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April Junior Varsity Problem 1

 
 
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April Junior Varsity Problem 1
by Daniel Xu - Wednesday, 10 April 2019, 10:37 PM
 




16 people are seated evenly around a circular table. To choose teams for a game, 8 (identical) white cards and 8 (identical) black cards are distributed among the people. If each person gets the same color as the person seated directly across from them, how many was can the cards be distributed?

Your answer for this problem was 70, because you assumed that for any given half, there must be 4 white cards and 4 black cards. However, this is false, for example, if one half of the table has all black cards, and the other half has all white cards. Since you can deal one of two colors to each person in half of the table, and the other half of the table is determined, then the answer is 2^8=256. Because algebra, there is 8 of each type of card dealt.

Thanks,

Daniel

 
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Re: April Junior Varsity Problem 1
by John Lensmire - Thursday, 11 April 2019, 10:51 AM
 

Thanks for the question Daniel.

I might be misreading your question, but I think you're missing the "each person gets the same color as the person seated directly across from them". Because of this, any way we cut the circle in half, the set of people on one side are always directly across from the set of people on the other side, so both halves must have $4$ white and $4$ black cards. In your example of all black cards for one half, then all the cards on the other half would also need to be black, which contradicts the $8$ of each color portion of the problem.

Let us know if you have any other questions. Thanks!