Full Breakdown
The “Nose-to-Tail” Puzzle: Finding the Smallest Number Starting with 4 Whose Quarter Is Its Rear-Shifted Form
6/9/2026, 5:09:27 AM
Puzzle Overview – Core Challenge
On 8 June 2026, *The Guardian* published a number-theory puzzle titled “Nose to tail.” The problem asks for the smallest integer N that begins with the digit 4 such that moving this leading 4 to the end of the number yields a new integer equal to one-quarter of N. Formally, if N = 4 […] then N ÷ 4 = […]4. The puzzle is sourced from the Moscow Mathematical Olympiad 1983 and was presented by puzzle columnist Kevin Gately.
Step-by-Step Solution Process
The article’s author proceeds by incrementally increasing the digit length of N until the condition holds.
1. Two-digit trial – With N = 4?, the only feasible second digit is 1 (since ¼ × 4 = 1). However, 14 != ¼ × 41, so two digits are insufficient.
2. Three-digit trial – Setting N = 41?, the quarter-condition implies 4 × ?4 = 41?. The final digit must be 6 (because 4 × 4 = 16), yet 416 ÷ 4 = 104, not 164.
3. Four-digit trial – Extending to N = 41?6, the quarter-condition yields 4 × ?64 = 41?6. The penultimate digit must be 5 (since 4 × 64 = 256), but 4156 ÷ 4 = 1039, not 1564.
4. Five-digit trial – With N = 41?56, the equation 4 × ?564 = 41?56 forces the ante-penultimate digit to be 2 (4 × 564 = 2256). Yet 41256 ÷ 4 = 10314, not 12564.
5. Six-digit trial – For N = 41?256, the relation 4 × ?2564 = 41?256 requires the missing digit to be 0 (4 × 2564 = 10256). This satisfies the condition:
N = 410256,\qquad \frac{N}{4}=102564,\qquad \text{and }102564\text{ with the leading 4 moved to the end becomes }1025644 = 410256/4.
Thus the smallest solution is N = 410 256.
Data & Statistics
- Solution: 410 256
- Digits: 6
- Quarter: 102 564 (the rearranged form of N)
Background & Context – The Guardian Puzzle Series
Kevin Gately notes that the puzzle appears in his regular column, which “has been setting a puzzle here on alternate Mondays since 2015.” The column invites readers to discuss the “snout” theme rather than share spoilers, fostering community interaction around mathematical challenges. The puzzle’s provenance from the 1983 Moscow Mathematical Olympiad underscores its longstanding appeal in competitive mathematics.
Why It Matters – Educational Impact
The “Nose-to-Tail” puzzle illustrates elementary concepts of place value, modular arithmetic, and systematic trial-and-error reasoning. By publishing such problems, *The Guardian* promotes public engagement with mathematics, encouraging readers to develop problem-solving skills through accessible yet non-trivial challenges.
Verbatim Quotes
- “Nose to tail There is a number N beginning with 4 such that moving the 4 to the end of it creates a new number that is a quarter of N.” — *The Guardian* puzzle description
- “Source: Moscow Mathematical Olympiad 1983, via @mathematicsproblems and Kevin Gately I’ve been setting a puzzle here on alternate Mondays since 2015.” — Kevin Gately
- “Instead discuss snouts.” — Puzzle introduction note
Conflicting Reports & Gaps
All source excerpts present a consistent solution and reasoning path; no discrepancies or missing data are identified.
