Counting Figures

Count each complete straight path once, even if it is drawn as many pieces. Two pieces on the same path, touching end to end, make one line. Two pieces with a gap between them make two lines.

Think of a road crossing ten junctions. It is still one road. Only a new direction, or a gap, starts a new line.

Count one direction at a time. This makes double counting impossible.

1. Count horizontal lines, from top to bottom.
2. Count vertical lines, from left to right.
3. Count each slant direction separately: down-to-right first, then down-to-left.
4. Add the groups.

A square with both diagonals and both midlines: 3 horizontal + 3 vertical + 2 slanting = 8 lines.

A grid of m columns and n rows has m + 1 vertical lines and n + 1 horizontal lines:

A 3 × 2 grid needs $4+3=7$ lines. A 5 × 4 grid needs $6+5=11$ lines.

If the long diagonals of the whole grid are also drawn, add them as extra slanting lines.

The classic trap: a grid where every small cell has its own diagonals. Pieces sitting on one path join into a single line.

Take a 2 × 2 grid with both diagonals drawn in all 4 cells. That is 6 grid lines plus 8 cell-diagonal pieces. On the main path, two cell diagonals touch end to end and join into one long line. The same happens on the other path. So the 8 pieces form only 6 lines: 4 short ones plus 2 long ones.

Total: $6+6=12$ lines, not $6+8=14$.

Star figures are built from long lines that cross each other.

• A five-point star (pentagram) is drawn with 5 lines.
• A six-point star made of two triangles has 6 lines.
• A hexagon with its 3 long diagonals has $6+3=9$ lines.
• A triangle cut into rows has 3 directions, each with one line per row: 3 rows give $3\times3=9$ lines.

A straight line has no bend anywhere. Circles, arcs and wavy strokes are not straight lines, so skip them. If the question asks for one direction only, count just that direction and ignore everything else.