Folding Polygons
Fold corner after corner across the line between its neighbours, and collect every shape the paper can end up as.
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- Grades
- 2–6
- Subjects
- Shapes & Space, Logic
- Kind
- Puzzle
What it is
A regular polygon of paper, coloured on top and white underneath. Tap a corner and it folds in over the straight line through the two corners beside it, landing white side up inside the shape. A fold that would land on the edge of the shape, or past it, is refused, and a corner that sticks out again may fold again. Keep folding until no corner can fold: that is a finished shape. In First Folds the task is to find every finished shape the polygon has, a shape turned round or flipped over counting as the same one. In Mirrors and Turns it is only the shapes with a mirror line, or a turn that brings them back onto themselves. In Smallest and Biggest it is the finished shape that covers the least of the polygon, and the one that covers the most, and there a fold may land right on a corner or an edge, as long as nothing crosses. A switch beside the Done button shows every fold as a coloured rhombus laid on the polygon instead of as folded paper.
The rules
What it gives you to practise
A fold is a reflection: the corner lands as far on the other side of the crease as it was before, and seeing where it will land before tapping is the first thing the board teaches. On a hexagon every corner folds exactly onto the centre, so one fold is all there is. Two finished shapes are the same if one can be turned or flipped onto the other, so deciding whether a shape is new is a question about rotations and reflections, and the board answers it by turning or flipping the shape until it lies on the one already found. Then the search: the octagon has six different finished shapes and the nonagon forty-seven, and being sure none is missing means keeping track of which orders of folds have been tried. Smallest and Biggest asks about area instead, as a percentage of the whole polygon. The piece a fold takes away is always a rhombus, all four sides as long as the polygon’s, so folding is tiling: the octagon, the decagon and the 12-sided polygon can be covered by rhombi completely, down to 0 per cent, while a polygon with an odd number of corners always keeps some of its paper.
In a classroom
The fold is a real one, so the puzzle works on paper too: cut a regular hexagon or octagon out of coloured paper and fold a corner in so that the crease runs straight from one neighbouring corner to the other. A class can share out the hunt for the octagon’s six shapes, check whether two are the same by turning and flipping the paper, and pin up one of each. The polygons are named by their number of corners first, 5 Pentagon, 6 Hexagon and on to 12-sided, so nobody needs the long names to begin. Printed, a board is its polygon in outline with the corners marked, which can be cut out and folded for real.
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