (4 × 564 = 2256), yet 41256 still fails.
  • Six‑digit trial finally satisfies 4 × 102564 = 410256, confirming N = 410256 as the minimal solution.

Thus, 410 256 = 4 × 102 564 and moving the leading 4 produces 102 564, exactly one‑quarter of the original number.

Why This Classic Olympiad Puzzle Still Captivates

The appeal lies in its blend of elementary arithmetic and logical deduction, making it accessible to a broad audience while rewarding systematic problem‑solving. It also highlights a broader theme in number theory: how digit‑manipulation constraints can define unique integer solutions. Educators frequently reuse the puzzle to illustrate modular reasoning and the power of incremental hypothesis testing.

What’s Next for Numerical Brain‑Teasers

Given the resurgence of interest on platforms like Instagram and puzzle newsletters, similar digit‑rotation challenges are likely to reappear, often with added twists such as different bases or multiple‑step transformations. Enthusiasts can expect new variants that push the boundary between recreational math and formal combinatorial research.