Back to Headlines
Science
Jun 15, 2026
Analyzed by GPT OSS 120B

The 410256 Puzzle: A Classic Cyclic Number Challenge

AI Summary
A classic number puzzle asks for the smallest integer beginning with 4 that becomes a quarter of itself when the leading digit is moved to the end. The solution, 410256, showcases a rare cyclic property and originates from the 1983 Moscow Mathematical Olympiad.

Quick Overview of the Challenge

A number N starts with the digit 4. When the leading 4 is shifted to the end, the resulting number equals N ÷ 4. The task is to find the smallest possible N.

Defining the Puzzle Mechanics

The condition can be written as:

  • N = 4 × X
  • Moving the leading 4 to the end yields X4
  • Therefore, 4 × X4 = N

Where X represents the remaining digit sequence.

Step‑by‑Step Search for the Smallest N

The author’s systematic approach examines increasing digit lengths:

  • Two‑digit attempt: 4? → only ?=1 works, but 14 ≠ 41/4.
  • Three‑digit attempt: 41? → ? must be 6 (since 4 × 4 = 16), yet 416/4 ≠ 164.
  • Four‑digit attempt: 41?6 → middle digit must be 5 (4 × 64 = 256), but 4156/4 ≠ 1564.
  • Five‑digit attempt: 41?56 → next digit must be 2 (4 × 564 = 2256), yet 41256/4 ≠ 12564.
  • Six‑digit attempt: 41?256 → the remaining digit must be 0 because 4 × 2564 = 10256, giving the valid number 410256.

Why 410256 Is Mathematically Special

The solution is an example of a cyclic number: multiplying it by 4 produces a rotation of its digits (102564), and moving the leading 4 to the end yields exactly 102564. Such numbers are rare; the most famous is 142857 (the 1/7 repetend).

Broader Significance and Puzzle Legacy

First published in the 1983 Moscow Mathematical Olympiad, the puzzle has resurfaced in modern puzzle columns, illustrating how elementary digit manipulation can reveal deep number‑theoretic properties. It continues to engage enthusiasts, prompting new generations to explore cyclic phenomena without advanced mathematics.