Friday, August 10, 2018

mathematics - Red cells in a 4x4 table


We say that two cells in a 4×4 table are interconnected, if they are in the same row or in the same column. No cell is interconnected with itself. There are r cells in the table that are colored red, and each of the 16 cells in the table is interconnected with at least two red cells.



Determine the smallest possible value for the number r of cells in the table.





Answer




I think r=6

If X is a red cell and 0 a white cell

X000X000X0000XXX
Number of interconnections for each cell :
2222222222226222



This is the minimum because



We need 32 interconnections (2 per cell for 16 cells)

1 red cell create 6 interconnections (3 in the line and 3 in the column)
The first multiple of 6 bigger than 32 is 36=6*6 so we need at least 6 red cells



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