Number of small boxes in a large box: Easy way of solving

Align in this pattern:

Bigger box : 35 : 19 : 18
smaller box: 5 : 3 : 2
counts: 7: 6 : 9 ( multipl 7*6*9 = 378)

IMO. this could be shortest.

hi
can u pls explain more
the alignment

Bigger box : 35 : 19 : 18
smaller box: 5 : 3 : 2

can be done in 6 ways ... how to decide/judge that he alignment taken above will give " maximum number of rectangular boxes" ????

[quote="lav"]hi
can u pls explain more
the alignment

Bigger box : 35 : 19 : 18
smaller box: 5 : 3 : 2

can be done in 6 ways ... how to decide/judge that he alignment taken above will give " maximum number of rectangular boxes" ????[/quote]

I actually computed for all 6 combinations and found that 378 is the max number of boxes by ignoring the decimals and not rounding off. It took some time to do the calculations but looks like there is no easier way (any thoughts guys!)

You get 378 for 2 combinations, one is as stated above and other one is as below:

Bigger box : 35 : 19 : 18
smaller box: 5 : 2 : 3 (7 : 9 : 6)

lav wrote:hi
can u pls explain more
the alignment

Bigger box : 35 : 19 : 18
smaller box: 5 : 3 : 2

can be done in 6 ways ... how to decide/judge that he alignment taken above will give " maximum number of rectangular boxes" ????

To maximize the number of boxes, we want to minimize the wasted space in the big container. Accordingly, the closer a fit we can get, the better.

Arraging the sides as suggested, we have two exact fits (35:5 and 18:2) and one side that's off by 1 (19:3... 18:3 would be a perfect fit).

If we flip the last two and arrange as follows:

35 : 19 : 18
5 : 2 : 3

We also minimize the wasted space.

In the first example our # of boxes would be 7*6*9; in the second our # of boxes would be 7*9*6. Of course, both give us the same answer to the question.