1. (4 pts) Test A: (x6)^2; Test B: x8.

2. (12 pts) part a) 3 pts for building matching network with edge capacities and initial flows;

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

3. (15 pts,3,3,3,7) Test A: a) 7, b) remove 1 from 2nd,c) remove 4 from pile 3 (Test A) or pile 1 (Test B), d) in the kernel- no move to kernel

4. (12 pts) 1/12[(x1)^18 + 2(x6)^3 + 2(x3)(6 + (x2)^9 + 3(x1)^2(x2)^8 + 3(x1)^4(x2)^7] , xi = (b^i + w^i);

5. (8 pts) Test A: supplies L 1, T 1, V 3: L & T each win one game in their series; V wins all the games in the other two series -- possible.

Test B: interchange L and V in the Test A solution-- possible.

6. (10 pts: 2,2,2,4): Test A: a) x11=20,x12=30,x22=20,x23=20,x33=40, b) u1=10, u2=8, u3=9, v1=14,v2=14,v3=14, c) increase x13, New solution obtained by increasing x13 and x32 by 30, reduce x12 and x33 by 30.

Test B: a) x11=30,x12=30,x22=20,x23=20,x33=30, b) u1=10, u2=9, u3=10, v1=15,v2=14,v3=15, c) increase x13, New solution obtained by increasing x13 and x32 by 30, reduce x21 and x33 by 30.

1. (4 pts) (x5)^2;

2. (12 pts) part a) 3 pts for building matching network with edge capacities and initial flows;

part b) 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). New match b-i, c-g, d-h, e-j, f-k.

3. (15 pts,3,3,3,6) 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.

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) supplies L 1, T 4, V 1: L & V play 3 games but each can only win once-- impossible.

6. (10 pts: 2,2,2,4): 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.

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) (1/6)[(x1)^9+ 2(x3)^3 + 3(x1)(x2)^4], xi = (b^i + w^i);

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

6. (10 pts: 2,2,2,4): 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.