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Probability and Cumulative Dice Sums

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Interview Doubler


Problem:

Find the smallest positive number that doubles when you move the last digit to the front.

Solution:

The answer is \( 105263157894736842 \), and the solution is similar to Twisty Temperature. Let this number be \[ x = x_{n}\cdot 10^{n-1} + ... + x_{1}\cdot 10^1 + x_{0} \] with \( 0 < x_{n} < 10 \), then after moving the last digit to the front we get \[ y = x_{0}\cdot 10^{n-1} + (x-x_{0})/10. \] We also have that \( y = 2 x \), so our equation becomes \[ 19\cdot x = x_0 \cdot (10^{n}-1). \] Now 10 is a primitive root modulo 19, and so it follows that \( x \) is integral if and only if \( n \) is of the form \( 18\cdot m \). Also note that we need \( 2 \le x_0 \le 9 \) since we require \( 0< x_{n} < 10\). When \( m=1 \) and \( x_0 = 2\) we get \( x  =105263157894736842 \); when \( m=2\) and \(x_0 = 2\) we get \[ 2 (10^{36}-1)/19 = 105263157894736842105263157894736842.\] That this number indeed doubles when you move the last digit to the front can be verified by WolframAlpha.

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