AMS 303 SECOND TEST REVIEW Solutions to Practice Tests
2009 AMS 303 Second Test
1. xsub9.
2. a (-,oo), b (a+,1) c (j-,1), d (h-,1), e (i-,1), f (k-,1), g (c+,1)or(f+,1),
h (b+,1), i (b+,1), j (d+1), k (e+,1), z (g+,1),
New match: b-h, c-g, d-j, e-i, f-k or b-i, c-j, d-h, e-k, f-g.
3. a) Gr = 2, b) remove 2 from pile 3 or 4, c) already at Gr = 2, d) remove 2
from first pile.
4. (1/4)[(b+w)^8 + (b^2+w^2)^4 + 2(b+w)^2(b^2+w^2)^3]
5. Not possible, T + T can win only one more game but they play each other
3 times.
6a) Initial Northwest corner solution: x11=40, x12=30, x22=10, x32=20, x33=40;
b) u1=$10, u2=$9, u3=$11, v1=$15, v2=$14, v3=$15, c) x23 saves $2 a unit;
d) x11=40, x12=0, x13=30, x22=10, x32=50, x33=10.
2008 AMS 303 Second Test
1. xsub8.
2. a(-,oo), b(a+,1), c(i-,1), d(g-,1), e(j-,1), f(h-,1), g(b+,1), h(b+,1), i(g+,1), j(f+,1), k(c+,1) or (j+,1)
3. a) Gr # = 5, b) remove 3 from pile 2, c) remove 4 from pile 2, d) several possible moves, e.g. remove 5 from pile 3 or 4.
4. (1/10)[(b+w)^10 + 4(b^5+w^5)^2 + 5(b+w)^2(b^2+w^2)^4].
5. I. Not possible. Not enough wins available for required series.
II. Show K'={x: g(x)=1} is kernel in G-K. i) Show no successor of an x in K' can be in K'-- follows since no successor of an x with g(x)=1 can have g(x)=1;
ii) Show is x not in K', x has a successsor in K'-- x not in K' means g(x) >1 and so my definition of Gr. function, x has a successor y with g(y)=1.
2007 AMS 303 Second Test
1. (xsub3)^4.
2. 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+,1) [or (e+,1)], z (k+,1),
New match: b-h, c-g, d-i, e-k, f-j or b-g, c-j, d-k, e-h, f-i.
3. Gr fn = 3, take away 3 from pile 2, b) Gr fn = 1, many moves possible, e.g.
take 1 away from pile 1.
4. (1/8)[(b+w)^12 + 2(b^3+w^3)^4 + 3(b^2+w^2)^6 + 2(b+w)^2(b^2+w^2)^5].
5. B. Easy for Vikings to be co-champions. Also can just barely be champions.