Using dot (·) patterns, which number can be arranged as a line, a triangle and a rectangle?
9
10
11
12
Idea: We need one number that can make all three shapes with dots: a line, a triangle, and a rectangle.
1) Line: Any number of dots can form a line (just put all dots in one row). So the line is always possible.
2) Triangle (triangular number): Triangular numbers have the form
\(T_k = \dfrac{k(k+1)}{2}\)
Now list small values:
\(T_1 = \dfrac{1\cdot2}{2} = 1\)
\(T_2 = \dfrac{2\cdot3}{2} = 3\)
\(T_3 = \dfrac{3\cdot4}{2} = 6\)
\(T_4 = \dfrac{4\cdot5}{2} = 10\)
So, \(10\) is a triangular number.
3) Rectangle: To make a rectangle with dots, the number must factor into two whole numbers, both greater than 1 (rows × columns).
For \(10\): \(10 = 2 \times 5\) → we can make a \(2\times5\) rectangle.
Check each option quickly:
• 9: Not triangular (\(T_3=6\), \(T_4=10\)), rectangle yes (\(3\times3\)), but triangle fails.
• 10: Triangular (\(T_4=10\)), rectangle (\(2\times5\)), line always → ✓ works.
• 11: Not triangular (between 10 and 15), rectangle no (only \(1\times11\), not a proper rectangle), line yes → fails.
• 12: Not triangular (between 10 and 15), rectangle yes (\(3\times4\)), line yes → triangle fails.
Answer: 10 (option B).