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

Source Work

Arithmetical and Algebraical Problems

Various Arithmetical and Algebraical Problems

$174$. -- More Curious Multiplication

Arithmetical and Algebraical Problems

Miscellaneous Puzzles

$238$. More Curious Multiplication

The number is $987,654,321$, which, when multiplied by $18$, gives $17,777,777,778$, with $1$ and $8$ at the beginning and end.

And so on with the other multipliers, except $90$, where the product is $88,888,888,890$, with $90$ at the end.

Dudeney completely missed all numbers of the form $\sqbrk {1 0 d_1 d_2 \ldots d_n 0 1}_{10}$.

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 $991$.