1230.65 – Identify the card

Take all the face cards out of a deck of playing cards. Keep all the numbered cards, counting ace as 1, and hand them to a friend. Your friend takes one card out of the deck and gives you back the rest.

Given only the sum total of the values of the remaining cards, how can you tell which numbered card is gone?

Solution

There should be four 1's, four 2's, and so on up to four 10's. This totals

4×(1+2+3+⋯+10)=4×(10×112)=4×55=220\begin{align*}4 \times (1+2+3+\cdots+10) &= 4 \times \left(\dfrac{10 \times 11}{2}\right) \\[1em]&= 4 \times 55 \\[0.5em]&= 220\end{align*}

Add up the cards that you have, and you will be a bit short of that 220. For example, if your sum is 216 the missing card is a 4.