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## Count Dracula Says 6, You Say....

You challenge Count Dracula to play a numbers game: "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 Count Dracula. "I want to win, so I will start."

What number should the Count say at the start?

Two variations to explore...Can you adjust the Count's strategy?

• Goal number is N rather than 50. (Why is N = 100 special?...any problems occur when N > 100?)
• First person says a number between 1 and M...and other person says a number that is at least 1 higher than that number, and at most M higher....with goal being to say the number N

Source: Adapted from a problem found on-line, website unknown

Hint: Play the game multiple times with someone. Watch what they do, and how someone wins.

Now, try to think backwards through the various moves made...how could they have been improved.

Solution Commentary: The suggested strategy for the given version of the game: Count Dracula should begin with 6.

Then, in sequence he will say the numbers 6, 17, 28, 39, 50.

Since the Count wants to say 50, he needs you to say a number 40-49, inclusive. Therefore, he needs to say 39 to force your hand.

Then, to say 39, the Count needs you to say a number 29-38, inclusive; so, he will say 28. Following this same backwards reasoning, the Count wants to say 17, and thus must say 6 to start the game...and victory is his.

Now, how can you adjust this strategy for the two suggested variations?