SOLUTION: What is the maximum number of bottles, each of diameter 8 cm, that can be packed into a box with a square base measuring 632 cm by 632 cm? (Using the hexagonal packaging pattern)

Usual "square cells" paking scheme gives the answer   =  = 6241 bottles.



"Equilateral triangle cells" packing scheme has the distance (the shift) between the lines of centers of every two adjacent rows of bottles  

    h =  = 6.9282 cm;



Hence, the number of such lines of centers is

     =  =  = 90.06668399 (rounded to integer number 90).



It means that we have the sequence of numbers of bottles in paired rows as

    (78 + 79) + (78 + 79) + . . . + (78 + 79).      (1)


Each sum in parentheses in (1) is 78+79 = 157 bottles and the number of such groups of 157 bottles in (1) is 90/2 = 45.


Hence, the total number of bottles at such packing is  157*45 = 7065 bottles.    ANSWER


For completeness, compare the two numbers for two packing schemes.