**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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