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Sweet Sixteen (Posted on 2005-06-12) Difficulty: 3 of 5
16 checkers are placed on an 8 by 8 checkerboard, no two checkers on the same square. Show that some four of the 16 checkers are on the vertices of a parallelogram with positive area.

  Submitted by McWorter    
Rating: 4.0000 (1 votes)
Solution: (Hide)
Let a_i, i=1,...,8, be the number of checkers in the i-th column of the checkerboard. If there are no checkers in the i-th column, set a_i=1 (so that for such i, a_i-1 contributes nothing to the sum below). There are a_i-1 different distances between the highest checker in the i-th column and the other checkers in that column. Hence there are

a_1-1+a_2-1+...+a_8-1=16-8=8

such distances, not necessarily all distinct. However, those distances must be in the range 1 square up to 7 squares. Hence the top checker, A, in one column must be the same distance from another checker, B, in that column as the top checker, C, in some other column is from another checker, D, in that column. But then the checkers A, B, C, and D form the vertices of a parallelogram with vertical sides AB and CD and parallel, but not vertical, sides AC and BD. This parallelogram has area, in squares, the distance between A and B times the distance between the two columns containing A and C.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: oopsMcWorter2005-07-31 22:37:28
oopsTan Kiat Chuan2005-07-31 19:17:36
re: I think this is a solution...McWorter2005-07-31 18:15:11
SolutionI think this is a solution...Tan Kiat Chuan2005-07-31 16:28:40
Some Thoughtsre: How about this?McWorter2005-07-24 22:43:02
Some ThoughtsHow about this?Liz2005-07-19 03:48:15
Solution idea, but not there yet.Larry2005-07-13 02:38:42
I'm not Jonathan Chang --- Really!McWorter2005-06-21 17:10:06
15-piece counter-exampleTristan2005-06-12 19:36:20
re: Direct me to a WebsiteMcWorter2005-06-12 18:48:20
QuestionDirect me to a WebsiteCharley2005-06-12 18:05:57
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