Henry Ernest Dudeney/Puzzles and Curious Problems/168 - Mental Arithmetic/Mistake

Source Work

Arithmetical and Algebraical Problems

Various Arithmetical and Algebraical Problems

$168$ -- Mental Arithmetic

Arithmetical and Algebraical Problems

Miscellaneous Puzzles

$233$ -- Mental Arithmetic

Calling the numbers $a$ and $b$, we have:

$a^2 + b^2 + a b = \Box = /a - m b/^2 = a^2 = 2 a m b + b^2 m^2$.

$\therefore b + a = -2 a m + b m^2$,

$\therefore b = \dfrac {a \paren {2 m + 1} } {m^2 - 1}$

in which $m$ may be any whole number greater than $1$, and $a$ is chosen to make $b$ rational.

$b$ is already guaranteed to be rational.

What we do is choose $a$ to be an integer.