the optimal score, in a form that can be followed clearly. For example, this table is given:

The first number chosen, 7, is justified on the basis that it is the largest prime on the list, or on a basis that amounts to a complete analysis of the game (e.g., "I chose the 7 first because if I chose the 10, then …; and if I chose the 9, then …; and if I chose the 8, then .…").

No better game is possible because "You" has captured the five largest numbers on the list, while giving the Taxman the five smallest numbers.

For the game of Taxman-95, 89 is chosen as the first move. The justification asserts that if any prime is to be chosen at any point in the game, it must be on the first turn. Since 89 is the largest prime number in the list, it's the best first move.

A somewhat less advanced response is simply that 89 is the largest number on the list that has only one factor on the list.

An even less satisfactory response is to declare that the largest prime is the optimal opening move, but then to misidentify 91, 93, or 95 as a prime.

Characteristics of the medium response:

A winning game is described, although it need not be an optimal one.

The first number chosen is justified simply on the basis that it works out to be a winning first move.

A correct answer, with some justification, is given to Question 3. (Of course, this response will have



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