One two-person envy-free division method is the moving-knife procedure. It was developed...

One two-person envy-free division method is the moving-knife procedure. It was developed by A. K. Austin and published in the Mathematical Gazette in 1982. The result of this division method gives both players portions that each values at one half the total worth of the item. To see how it works, suppose two people want to split a candy bar.

STEP 1: Player 1 holds two knives over the candy bar with one positioned at the left edge so that the portion in between the knives is exactly one half according to his or her values. If player 2 agrees that this represents exactly half, then the procedure ends. Player 1 gets the portion between the knives while player 2 gets the remaining portion.

STEP 2: If the players do not agree, player 1 will move the knives across the candy bar from left to right, keeping the portion between the knives exactly half according to his or her values, until player 2 agrees it is exactly half.

Player 1 keeps the portion between the knives, and player 2 keeps the pieces outside of the knives.

a. Explain why both players end up with portions that are fair shares according to their values.

b. Explain why the division is envy-free.

c. Player 1 begins with one knife at the left end of the candy bar. By the time the right-hand knife reaches the right side of the candy bar, where will the left knife be positioned? Explain.

d. As player 1 moves the knives over the candy bar, player 2 is watching for the point at which he or she feels exactly one half of the value is between the knives. Explain why such a point must exist.