Dancing Shapes II

Probability Level 4

Imagine the same scenario as in previous problem , but now the dance floor is marked with a hexagon, as shown. The internal cells may be hexagons or triangles, but the rules of the dance are exactly the same as before: Dance in your cell until you hear the drumbeat, at which time you cross over a line into an adjacent cell. Again, what is the minimum number of dancers for whom the club will have to spring for drinks at the end of the song, if the dancers follow the instructions correctly? Can you find a general formula for any hexagon having $k$ internal lines parallel to each side? (The figure shown has four.)

18 17

1 solution

Brian Lie
May 28, 2019

For this problem also, the key is to find the right coloring. As shown, the hexagons can be colored light, and the triangles dark. Since there are 19 hexagons and 36 triangles, at the end of the song, at least 17 triangles will be empty. For this problem there was a slight twist in the tail -- the kind of twist that math teachers add in order to see which students are really attentive to detail. Since 17 is an odd number, at least one of the partners chosen by the dancers whose original cells are empty has to be someone outside this group. So the club will have to spring for drinks for at least $\boxed{18}$ dancers.

When there are $k$ internal lines ( $k$ must be even), denote $m = k/2$ ; there are $6m(m+1)$ triangles and $3m(m+1)+1$ hexagons. The minimum number of empty cells is $3m(m+1)-1$ .

Um......Excellent. Did I publish your problem before you did?

Carwaniwer Qee - 2 years ago

Dancing Shapes II

1 solution

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