181. Card Trick. Take a regular deck of cards (this time, let's consider all jacks, queens, and kings as tens). Glancing at the top card (let's say it's a seven), place it face down on the table, then, continuing to count out loud in order: "Eight, nine, ten..." and so on up to 12, lay out other cards from the deck on top of it. Since the bottom card is a seven, a stack of 6 cards will form on the table.

Take another look at the top card of the remaining part of the deck (let's say, for example, it's a "former queen" - now a ten), place it face down on the table, and, continuing to count in order up to 12, lay out one card from the deck on the table with each count. This time, the stack will have 3 cards (10, 11, 12). Continue doing this until you exhaust the entire deck. If at the end of the layout there are not enough cards in the deck to complete a full stack (up to the count of 12), set the incomplete stack aside.

Now tell me how many stacks you have and how many cards you set aside, and I will immediately tell you the sum of the values of the bottom cards in all the stacks. To do this, I simply multiply 13 by the number of stacks minus 4, and add the number of cards set aside. For example, if there are 6 stacks and 5 extra cards, then 13, multiplied by 2 (6 minus 4), plus 5 equals 31, the sum of the bottom cards.

Why does this work?
181. In each stack, the number of cards must equal 13 minus the value of the lowest one. Therefore, 13, multiplied by the number of stacks, minus the sum of the bottom cards and plus the number of remaining cards should equal the total number of cards in the deck, which is 52. Thus, the sum of the bottom cards equals 13, multiplied by the number of stacks, minus 52 and plus the number of remaining cards. But this is the same as 13, multiplied by the number of stacks minus 4, plus the number of remaining cards. A reader with an algebraic inclination will easily be able to express all this in the language of familiar symbols.