AoPS - Problem

Cauchy the magician has a new card trick. He takes a standard deck(which consists of 52 cards with 13 denominations in each 4 suits) and let Schwartz to shuffle randomly. Schwartz is told to take $ m $ cards not more than $ \frac{1}{3} $ form the top of the deck. Then, Cauchy take $ 18 $ cards one by one from the top of the remaining deck and show it to Schwartz with the second card is placed in front of the first card (from Schwartz view) and so on. He ask Schwartz to memorize the $ m-th $ card when showing the cards. Let it be $ C_1 $. After that, Cauchy places the $ 18 $ cards and the $ m $ cards on the bottom of the deck with the $ m $ cards are placed lower than the $ 18 $ cards. Now, Cauchy distributes and flip the cards on the table from the top of the deck while shouting the numbers $ 10 $ until $ 1 $ with the following operation: a) When a card flipped has the number which is same as the number shouted by Cauchy, stop the distribution and continue with another set. b) When $ 10 $ cards are flipped and none of the cards flipped has the number which is same as the number shouted by Cauchy, take a card from the top of the deck and place it on top of the set with backside(the site which has no value) facing up. Then continue with another set. Cauchy stops when 3 sets of cards are placed. Then, he adds up all the numbers on top of each sets of cards( backside is consider $ 0 $ ). Let $ k $ be the sum. He placed another $ k $ cards to the table from the top of the remaining deck. Finally, he shows the first card on top of the remaining deck to Schwartz. Let it be $ C_2 $. Show that $ C_1 = C_2 $.

Very great problem about cards! Now, about the solution... Starting from the time Cauchy "distributes and flip the cards..." to the time Cauchy "places another $k$ cards...", it is known that he draws 33 cards in total. Why? For example, if Cauchy stops when shouting number $7$, it means that he has drawn $4$ cards before that, and the $4\text{th}$ one is number $7$, so later on he will draw $7$ cards. Or if Cauchy draws $10$ cards and then put a card considered as number $0$ on it, in total he draws $11$ cards. Multiply by $3$ and we have $33$. So card $C_2$ is the $34\text{th}$ card, counting from the top. To prove that card $C_1$ is put at the $34\text{th}$ position, it is known that Cauchy and Schwartz in total draw $m+18$ cards, so there are left with $34-m$ cards. When putting the $m+18$ cards again to the deck, the $C_1$ card is put at the $34-m+m=34\text{th}$ position. Yeah! Proof is complete.