Problem of the Week 15

In the game of UNDER, two dice are thrown.  If the sum of the dots is under seven, the player receives 8 points.  If the sum of the dots is seven or greater, the player loses 6 points.  How many points can a player expect to win or lose if 100 games are played?  Write the answer to the nearest point and indicate whether it is a gain or a loss.


Alternate Problem 15

Twenty cards are chosen from a standard deck of playing cards–ten black cards, ace to ten, and ten red cards, ace to ten.  The cards are arranged in a certain order so that, after completing the steps listed below, the cards turn up in order–black, ace to ten, and then red, ace to ten.

STEP 1:  Hold the pack face down.  Make two piles of cards by alternately placing one card up and one card down.  Start with one card up.

STEP 2:  Pick up the cards that are face down.  Repeat STEP 1, placing the face-up cards on the first face-up pile.

STEP 3:  Repeat the process described in STEPS 1 and 2 until all the cards are face up.

What is the original order of the twenty cards?


Extension Problem 15

Suppose 3 and 8 are the only individual scores that can be made in a certain game.  How many different scores are impossible for a person to attain.