Question

Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are: $\bullet$ Carolyn always has the first turn. $\bullet$ Carolyn and Paul alternate turns. $\bullet$ On each of her turns, Carolyn must remove one number from the list such that this number has at least one positive divisor other than itself remaining in the list. $\bullet$ On each of his turns, Paul must remove from the list all of the positive divisors of the number that Carolyn has just removed. $\bullet$ If Carolyn cannot remove any more numbers, then Paul removes the rest of the numbers. For example, if $n=6,$ a possible sequence of moves is shown in this chart: \begin{tabular}c \hline Player & Removed \# & \# remaining \\ \hline Carolyn & 4 & 1, 2, 3, 5, 6 \\ \hline Paul & 1, 2 & 3, 5, 6 \\ \hline Carolyn & 6 & 3, 5 \\ \hline Paul & 3 & 5 \\ \hline Carolyn & None & 5 \\ \hline Paul & 5 & None \\ \hline \end{tabular} Note that Carolyn can't remove $3$ or $5$ on her second turn, and can't remove any number on her third turn. In this example, the sum of the numbers removed by Carolyn is $4+6=10$ and the sum of the numbers removed by Paul is $1+2+3+5=11.$ Suppose that $n=6$ and Carolyn removes the integer $2$ on her first turn. Determine the sum of the numbers that Carolyn removes.

Answer 1

I believe the answer is 8, but I am not sure.

Answer 2

The sum of the numbers that Carolyn removes is 5.

Explanation:

The provided instruction for the game are:

• Carolyn always has the first turn.
• Carolyn and Paul alternate turns.
• On each of her turns, Carolyn must remove one number from the list such that this number has at least one positive divisor other than itself remaining in the list.
• On each of his turns, Paul must remove from the list all of the positive divisors of the number that Carolyn has just removed.
• If Carolyn cannot remove any more numbers, then Paul removes the rest of the numbers.

The value of n is supposed as 6.

And it is also provided that Carolyn removes the integer 2 on her first turn.

The table displaying the outcomes of the game are as follows:

Player Removed Remaining

Carolyn 2 1, 3, 4, 5, 6

Paul 1 3, 4, 5, 6

Carolyn 3 4, 5, 6

Paul 6 4, 5

Carolyn None 4, 5

Paul 4, 5 None

The sum of the numbers that Carolyn removes is:

S = 2 + 3 = 5

Thus, the sum of the numbers that Carolyn removes is 5.