1. (4 pts) Test A: (x5)^2; Test B: x9.

2. (15 pts,3,3,3,6) Test A: a) 3, b) remove 3 from 1st, 3rd or 4th pile, c) remove 2 from one of those piles, d) remove 4 from pile 2 (moving from Grundy value 2 to Grundy value 0.

Test B: a) 5, b) remove 5 from 3rd pile, c) remove 4 from pile 2, 3 or 4, d) remove 5 from pile 2 or 3 (moving from Grundy value 3 to Grundy value 0.

3. (12 pts) part a) 3 pts for building matching network with edge capacities and initial flows;

part b)9 pts: a (-,oo), b(g-,1), c(h-,1), d(a+,1), e(j-,1), f(k-,1), g(c+,1) h(d +,1), i(b+ or e+,1), j(f+,1), k(d+,1), z(i+,1). One new solution using b-i is b-i, c-g, d-h, e-j, f-k,

4. (12 pts) (1/8)[(x1)^12+ 2(x4)^3 + 3(x2)^6 +2(x1)^2(x2)^5], xi = (b^i + w^i);

5. (8 pts) Test A: supplies L 1, T 4, V 1: L & V play 3 games but each can only win once-- impossible.

Test B: supplies L 2, T 1, V 5, L & T play 4 games but together have only 3 wins-- impossible.

6. (10 pts: 2,2,2,4): Test A: a) x11=20,x21=10,x22=20,x23=10,x33=30, b) u1=10, u2=8, u3=10, v1=16,v2=17,v3=15, c) increase x12, New solution obtained by increasing x12 and x21 by 20, reduce x11 and x22 by 20.

Test B: a) x11=20,x22=10,x22=10,x23=10,x33=30, b) u1=10, u2=11, u3=13, v1=14,v2=17,v3=18, c) increase x13, New solution obtained by increasing x13 and x32 by 10, reduce x21 and x33 by 10.

1. (4 pts) (x4)^2.

2. (15 pts,3,3,3,6) a) 3, b) remove 3 from 3rd pile, b) could use pile 2 or 3 or 4- remove 1, d) Gr = 2, remove 4 from pile 4.

3. (12 pts) a (-,oo), b(g-,1), c(h-,1), d(a+,1), e(j-,1), f(k-,1), g(c+,1) h(d +,1), i(b+ or f+,1), j(d+,1), k(e+,1), z(i+,1). One new solution using b-i is b-i,

4. (12 pts) (1/8)[(x1)^12+ 2(x4)^3 + 3(x2)^6 + 2(x1)^2(x2)^5], xi = (b^i + w^i);

5. (8 pts) Test A: supplies L 1, T 5, V 1: L and V must play 3 times but each can win only 1 game-- impossible:

Test B: supplies L 2. T 1, V 5: L and T must play 4 times but L has only 2 wins and T 1 win - impossible.

6. (10 pts: 2,2,2,4):Test A: a) x11=20,x21=10,x22=20,x23=10,x33=30, b) u1=10, u2=8, u3=10, v1=16,v2=17,v3=16, c) increase x12,(new solution obtained by increasing x12 and x21 by 20, reduce x11 and x22 by 20.

Test B: a) x11=10,x12=10,x22=q0,x23=10,x33=30, b) u1=10, u2=11,u3=13, v1=14,v2=17,v3=18, c) increase x13, new solution obtained by increasing x13 and x32 by 20, reduce x21 and x33 by 20.

1.(4 pts) : (x sub 8).

2. (12 pts) a (-,oo), b(g-,1), c(i-,1), d(j-,1), e(k-,1), f(a+,1), g(d+,1) h(b+ or c+,1), i(e+,1), j(f+,1), k(f+,1), z(h+,1). One new solution using b-h is b-h, c-i, d-g, e-k, f-j.

3. (15 pts) a) initial digital sum is 011 = 3; b) remove 3 from pile 3; c) remo ve 1 from pile 2 or 3 or 4; d) g(0)=0, g(1)=1, g(2)=0, g(3)=1, g(4)=2, g(5)=3, g(6)=2, g(7) = 0, sum is 2, remove 4 from pile 4.

4. (12 pts) (1/4)[(b+w)^11 + 2(b+w)(b^2+w^2)^5 + (b+w)^3(b^2 + w^2)^4].

5. (8 pts) Possible. Number of available wins (L 2, T 2, V 4). Solution has L beat T twice, T beat L twice, and V beat L and T twice each.

6. (10 pts) Initial solution: x11=20, x21=10, x22=20, x23=10, x33=30. Prices u1=$10, u2=$8, u3=$10,v1=$14, v2=$15, v3=$13. Add x12: increase x12, x21 by 20, decrease x11 and x22 by 20.