In some games, the winner is not the one who rushes fastest toward the goal, but the one who knows from which squares the goal is already controlled. The key is not to start from the beginning, but to work backward from the end.
The Game of Reaching 100
Riddle statement
Two players say numbers aloud.
The first player says a number from 1 to 10.
Then, taking turns, each player adds a number from 1 to 10 to the current total.
The player who says exactly:
wins.
Does either player have a winning strategy?
Show solution
Solution
Yes: the first player can win.
The key is to reach 100 by first leaving your opponent in controlled positions.
If you want to say 100 on your final turn, you would like your opponent to leave you a number between 90 and 99. Then you can add exactly what is needed to reach 100.
Therefore, the truly important number before 100 is:
If you say 89, your opponent must add between 1 and 10. They will leave you a number between 90 and 99, and you can complete it to 100.
Now apply the same idea backward.
To be able to say 89, you want to have said:
earlier.
Because if you say 78, your opponent adds between 1 and 10, and you can add what is needed to reach 89.
The key numbers are:
Each one is 11 away from the next.
The first player's strategy is to begin by saying:
Then, whenever the opponent adds a number, you add whatever is needed so that the two additions together make 11.
For example:
if the opponent adds 4, you add 7; if the opponent adds 10, you add 1; if the opponent adds 1, you add 10.
In this way you always say the key numbers:
Answer: yes. The first player wins by starting with 1 and always completing pairs that add up to 11.