Henry Ernest Dudeney/Puzzles and Curious Problems/174 - More Curious Multiplication/Solution 2
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Puzzles and Curious Problems by Henry Ernest Dudeney: $174$
- More Curious Multiplication
- 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?
Solution
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.
Sources
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $238$. More Curious Multiplication