Map folding
Problem
Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles.
Label each quarter with same letter on both sides of the paper.
How many different ways can you fold it up?
Keep track of the ways by writing down the orders of letters from the top rectangle through to the bottom rectangle.
Now try it with a set of six rectangles!
Getting Started
Have you got a system for making sure you find all the ways?
Student Solutions
Emily (Tattingstone Primary UK) is the champion map folder. She explains:
I worked this problem out by folding the piece of paper and then writing down the sequence. I then wrote the sequence backwards as, if I turned the paper upside down, it would be the same. I kept doing this until I came up with a sequence that I had already found, and couldn't find another way to do it.
This is what I came up with: BACD, DCAB, CDBA, ABDC, DBAC, CABD, ACDB, BDCA.
So that makes eight different ways to fold the map.
Emily also worked on finding the fold sequences for the larger map and found these twelve sequences:
CBADEF, FEDABC, DEFCBA, ABCFED, EFDACB, BCADFE, DFEBCA, ACBEFD, DCFEDA, ADEFCB, DABCFE, EFCBAD.
BUT ... there are more ways than this to fold the map! Can anyone find them all?