Ninety-nine non-negative numbers

Question

Alice and Bob play the following game with $99$ numbers $M_1,\ldots,M_{99}$.

• In the beginning, the $99$ numbers take the values $1,2,4,8,16,\ldots,2^{97},2^{98}$.
• Then Alice and Bob alternatingly make moves; Alice makes the first move.
• In every move, the active player picks five numbers $M_a,M_b,M_c,M_d,M_e$ (with five distinct indices $a,b,c,d,e$) and decreases each of them by $1$.
• If one of the numbers becomes negative, the game ends. The player who has caused this loses, and the other player is declared winner.

Question: Which player has a winning strategy? (As usual, we assume that Alice and Bob use optimal strategies.)

Answer 1

TL;DR

The first player should decrease $M_1$, $M_{96}$, $M_{97}$, $M_{98}$, $M_{99}$ in the first move. Afterwards he can repeat all moves of the second player to win.

This is guaranteed to work because the numbers $M_1 .. M_{95}$ are all even after the first move, and each of the numbers $M_{96} .. M_{99}$ is bigger than the sum of all numbers $M_1 .. M_{95}$. At the end all numbers $M_1 .. M_{95}$ will be $0$, and the numbers $M_{96} .. M_{99}$ will be positive with the second player to move.

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The rest of the post describes how I came to the conclusion above.

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Closer look

Let's make two groups of numbers $M_1 .. M_{95}$ and $M_{96} .. M_{99}$:

$\begin{array}{} M_1 & M_2 & M_3 & ... & M_{95} & \qquad\qquad & M_{96} & M_{97} & M_{98} & M_{99} \\ 1 & 2 & 4 & ... & 2^{94} & \qquad\qquad & 2^{95} & 2^{96} & 2^{97} & 2^{98} \end{array}$

The sum of the numbers $M_1 .. M_{95}$ is $2^{95}-1$ which is smaller than $M_{96}=2^{95}$. Even if we would decrease all numbers $M_{96} .. M_{99}$ in each move, they will stay positive until the end of the game.

Changing the rules

This allows us to create an equivalent game with slightly different rules. First we get rid of the numbers $M_{96} .. M_{99}$. Second we change the move rule to the following:

- In every move, the active player picks between one and five numbers and decreases each of them by $1$.

This game is equivalent to the original game, because we can decrease the missing one to four numbers from the group $M_{96} .. M_{99}$ without changing the outcome of the game.

Winning position

Let us assume we have a position where all numbers are even numbers. In such a position it is always possible to repeat the move done by the previous player reaching again a position where all numbers are even numbers. At some point the first player will run out of possible moves and the second player wins.

How to win?

The first player can decrease the number $M_1=1$ (and $M_{96} .. M_{99}$ in the original game) and reach the winning position from the previous chapter (he will be then the second player).

Answer 2

Second (but completed) answer, once again building of Sleafar's answer.

Alice wins

How?

If we disregard M96-M97-M98-M99, and therefor consider each turn as removing 'one to five' units from the remainder, eventually said remainder will run out. If someone gets to a point where the remainder is dividible by six, they just let the other player pick 1 to 5 from the 'remainder', and on their pick, they pick 6 - (whatever the other did), so after each two turns, the net value of the remainder goes down by 6, therefor always remaining dividable by 6.

In the limit case, all M1 to M95 will be depleted after a such move, meaning the other player only gets to pick 4 from M96-M99, and needs to turn one negative.

Now can Alice force the wining postion on the first try?

YES. The only way she couldn't, is if the initial amount were dividable by 6. Given it's an odd number (due to M1), she can always force the win.