Problem 4: Cubes, Toast and Chess

Monday, November 9, 2009

Question 1: Chessboard Rectangles

How many rectangles are there on a chessboard?

Problem 2: Quick and Toasty

Three slices of bread are to be toasted under a grill. The grill can hold two slices at once but only one side is toasted at a time. It takes 30 seconds to toast one side of a piece of bread, 5 seconds to put a piece in or take a piece out and 3 seconds to turn a piece over. What is the shortest time in which the three slices can be toasted?

Question 3: Cubes Cubed

I have eight cubes. Two of them are painted red, two white, two blue and two yellow but otherwise they are indistinguishable. I wish to assemble them into one large cube with each colour appearing on each face. In how many different way can I assemble the cube?

Stuck?

Question 1: Chessboard Rectangles

- Try a small chessboard first.
- What systematic way of counting the rectangles will be best?
- Examine the method used to count the squares on a
chessboard and generalize.

Question 2: Quick and Toasty

- Watch out for ambiguity!
- It can be done in less than 140 seconds.
- Which time periods should you concentrate on minimizing?

Question 3: Cubed Cubes

- Can you find one arrangement?
- What do you KNOW? Are the conditions clear?
- What do you WANT? When are two large cubes different?
- What presentation can you INTRODUCE? Before using 8
coloured cubes try drawing a cube or diagrams of its faces or
try looking at a convenient box - it may help you visualize
the situation even if you do not touch it.
- INTRODUCE a way of describing or recording different
solutions.