I don't understand... In the video, I thought the line was a number line?

[divinedolphin]

In the video, I thought the line was a number line? Meaning in this picture, a has a value of around 2, b has a value of 3, and c is around 5. But then later in the video, Prof. Loh says that if the lines for a and b are together, then b has a value of 0?

[debbie]

@divinedolphin Thanks for asking; this is a really neat solution method that reframes the question as something even more simple to calculate! This number line is a number line, but the locations of the $a, b$ and $c$ are shiftable, as if there are movable partitions placed at the $\textcolor{blue}{\text{ blue }}$ lines in the picture.

So, the diagram that he drew might correspond to $a = 2, b = 3, c = 4.5,$ but there are many other possible values for $a, b,$ and $c,$ and this is exactly what we desire to solve for! The only requirements here are:

• You must keep the partitions in this order, with $a$ first, followed by $b,$ and followed by $c.$ (You cannot have $b$ first, followed by $c$ followed by $a.$ )

• The partitions must lie at integer values on the number line ($1, 2, 3, 4, 5,$ or $6.$ )

Since $0 \leq a \leq b \leq c \leq 6,$ we can recast this in an amazingly simple way:

• The quantity $a - 0$ equals the number of coins allocated to the first pirate
• The quantity $b-a$ equals the number of coins allocated to the second pirate
• The quantity $c-b$ equals the number of coins allocated to the third pirate

Here are some examples of different ways:

 

It is possible for $a = 0,$ meaning the first pirate gets none. It's also possible for $b = a,$ corresponding to the second pirate gets none. And it's possible for $c = b,$ corresponding to the third pirate gets none. If the question had stated instead that $0 < a < b < c < 6,$ then it wouldn't be as easy to turn this into a pirates-sharing-gold problem. It would still be doable, though! We would just need to make sure that each pirate get a gold coin to start with, and then apportion out the remaining coins.

I hope this helped to make this more clear! Please ask if you have any more questions and I'll be more than happy to answer.