This quiz lasts 70 minutes and covers materials from lessons 1-9.

- Use the simplicity rule to compute the values of the following games. (10 points)
- {1/2 | }
- {-1/4 | }
- {1/8 | 3/8}
- {0 | 7/8}
- {1/8 | 9/16}
- { | }

- Compare the following games by writing
*G*<*H*,*G*=*H*,*G*>*H*or*G*||*H*. (12 points)*G*= 1 + ↑,*H*= 1/2 + 10↑*G*= ↑,*H*= *10*G*= 3↑,*H*= *10*G*= -1/2 + 3↑ + (*8),*H*= -1/2 + 3↑ + (*9)*G*= -1/2 + 3↑ + (*8),*H*= -1/2 + 4↑ + (*9)*G*= -1/2 + 3↑ + (*5),*H*= -1/2 + 4↑ + (*6)

- The game of
**Maundy’s Cutcakes**is played as follows. Take a cake, divided into*m*×*n*unit squares. At each player’s turn, he takes a piece of cake and makes any number of slices along the edges of the squares, dividing the piece into two or more pieces of*exactly the same dimension*. As before,*Left*can only cut vertically, while*Right*can only cut horizontally. A typical game may proceed as follows if*Left*starts:

It turns out this game is a number for all *m* × *n* cakes. Find the numbers for all 1 ≤ *m* ≤ 6, and 1 ≤ *n* ≤ 6. (16 points)

- Compute the values of the following Toads-and-Frogs games (some of them are given). Note that
*Left*has more than one possible move in some of the games. (18 points)

- Decide whether each of the following statements is true or false. For false statements, find a counterexample. (18 points)
- If the first player wins in
*G*, and the first player wins in*H*, then the first player wins in*G+H*. - If the first player wins in
*G*, and the second player wins in*H*, then the first player wins in*G+H*. - If the first player wins in
*G*, and*Left*wins in*H*, then*Left*wins in*G+H*. - If
*x*,*y*are any numbers and*m*= {*x*|*y*}, then*x*<*m*<*y*. - If
*G*+*G*= 2, then either*G*= 1 or*G*= 1* = 1+*. - If
*G*and*H*are games which are numbers, then we cannot have*G*||*H*. - If
*A*< 0 and*X*|| 0, then the game*G*= {*A*|*X*} equals zero. - If
*G*+*G*= 1, then*G*cannot be negative. - If
*G*+*G*= 1, then*G*cannot be fuzzy.

- If the first player wins in
- Find the values of the following six Hackenbush diagrams:

Find the value of the following Hackenbush game when the length tends to infinity (no rigourous proof required). (16 points in all)

### Note

Out of a class of >20 students, the average score was 72.4/90, with a standard deviation of 11.5. This time, I think the quiz is a little simple. 😛