5 Holiday Math Puzzles

Happy Holidays! Here are a couple Christmas and Hanukkah puzzles just for you:

The 2n-Menorah: In an alternate universe, instead of the flask in the temple lasting for just 8 days, it lasted for 2n days for a positive integer n. Thus, Hanukkah in that universe is celebrated for 2n days, and the menorah has 2n+1 candles (including the shammash):

A 2n-Menorah

On the first day, 1 regular candle is lit and also the shammash. On the second, 2 regular candles and the shammash. And so on until the 2n-th day where 2n regular candles are lit alongside the shammash. How many candles do you need for the whole of Hanukkah in that universe?

(Note that the shammash is just the name for the candle put in the center, which is traditionally used to light the other ones. No candle is reused)

Lazy Santa: Santa got lazy and forgot to write a nice-naughty list. So when giving out presents in a linear street with 50 houses he does the following: For each house, Santa either gives them a present, or skips directly to the next house instead. Additionally, Santa does not skip a house twice in a row, so if he skips one house he must give a present to the next one

How many total ways are there for Santa to give out presents this way?

Circle of Dreidels: One of the most iconic symbols of Hanukkah is the dreidel, a spinning top with four faces. When spun, the top has a uniform chance to land on each face, which are marked by Hebrew letters (נ,ג,ה,ש or, in Israel, נ,ג,ה,פ)

Suppose you put 8 drediels in a circle as follows:

All 8 are then spun together. What’s the expected number of dreidels whose face they land on is the same of one of their neighbors?

Ornament Count: A Christmas tree is decorated the following way: There are 3 rings of ornaments one above the other, each holds 5 ornaments. Additionally all the rings have simultaneous rotational symmetry (of the positions of ornaments, not their colors). If each ornament can be either red, blue, or yellow, how many different ways are there to decorate the tree?

Note that the tree is three dimensional. If you can rotate it to get from one way of decoration to the other, those two ways of decorating are the same. Here is a top down view of the tree for refrence:

Chocolate Coin Toss: A chocolate coin is a coin shaped piece of chocolate covered in tinfoil making it look like a golden coin. Suppose you start with 5 chocolate coins, and you toss all of them. Then you eat any that landed on tails, and toss the others again. You repeat this process until you have no coins left. What’s the expected number of steps (tossing all the coins you have and eating those that landed on tails) it will take for you to eat all your coins? Assume that each of the coin tosses is fair

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