Regions Cut by 5 Planes

for

Shanti L. Howard

How many regions or what is the maximum number spaces that can be made by using 5 planes?

 

 n

 Point by n

 Line by n POINTS

 2 Space by n LINES

 3 Space by n PLANES

 4 Space by 3 SPACE

 0

 1

 1

 1

 1

 1

 1

 1

 2

 2

 2

 2

 2

 1

 3

 4

 4

 4

 3

 1

 4

 7

 8

 8

 4

 1

 5

 11

 15?

 16

 5

 1

 6

 16

 26?

 31

 6

 1

 7

 22

 42?

 57

 7

 1

 8

 29

 64?

 99

By using a spreadsheet or just a handwritten make-up model of these 5 planes, it looks like we have the pattern:

B1 + C1 = C2 or D3 + E3 = E4. All the cells seem to act in the very same manner.

 

What can you conclude about this? Is there a good explanation or conjecture for the action that is taking place within the cells?

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