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Break it Up!

Age 5 to 11 Challenge Level:

Uygor from FMV Erenkoy Isik School in Turkey sent us a very clearly reasoned solution. He says:

I can break the first stick (that consists of $7$ blocks) as:

$1$ 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.