Project #50525 - Creative Problem Solving

I NEED FULL SOLUTIONS (step-by-step) FOR THESE PROBLEMS. Solutions cannot include solutions that you find in the internet. 

 

1. Two players write positive integers not exceeding a given number N on a board. The person forced to write a number which is the divisor of a previously recorded number loses. Which player has a winning strategy?

(Hint: Think about using 1.)

 

2. I ask you to shuffle a deck of card thorougly. Then I ask for them back (face down). Carefully examining the backs of the cards, I separate them into two piles. I then claim that, through the power of magic, I have made sure that the nunber of black cards in the first pile is the number of red cards in the second pile. 

 

3. There are six plates next to each other in a row on a table. Bob grabs two plates and moves each plate one position to the left or to the right (if there is one plate or more at the target position then he puts the plate on the top). Repeating this procedure, can he stack the plates into one single pile?

 

4. There are three grasshoppers in a row. They jump over each other: at every second one of the grasshoppers jumps over one of her neighbors (they still stay on the same straight line). Can they be back in their original positions after 1999 jumps?

Subject Mathematics
Due By (Pacific Time) 12/06/2014 07:00 pm
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