This activity investigates how you might make squares and pentominoes from Polydron.
If you had 36 cubes, what different cuboids could you make?
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
Uygor from FMV Erenkoy Isik School in Turkey sent us a very clearly reasoned solution. He says:
$1$ block + $6$ blocks
$2$ blocks + $5$ blocks
$3$ blocks + $4$ blocks
$4$ blocks + $3$ blocks
$5$ blocks + $2$ blocks
$6$ blocks + $1$ block
I found $6$ different styles for a $7$-block stick.
Then I tried a $6$-block stick. I found the following breaking styles:
$1$ block + $5$ blocks
$2$ blocks + $4$ blocks
$3$ blocks + $3$ blocks
$4$ blocks + $2$ blocks
$5$ blocks + $1$ block
So I found $5$ different styles for a $6$-block stick. I predict I can find $7$ styles for an $8$-block stick.
The number of ways is $1$ less than the number of blocks of the stick.
So for $5$ cubes, I should find $4$ ways by doing it like this:
$1$ block + $4$ blocks
$2$ blocks + $3$ blocks
$3$ blocks + $2$ blocks
$4$ blocks + $1$ block
I saw that my prediction is right!
For $20$ cubes, I can find $19$ ways
For $50$ cubes, I can find $49$ ways
For $100$ cubes, I can find $99$ ways
On the other hand, if the cubes are the same colour:
For $6$ cubes, we have $3$ ways
For $7$ cubes, we have $3$ ways
For $8$ cubes , we have $4$ ways
For $9$ cubes, we have $4$ ways ...
In this case, we can find the number of ways by dividing the number of cubes by $2$ then we only look at the quotient.
(By this Uygor means to ignore any remainder.) This is very well explained, thank you. Thomas from New York looked at the last part of the problem in a slightly different way. If n is the number of cubes then:
To find out how many ways any number of cubes of the same color can be split, the formula is $(n-1)/2$ if $n$ is odd, and $n/2$ if $n$ is even.
Well done both of you.