How many objects measuring 43mm by 49mm can fit into an object measuring 457mm by 330mm?

Answer 1

I assume that the smaller objects must not overlap, for otherwise, infiniely many of them will fit.

It would be incorrect to calculate the area of the larger object and divide that by the area of the small objects. This is because these objects may not fir end to end.

For example, you can get only one 2mm*2mm object into a 3mmx3mm object even though the larger area is 9 mm2, more than twice as big.

This question is, I think, called the knapsack problem which does not have a simple solution.

You can start looking at how many of the smaller objects will fit in, using the two possible orientations:

Thus 457/43 = 10.63

and 330/49 = 6.73

so in that orientation, you can get 10*6 = 60 smaller objects.

Alternatively,

457/49 = 9.33

and 330/43 = 7.67

so in this orientation you can get 9*7 = 63 smaller objects.

So far 63 is the largest number.

In fact, it will be possible to increase the number from 63 by changing the orientation of some of the smaller objects.