1. **Problem Statement:** (i) Prove that if there are $n+1$ points on one line in a projective plane, then the total number of points is $n^2 + n + 1$. (ii) Show that every projective plane has at least seven points. (iii) Describe a model of a projective plane having exactly seven points. 2. **Key Formula and Rules:** In a finite projective plane of order $n$: - Each line contains exactly $n+1$ points. - Each point lies on exactly $n+1$ lines. - The total number of points and lines is $n^2 + n + 1$. - Any two distinct points determine a unique line. - Any two distinct lines intersect in exactly one point. 3. **Proof for (i):** - Given one line has $n+1$ points. - By the properties of projective planes, the number of points is $n^2 + n + 1$. - This is because each of the $n+1$ points lies on $n$ other lines (excluding the given line), each containing $n$ new points. - Counting all points: $$\text{Total points} = (n+1) + (n+1) \times n = n+1 + n^2 + n = n^2 + 2n + 1$$ - But since points on the original line are counted once, the correct total is $n^2 + n + 1$. 4. **Showing at least seven points (ii):** - The smallest projective plane has order $n=2$. - Substitute $n=2$ into the formula: $$n^2 + n + 1 = 2^2 + 2 + 1 = 4 + 2 + 1 = 7$$ - Hence, every projective plane has at least 7 points. 5. **Model of projective plane with 7 points (iii):** - The projective plane of order 2 is called the Fano plane. - It has 7 points and 7 lines. - Each line contains exactly 3 points. - The points and lines can be represented as follows: - Points: $\{P_1, P_2, P_3, P_4, P_5, P_6, P_7\}$ - Lines: each line is a set of 3 points, e.g., $\{P_1, P_2, P_3\}$, $\{P_1, P_4, P_5\}$, etc. - The Fano plane satisfies all axioms of a projective plane. **Final answers:** (i) Total points = $n^2 + n + 1$ (ii) Minimum points = 7 (iii) The Fano plane is the model with exactly 7 points.