# Problem of the Week Problem C Fair Game?

For a school mathematics project, Zesiro and Magomu created a game that uses two special decks of six cards each. The cards in one deck are labelled with the even numbers $$2$$, $$4$$, $$6$$, $$8$$, $$10$$, and $$12$$, and the cards in the other deck are labelled with the odd numbers $$1$$, $$3$$, $$5$$, $$7$$, $$9$$, and $$11$$.

A turn consists of Zesiro randomly choosing a card from the deck with even-numbered labels and Magumo randomly choosing a card from the deck with odd-numbered labels. These two cards make a pair of cards. After a pair of cards is chosen, they perform the following steps.

1. They determine the sum, $$S$$, of the numbers on the cards. For example, if Zesiro chooses the card labelled with a $$6$$ and Magumo chooses the card labelled with a $$3$$, then $$S=6+3 = 9$$.

2. Using $$S$$, they determine, $$D$$, the digit sum. If $$S$$ is a single digit number, then $$D$$ is equal to $$S$$. If $$S$$ is a two-digit number, then $$D$$ is the sum of the two digits of $$S$$. For example, if Zesiro chooses the card labelled with a $$6$$ and Magumo chooses the card labelled with a $$3$$, then the sum and the digit sum are both $$9$$. If Zesiro chooses the card labelled with a $$10$$ and Magumo chooses the card labelled with a $$5$$, then the sum is $$S = 10 + 5 = 15$$ and the digit sum is $$D = 1+5=6$$. If Zesiro chooses the card labelled with a $$10$$ and Magumo chooses the card labelled with a $$9$$, then the sum is $$S = 10 + 9 = 19$$ and the digit sum is $$D = 1+9=10$$.

Zesiro gets a point if the digit sum, $$D$$, is a multiple of $$4$$.

Magomu gets a point if the number on one of the cards is a multiple of the number on the other card.

Is this game fair? That is, do Zesiro and Magomu have the same probability of getting a point on any turn? Justify your answer.

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