You and a friend play “first to 100”, a game in which you start with 0, and you each take turns adding an integer between 1 and 10 to the sum. Whoever makes the sum reach 100 is the winner.

Is there a winning strategy? If so, what is it?

#### Solution

Yes, the first player should add 1, then after each of the second player’s turns, add the integer that will bring the sum to exactly 12, 23, 34, 45, 56, 67, 78, and 89. In other words, if the second player adds *x*, the first player should add 11 – *x*.

To figure out this strategy, work backwards from 100:

- Leaving the sum at 90-99 means you lose, because the other player can immediately reach 100.
- But this means leaving the sum at 89 guarantees a win, because the other play will be forced to leave the sum at 90-99 afterward.
- Using this same logic, 78 also guarantees a win, and any number of the form 11
*n*+ 1.

start at 1 -> 12 -> 23 -> 34 -> 89

You say 1 let’s just say they say 6 then what you do is that you minus 6 from 11 which makes 5 so you say five and do this until the opponent reaches 89 you win.