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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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[ACTION NAME]

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# Your action step 2

# Your action step 3

...

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system
Evin and Jerry are playing a game with a pile of marbles. On each players' turn, they can remove  $2$ ,  $3$ ,  $7$ , or  $8$ marbles. If they can’t make a move, because there's  $0$  or  $1$  marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of  $n$  from  $1$  to  $1434$  does Evin lose the game?

*Proposed by Evin Liang*

<details><summary>Solution</summary>*Solution.*  $\boxed{573}$ Observe that no matter how many marbles a one of them removes, the next player can always remove marbles such
that the total number of marbles removed is  $10$ . Thus, when the number of marbles is a multiple of  $10$ , the first player loses the game. We analyse this game based on the number of marbles modulo  $10$ :

If the number of marbles is  $0$  modulo  $10$ , the first player loses the game 

If the number of marbles is  $2$ ,  $3$ ,  $7$ , or  $8$  modulo  $10$ , the first player wins the game by moving to  $0$  modulo 10 

If the number of marbles is  $5$  modulo  $10$ , the first player loses the game because every move leads to  $2$ ,  $3$ ,  $7$ , or  $8$  modulo  $10$ In summary, the first player loses if it is  $0$  mod 5, and wins if it is  $2$  or  $3$  mod  $5$ . Now we solve the remaining cases by induction. The first player loses when it is  $1$  modulo  $5$  and wins when it is  $4$  modulo  $5$ . The base case is when there is  $1$  marble, where the first player loses because there is no move. When it is  $4$  modulo  $5$ , then the first player can always remove  $3$  marbles and win by the inductive hypothesis. When it is  $1$  modulo  $5$ , every move results in  $3$  or  $4$  modulo  $5$ , which allows the other player to win by the inductive hypothesis.
Thus, Evin loses the game if n is  $0$  or  $1$  modulo  $5$ . There are  $\boxed{573}$  such values of  $n$  from  $1$  to  $1434$ .</details>

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