Trominoes
Problem
A tromino is a $3 \times 1$ block:
What sized rectangles may be made using trominoes? You can print off and cut out trominoes from this sheet . Alternatively, if you have a set of the game Jenga, then use the blocks as trominoes (but check they are $3 \times 1$ first)
Can you cover $63$ squares of an $8\times 8$ chessboard using trominoes? (Why can't you cover $64$?). If so, which square remains uncovered, and are there other arrangements of the trominoes which would lead to a different square being left uncovered?
Think of some other squares which cannot be covered with trominoes. Can you "almost" cover them, leaving only one hole? When can an $n \times n$ square be covered or "almost" covered?
Getting Started
Think about colouring in the $8 \times 8$ chessboard with three colours.
How many squares of each colour are there?
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem :
This problem can be a good way to demonstrate mathematics emerging out of play.Possible approach :
Using wood blocks (trominoes from three squares, like dominoes from two), arranged as a tower, a game of coordination and experimentation begins as players try not to topple the stack.
Pick up one of the pieces of wood and begin posing some questions about it.
- Can these blocks form a square ? - other than the three side by side used for each layer in the stacking game.
Let the questions flow and get pursued. Try to encourage adjustment rather than abandonment when a line of questioning appears to run out. For example a five by five square can't be made because each block contributed three squares of area and there are 25 to be covered, so there must be a hole. OK, given there's a hole, can you make a square ?
Key questions :
- What questions can you pose ? (offer an example : what rectangles can and cannot be made ?)
- What others ?
- Which seem like good questions to pursue ?
- (later) What were our questions ? Where has each question taken us in what we now see or understand ? What came out of this that we didn't know already or didn't expect ?
Possible extension :
Equal Equilateral Triangles is a good next step.Possible support :
Create a competition to produce all rectangles up to side length 10.
Perhaps use a digital camera or the camera in a mobile phone to catch arrangements quickly. A drawing record may sap energy that might be used more effectively pursuing the task.