Henry Ernest Dudeney/Puzzles and Curious Problems/174 - More Curious Multiplication/Mistake
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Source Work
1932: Henry Ernest Dudeney: Puzzles and Curious Problems:
- Arithmetical and Algebraical Problems
- Various Arithmetical and Algebraical Problems
- $174$. -- More Curious Multiplication
- Various Arithmetical and Algebraical Problems
1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems:
- Arithmetical and Algebraical Problems
- Miscellaneous Puzzles
- $238$. More Curious Multiplication
- Miscellaneous Puzzles
Mistake
- 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.
Correction
Dudeney completely missed all numbers of the form $\sqbrk {1 0 d_1 d_2 \ldots d_n 0 1}_{10}$.
He also missed:
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 $991$.
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $174$. -- More Curious Multiplication
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $238$. More Curious Multiplication