I need help to solve this Rules for bulding towers of cubes
|rule 1|| The number of cubes on the bottom layer is always one less than the number of squares on the grid
|rule 2|| Each new layer is made with one cube less than the layer underneath it.
Maths" grade 11
- Investigate how many different arrangements there are of 4 cubes on top of 5 cubes on a two by three grid
- investigate the number of different arrangements of six cubes on top of seven cubes on a two by four grid
- investigate the relation between the number of arrangements of cubes and the size of the grid
- when there are two layers of cubes
- when there are more than two layers of cubes
I would approach this problem visually. Actually with two images:
The first image:
The second image is a basic counting principle. You have layers (in a
I think of them (for this problem) as a vertical stack of rectangles.
In each rectangle, I place the corresponding number from above.
The principle is that these numbers should be multiplied, because EACH
possibility counted at a lower layer had ALL the possibilities counted
the upper layer.
|(a)||For the first layer, I would imagine the grid and the blocks
How many ways are there to place 5 blocks on six squares?
Alternately, how many ways are there to pick the square with no block on
|(b)||Now imagine the five blocks placed in some patttern.
How many ways are there to pick the four blocks to place a block on top
Alternately, how many ways are there to pick the one block that will not
have a new block on top?
[Check that this number does NOT depend on
which five squares were covered in the first layer!]
|(c)||Proceed to the next layer (if there are more than two layers) in the
Therefore you multiple your answers to (a) x (b) [ x(c) if you have
I think you can figure it out from here. If you are stuck , image a 2x2
|(a)||How many ways to choose three? (Think of it as choosing one to omit.)
There are 4 ways.
|(b)||For each of these, how many ways are there to pick two of these to cover? (Or one to not cover) For each of the 4 options above there are 3 ways to pick two to cover. For example for the first of the 4 options above the three ways are.
Hence in total there are 4 x 3 = 12 towers.
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