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'Delia's Routes' printed from https://nrich.maths.org/
David and Fiona told us how they started to
tackle this problem:
First we tried different paths to find the shortest way for Delia
to get to the bird table. We tried lots of paths and found that
Delia must go along six tiles and up six tiles and so any path that
does that and does not go down or left is a shortest path.
Then we worked out that to avoid the pond, Delia must only run
along the edges of the outside tiles, because all the other edges
run into the pond.
Lizzy explained how she continued from
here:
First Delia can go up one square or right one square. That is two
choices. Then she can go up one square or right one square. That is
2 x 2 = 4 choices, because she can do this for both of her first
choices. Then she can go up one square or right one square. That is
4 x 2 = 8 choices, because she can do this for all four of her
second choices. Then if she is on the side of the garden she can
still go up one square or right one square, but if she is not then
she might run into the pond if she makes the wrong move. I made a
table of what she can do from here.
First move |
Second move |
Third move |
Fourth move |
Along |
Along |
Along |
Along |
Along |
Along |
Along |
Up |
Along |
Along |
Up |
Along |
Along |
Up |
Along |
Along |
Along |
Up |
Up |
Up |
Up |
Along |
Along |
Along |
Up |
Along |
Up |
Up |
Up |
Up |
Along |
Up |
Up |
Up |
Up |
Along |
Up |
Up |
Up |
Up |
I saw that she cannot go along two and up two in any order because
then she will be off the edges of outside squares and into the
pond.
Then I looked at the rest of the path and saw that if she starts by
going along two and up one or along one, up one and along one or up
one, along two then she must continue going along until she gets to
the edge of a tile that is on the righthand edge of the garden, and
if she starts going up two then along one or up one, along one, up
one or along up, up twothen she must go up until she gets to the
edge of a tile at the top of the garden.
Then I worked out that if she starts going along, along, along then
she must only go up once or not at all before getting to a tile on
the righthand edge of the garden. If she goes up, up, up then she
must only go along once or not at all before getting to a tile at
the top of the garden.
This way I broke everything down so I could be sure I was counting
all the paths she could take and none of them twice. I counted 74
different ways, so she must repeat every 75 days or less.
That's right,Lizzy! Thank you, Lizzy, David
and Fiona.