João has a deck of 52 cards numbered from 1 to 52. A set of three cards is called lucky if the sum of the digits on each card is the same. What is the minimum number of cards João has to take from the deck, without looking, so that among the cards he takes there are necessarily three cards that form a set of lucky cards?
First, observe that the sum of the digits of the cards is at most $4+9=13$, which only happens with the card 49. For sums that are between 1 and 12, there are at least two cards that satisfy each sum. Thus, by taking the card 49 plus two cards for each sum between 1 and 12, that is, $2 \times 12+1=25$ cards, we still do not have three cards that form a lucky set.

Now, if we take 26 cards, at least 25 of them have a sum of their digits between 1 and 12. Therefore, at least 3 cards have the same sum of their digits.

#