Good old pen and paper

As is often the case with Project Euler problems, the problem can be solved with pen and paper.

Before anything, know that \( [0-9] \) means any digit between \( 0 \) and \( 9 \).

We know that the number to beat is \( 918273645 \), so any solution we find must be greater than that number. We also know that the first multiplier is \( 1 \), so the first digit of our seed must start with \( 9 \). We can discard any number in the form \( 9[0-9], 9[0-9][0-9] \) or any number greater than \( 100000 \), because none of them will result in a 9-digit number when multiplied by (1, 2, 3) or (1, 2, 3, 4).

Therefore, the solution must be of the form \( 9[0-9][0-9][0-9] \).

The solution must contain different digits and no zeros, because of the *1 multiplier. It can not contain any 1 at all because it will be present twice. One with the *1 multiplier and another because the *2 multiplier will result with the number of the form \( 18[0-9][0-9] \).

Therefore, the solution must be of the form \( 9[2-9][2-9][2-9] \).

We can continue these eliminations rules:

\[ \begin{align} &9[5-9][2-9][2-9] * 2 = 19[0-9][0-9][0-9] \text{, the *1 multiplier already contains a 9.}\\ &94[5-9][2-9] * 2 = 189[0-9][0-9] \text{, the *1 multiplier already contains a 9.}\\ &94[2-4][2-9] * 2 = 188[0-9][0-9] \text{, the double 8 is obviously wrong.}\\ &937[2-7] * 2 = 187[0-9][0-9] \text{, the *1 multiplier already contains a 7.}\\ &936[2-7] * 2 = 18724, 18726, 18728, 18730, 18732 \text{ or } 18734 \text{. They always lack the number 5 except for 18730 which have the number 4.}\\ &935[2-7] * 2 = 187[0-1][0-9] \text{, contains either a 0 or a double 1.}\\ &934[2-7] * 2 = 186[8-9][0-9] \text{, contains either a 9 or a double 8.}\\ &933[2-7] \text{ is obviously wrong because of the double 3.}\\ &932[2-6] * 2 = 1864, 1866, 1868, 1870 \text{ or } 1872 \text{. They always lack the number 5 except for 1870 which lack the number 4.}\\ \end{align} \]

Thus, the number must be \( 9327 \). Indeed, \( 9327 + 18654 \) is the number we are looking for.