A security camera, taking pictures each half a second, films a
cyclist going by. In the film, the cyclist appears to go forward
while the wheels appear to go backwards. Why?
The triangle ABC is equilateral. The arc AB has centre C, the arc
BC has centre A and the arc CA has centre B. Explain how and why
this shape can roll along between two parallel tracks.
A triangle ABC resting on a horizontal line is "rolled" along the
line. Describe the paths of each of the vertices and the
relationships between them and the original triangle.
We have two splendid solutions here. The first is from Vassil
from Lawnswood High School, Leeds, and it is in a series of
diagrams that speaks for itself.
Here is a different solution from Edwin, The Leventhorpe School,
I'm on a roll (I hope). Here is my solution to the Tough Nut
"Weighty Problem". Your Hint suggests that only 5 moves are needed,
but I have used 6. I think I have made things too difficult for
myself by being too cunning. As my solution is essentially a series
of diagrams I have decided to include an account of how I found my
To work this puzzle out I had to use quite a bit of intuition.
The most shaky piece of intuition was coming up with moves 1 and 2,
which I choose because they flipped the cabinet, but did not put it
in an obvious place. I probably wanted it like this because I
assumed the solution would involve the cabinet being in complex
positions. I then ran out of inspiration for a third move and
decided to look for more clues. Recognising that the rectangle
could be thought of as two 3 by 4 triangles, I remembered the
Pythagorean triple 3 4 5 and realised that the diagonals of the
cabinet were exactly 5 squares long. I played around with moving
the shape using its diagonals and came up with moves 5 and 6, but
didn't have anywhere to use them. All that was left was a bit of
old fashioned guess work to try and connect the moves I had thought
of, and I eventually stumbled across moves 3 and 4, and thus the
In the following diagrams, the area in red represents the
position of the cabinet at the start of the move, and the area in
green represents the position of the cabinet at the end of the
move. The small orange square represents the corner about which the
cabinet is rotated. The arrows show which way the cabinet is
facing. All the pictures are based on the same 7 by 8 grid so you
can see all the moves relative to the initial position of the
(The picture above does not show the arrows on the green area,
but they would be facing left, so that the green area was a mirror
image of the red area in the first picture).
The cupboard can be turned around in 5 moves and we leave that
as a further challenge. Try turning it through 180° in
two moves then, taking symmetry into account, it can be shunted
into the final position in three moves.