Puzzle for Settle District U3A Mathematical Puzzles
From the Guardian Saturday 20 March 2010
by Chris Maslanka
Cut each of two identical regular octagons into 4 pieces in exactly the
same way so that the whole can be assembled to make one big octagon.
The new octogon will have a side
√2
times that of the smaller octogon.
One of the two original octogons is shown in green.
This is cut along the solid mauve lines,
to produce two triangles and two irregular hexagons.
The second octogon is cut similarly (as requried by the question).
The four hexagons are arranged as shown in the lower part of the diagram,
and the square in the centre is exactly filled with the four triangular pieces.
Food for thought
The subsequent week’s issue had a puzzle asking that three
regular hexagons be cut to produce a single hexagon.
This can be achieved by putting the three together and cutting
the outer corners.
The resulting triangular pieces cut off just make up
the larger hexagon.
Furthermore, if one takes two squares, and cuts each on a
diagonal, the resulting four pieces can be arranged into a larger square.
Returning to the octogon question, one can take four regular octogons,
and combine them to make two octogons, which can then be combined to make
a single octogon, equal in are to the sum of the original 4.
So we have a progression:
2 regular 4-sided polygons can be cut to form a jigsaw which makes a single regular 4-sided polygon.
3 regular 6-sided polygons can be cut to form a jigsaw which makes a single regular 6-sided polygon.
4 regular 8-sided polygons can be cut to form a jigsaw which makes a single regular 8-sided polygon.
Questions:
Can 5 regular 10-sided polygons can be cut to form a jigsaw which makes a single regular 10-sided polygon?
Can n regular 2n-sided polygons can be cut to form a jigsaw which makes a single regular 2n-sided polygon?
The new octogon will have a side
√2
times that of the smaller octogon.