Henry Ernest Dudeney/Modern Puzzles/98 - Curious Multiplication

by : $98$

 * Curious Multiplication


 * I have frequently been asked to explain the following, which will no doubt interest many readers who have not seen it.


 * If a person can add correctly but is incapable of multiplying or dividing by a number higher than $2$,
 * it is possible to obtain the product of any two numbers in this curious way.


 * Multiply $97$ by $23$.

97    23 48    (46) 24    (92) 12   (184)  6   (368)  3    736  1   1472      2231


 * In the first column we divide by $2$, rejecting the remainders, until $1$ is reached.
 * In the second column we multiply $23$ by $2$ the same number of times.
 * If we now strike out those products that are opposite ton the even numbers in the first column
 * (we have enclosed these in brackets for convenience in printing)
 * and add up the remaining numbers we get $2231$, which is the correct answer.


 * Why is this?