Holiday Math Problems!

While all of you are bored from sleeping in, sipping eggnog, and munching on fruitcake, try a few of these holiday math problems courtesy of MathCounts.

Ariel and Thomas are playing the Dreidel game.  In their version of the game players take turns spinning the dreidel and take various actions based on which symbol is face up when the dreidel stops spinning.  The four actions the spinner can take are (1) put two tokens in the pot, (2) take half of the tokens from the pot, (3) take all of the tokens from the pot (and all other players each deposit one token into the pot), or (4) take no action and the next player spins. Before they begin, the total number of tokens is divided evenly between them.  To start the game Ariel and Thomas each deposit one of their tokens into the pot.  Ariel spins the dreidel first and must place two of her tokens in the pot.  For Thomas’ first spin he does nothing.  In the second round Ariel spins and must take half the tokens from the pot, while Thomas’ spin then results in his placing two tokens in the pot.  During the third round Ariel does nothing after her spin.  When Thomas spins he takes all the tokens from the pot which is equal to 2/5 of the total number of tokens.  How many total tokens does Thomas have after taking these tokens?

Every year on the first day of Christmas my true love gives me one gift, a partridge in a pear tree. On the second day of Christmas my true love gives me a total of three gifts, two turtle doves and a partridge in a pear tree.  This will go on until the twelfth day of Christmas when my true love gives me a total of 78 gifts given: 12 lords a-leaping, 11 ladies dancing, 10 pipers piping, 9 drummers drumming, 8 maids a-milking, 7 swans a-swimming, 6 geese a-laying, 5 golden rings, 4 calling birds, 3 French hens, 2 turtle doves, and a partridge in a pear tree. Today I’m at home “alone” with the gifts from my true love.  Assuming each person, and each bird, has two legs what day of Christmas is it if there are now 200 legs in my home (not including mine)?

Each year Desmond and his family take part in the cultural celebration of Kwanzaa.  The observance centers around the seven principles of Kwanzaa: Umoja, Kujichagulia, Ujima, Ujamaa, Nia, Kuumba and Imani.  Suppose Desmond writes each of the five letters of the seventh principle, Imani (Swahili for Faith) on its own index card.  If Desmond places the five cards in a bag and randomly chooses three cards, what is the probability that he chooses the three letters of the fifth principle, Nia (Swahili for Purpose)? Express your answer as a common fraction.