With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
This is an interesting investigation with many
possible ways of solving and generalising the problem, and several
people wrote in giving an answer of 66 moves.
Isabelle from Lathallan School investigated
what happened with different sized grids, and spotted a pattern,
which she used to get an answer of 66.
The smallest numbers of moves for 99 squares is 66 moves. I
worked out the smallest number of moves for a 3 square grid which
was 4. I then worked out the number of moves for a 4 square (which
was 2) and a 5 square which was 4. The 6 square was also 4 and so
was the 7 square. And then because the moves have to go from a
white square and end up on a white square then the number of moves
cannot ever be an odd number. So then I tried 8, 9 and 10 square
grids which all needed6 moves. So then when I got to 15 squares I
needed 10 moves. So then I worked out that a 30 square needs 20
moves. The number of squares is 1.5 times the number of required
moves. I tested this at 45 squares (30 moves) and therefore at 99
squares it was 66 moves.
Emily from Durham Johnston Comprehensive
School and Tom from Bristol Grammar School both used algebra to
explain Isabelle's pattern. Here is Tom's solution: