The Solution to the Nose to Tail Number Puzzle

The Puzzle

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 Solution Strategy

The strategy to solve this puzzle involves starting with a two-digit N and gradually increasing the number of digits until the solution is found.

Two-Digit Solution Attempt

Say N = 4[?], where [?] is a digit. The only possible value for [?] is 1 because a quarter of 4[?] is [?]4 and a quarter of 4 is 1. However, 14 is not a quarter of 41.

Three-Digit Solution Attempt

Let N = 4[??]. For the same reason as before, the second digit of N must be 1. So N = 41[?]. The final digit of N must be 6, since 4 x 4 is 16. However, a quarter of 416 is not 164.

Four-Digit Solution Attempt

Let N = 4[???]. For the same reasons as above, N = 41[?]6. The penultimate digit of N must be 5, since 4 x 64 = 256. However, a quarter of 4156 is not 1564.

Five-Digit Solution Attempt

N = 41[?]56. The antepenultimate digit of N must be 2, since 4 x 564 = 2256. But a quarter of 41256 is not 12564.

Six-Digit Solution

N = 41[?]256. Since 4 x 2564 = 10256, we know that the [?] must be 0. This works! We have our answer: N = 410256 = 4 x 102564.