### Brain Teasers

# Counting to 50

Henry and Gretchen are going to play a game. Henry explains, "You and I will take turns saying numbers. The first person will say a number between 1 and 10. Then the other person will say a number that is at least 1 higher than that number, and at most 10 higher. We will keep going back and forth in this way until one of us says the number 50. That person wins. I'll start."

"Not so fast!" says Gretchen. "I want to win, so I will start."

What number should Gretchen say to start?

"Not so fast!" says Gretchen. "I want to win, so I will start."

What number should Gretchen say to start?

### Hint

By following this strategy, she is assured a victory. Think of all of the numbers she will want to say.### Answer

She wants to say, "6."The series of numbers she will say is 6, 17, 28, 39, 50.

Since she wants to say 50, she needs Henry to say a number between 40 and 49. Therefore, she wants to say 39.

Knowing she wants to say 39, she needs Henry to say a number between 29 and 38. So she wants to say 28. Following this same logic recursively, she will want to say 17, and she will want to say 6 to start the game, and be assured victory.

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