# Steiner’s regions of space problem

## How many regions of the plane with \(\normalsize{n}\) lines?

Clearly with \(\normalsize{0}\) lines we have \(\normalsize{1}\) region, with \(\normalsize{1}\) line we have \(\normalsize{2}\) regions, with \(\normalsize{3}\) lines we have \(\normalsize{7}\) regions. After that one has to look more carefully at the alternatives: with \(\normalsize{4}\) lines we have \(\normalsize{11}\) regions and here we see that with \(\normalsize{5}\) lines we can have \(\normalsize{16}\) regions. In general, the answer is that with \(\normalsize{n}\) lines you can create \[\Large{a(n)=\frac{n^2+n+2}{2}}\] regions. Here are some small values :$$\normalsize{n}$$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

$$\normalsize{a(n)}$$ | 1 | 2 | 4 | 7 | 11 | 16 | 22 | 29 | 37 |

Q1(M): Can you arrange \(\normalsize{6}\) lines in the plane to make \(\normalsize{22}\) regions?Q2(M): Can you see the pattern of these numbers? Could you predict the value for \(\normalsize{n=9}\) without the formula?

## The regions of space problem

$$\normalsize{n}$$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

$$\normalsize{b(n)}$$ | 1 | 2 | 4 | 8 | 15 | 26 | 42 | 64 | 93 |

Q3(C): Can you see a pattern with these numbers that would allow you to extend the table without knowing the formula? What for example is the next number in the pattern?

## Answers

A1.Here is an example:

A2.If we look at successive differences, you see that they grow by one with each step. So the difference between the 8th value and the 9th value ought to be one more than the difference between the 7th value and the 8th value, which is \(\normalsize{37-29=8}\). That means the 9th value ought to be \(\normalsize{37+9=46}\).A3.The trick is to look at the differences between successive elements in the sequence. These differences are \(\normalsize{1,2,4,7,11,16,22,29,\dots}\) Can you see a pattern here? The trick is to look at successive differences once again! These are \(\normalsize{1,2,3,4,5,6,7,\dots}\) Now the pattern is clearer. The next difference of differences must be \(\normalsize{8}\), so the next difference must be \(\normalsize{37}\), so the next number must be \(\normalsize{93+37=130}\). You can check the formula!

#### Maths for Humans: Linear, Quadratic & Inverse Relations

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