How do you divide a square into eighths four different ways?

Answer 1

1) Mark every 1/8 along opposite sides and join corresponding marks to split the square up into 8 rectangles of sides square length and 1/8 square length;

2) Joint the centre points of opposite sides and draw in the two diagonals to split the square up into 8 isosceles right angles triangles with the two shorter sides half the square length;

3) (variation on 2) Join the centre of the square to the centre of each side and each corner with a curved [spiral] line (all going in the same direction);

4) (variation on 1) Join the centre of one pair of opposite sides; mark every 1/4 along the other pair of opposite sides and join them to split the square up into 8 rectangles of sides half square length and quarter square length.

Answer 2

• Select any side of the square.
  
• Bisect the line and select one of the half-lines.
  

• Select any one point in this line-segment.
  
• Draw a straight line through this point and the centre of the square.
• Rotate the line through 45%, 90% and 135%.

These four lines will divide the square into eighths.

Since there are infinitely many points in the half-line you have not only 4 but infinitely many ways. And the manner of selecting the first point ensures that each point that is selected will lead to a different way.