You make a 9x9x9 cube by gluing together 9³=729 unit cubes. You machine a cone (base diameter= height= 9 units) out of the cube. How many unit cubes will stay undamaged inside the cone?

There are 9 layers of cubes.  Each cube will remain undamaged if the top of it is entirely within the circular section of the cone.  The 9 sections of interest have radii 4, 3.5, 3, 2.5, 2, 1.5, 1, .5, 0.

Consider circles of these radii centered at the middle of the center square.  The distances from outer corner of each square to the center will determine which squares cubes fit in each layer.

The center square has distance sqrt(.5^2 + .5^2) = .71 so it will fit in 7 of the layers.

Calling each square by its outermost corner I will summarize in a chart:

Point        Distance     Layers    Multiplicity    Total
(0.5,0.5)  0.71            7            1                 7
(1.5,0.5)  1.58            5            4               20
(2.5,0.5)  2.55            3            4               12
(3.5,0.5)  3.54            1            4                 4
(1.5,1.5)  2.12            4            4               16
(2.5,1.5)  2.92            3            8               24
(3.5,1.5)  3.81            1            8                 8
(2.5,2.5)  3.54            1            4                 4

For a grand total of 95.

I believe my method is sound, but I'm not sure I didn't miscount somewhere.

Posted by Jer on 2005-03-17 19:18:02