Henry Ernest Dudeney/Puzzles and Curious Problems/115 - The Eight Cards/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $115$

Rearrange these cards, moving as few as possible, so that the two columns add up alike.

Can it be done?

$\begin{array} {} \boxed 1 & \boxed 3 \\ \boxed 2 & \boxed 4 \\ \boxed 7 & \boxed 5 \\ \boxed 9 & \boxed 8 \\ \end{array}$

$\begin{array} {} \boxed 1 & \boxed 3 \\ \boxed 2 & \boxed 4 \\ \boxed 7 & \boxed 5 \\ \boxed 8 & \boxed 6 \\ \end{array}$

having inverted the $\boxed 9$ to make $\boxed 6$.

As the numbers add up to $39$, it is clear this cannot be done by keeping the cards in the same orientation.

However, a veteran of these puzzles will instantly consider turning the $\boxed 9$ card round to make $\boxed 6$.

Then we see the columns add to $16$ and $20$ respectively.

So $2$ needs to be added to column $1$ and subtracted from $2$.

This can be done by exchanging the $\boxed 9$, turning it round to make $\boxed 6$, and the $\boxed 8$.

$\blacksquare$