Henry Ernest Dudeney/Puzzles and Curious Problems/174 - More Curious Multiplication/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $174$

What number is it that, when multiplied by $18$, $27$, $36$, $45$, $54$, $63$, $72$, $81$ or $99$,

gives a product in which the first and last figures are the same as those in the multiplier,

but which when multiplied by $90$ gives a product in which the last two figures are the same as those in the multiplier?

$987 \, 654 \, 321$

As Martin Gardner points out: all numbers such as:

$1001$

$10101$

$100101$

that is, numbers made up of $1$ and $0$, with $1$ at each end and no two consecutive instances of $1$.

Matt Westwood notes that you can't rule out numbers of the form $[1 0 d_1 d_2 \ldots d_n 0 1]_{10}$.

However, at the left hand end you need to make sure your carry digit does not affect the second digit.

All $d$-digit numbers $n$ in the form of:

$n = \sqbrk {9 a_1 a_2 \dots a_{d - 2} 1}$

where $n \ge \dfrac 8 {81} \times 10^{d + 1}$ and $d \ge 3$.

The smallest solution of this form is $991$.