Counting Figures

What the question asks

"How many straight lines are there in the figure?" or "What is the least number of straight lines needed to draw it?" Both mean the same: count each complete straight line once, even if it passes through many shapes or is drawn as several touching pieces.

Think of a road: it may cross ten junctions, but it is still one road. Two pieces lying on the same path and touching end to end are one line. Two pieces on the same path with a gap between them are two lines.

The direction sweep

1. Count all horizontal lines (left to right), from top to bottom.
2. Count all vertical lines (top to bottom), from left to right.
3. Count slanting lines of each direction separately: first those going down to the right, then those going down to the left.
4. Add the groups.

Doing one direction at a time stops you from counting a line twice.

Example: a square with both diagonals and both midlines. Horizontal 3 (top side, midline, bottom side), vertical 3, slanting 2. Total = 8.

Grids

A grid of m columns and n rows has m + 1 vertical and n + 1 horizontal lines, so $L = (m+1) + (n+1)$ A 3 × 2 grid needs 4 + 3 = 7 lines.

If both long diagonals of the whole grid are also drawn, add 2. If every small cell has its own diagonals, cell diagonals that meet end to end on one path join into a single long line, so count paths, not pieces.

Stars and polygons

• A five-point star (pentagram) is drawn with only 5 lines.
• A six-point star made of two triangles has 6 lines.
• A regular hexagon with its 3 long diagonals has 6 + 3 = 9 lines.
• A triangle split into rows of small triangles: 3 directions, and each direction has as many lines as there are rows. So 3 rows give 3 × 3 = 9 lines.

Worked example

A 2 × 2 grid where every one of the 4 small cells has both its diagonals drawn.

• Horizontal: 3. Vertical: 3.
• Down to the right: the two cell diagonals on the main diagonal join into one long line, and there is one shorter line on each side of it, so 3.
• Down to the left: by the same logic, 3.
• Total = 3 + 3 + 3 + 3 = 12, not 6 + 8 = 14. The 8 cell diagonals are only 6 lines because 2 pairs join end to end.

Special cases

• A curve is not a straight line. Circles and arcs are ignored.
• Lines that only look close are different unless they are exactly in line.
• In "how many vertical lines" questions, count only that direction.

Quick revision

• One straight path = one line, however many pieces it is drawn in.
• Sweep: horizontal, vertical, then each slant direction.
• m × n grid: (m + 1) + (n + 1) lines.
• Square + diagonals + midlines = 8 lines.
• Pentagram 5, six-point star 6, hexagon with long diagonals 9.