It Must Be 2000

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

We're going to have a look at this number - $2000$.

We've used all sorts of numbers in the past, so let's use our detective skills to find things out about this number $2000$.

It looks like $2000$ because we count in tens and along in a straight line:-


Sometimes, yes some "TIMES" we count in a circle.


If we started counting around this "$12$" clock and counted up to $2000$ we would end up at the number $8$.

But of course we could count in a circle of any size with a different number of numbers around it.

So if the "clock" looked like this:


and we counted up to $18$ before returning to $1$, then we would end up at $2$ if we counted to $2000$.

Some of you will know about this kind of counting already, it is sometimes called MODULO arithmetic or CLOCK arithmetic.

Thinking about it is all to do with remainders. Like this:

$2000$ divided by $12$ will be $166$ remainder $8$

$2000$ divided by $18$ will be $111$ remainder $2$

So we could just look at remainders.

Well save your brains!! I used a spreadsheet on my computer and got the following table.

Column A is the number of numbers around the clock, or the modulo number.

Column B is the remainder when divided into $2000$!

And just for fun I added column A to column B to get column C.

Just explore and explore ... Be a detective ... Look, think ... write what you notice and what you think.

Some of you may have also thought about working in different bases. We usually work in base ten, and you may have explored base $2$.
If you are counting $2000$ dots then the number looks like this :-

Base Number
$2$ $110110000$
$3$ $2202002$
$4$ $133100$
$5$ $31000$
$6$ $13133$
$7$ $5555$
$8$ $3720$
$9$ $2662$
$10$ $20004

Well, what an interesting collection of $3$s and $1$s in bases $4$, $5$ and $6$. I found base $7$ to be a very big surprise.

I have not looked deeply into this, I thought I'd leave it up to you.

Talking of $2000$ dots.

This is what they look like in a $40$ by $50$ rectangle.

You might like to print this off and explore as I did.

I just drew a line and then explored the number on each side of that line.

I then thought that squares might be better to look at:-

In another investigation on this site, we looked at tiling a patio with square tiles. So what if we have to cover an area of $2000$ squares in a $40$ by $50$ arrangement with squares?

You could have:-

That used one $40$ by $40$ and four $10$ by $10$ tiles.

This one used one $30$ by $30$, two $20$ by $20$ and three $10$ by $10$ tiles.

This third one used two $25$s, three $15$s and three $5$s. {It's a bit easier to write that way!}

Lastly I made

This used one $25$, one $20$, one $15$, four $10$s and fourteen $5$s.

You could print off the $40$ by $50$ squares and try some of your own.


Be on the look out for patterns

Be on the look out for relationships between your results.

Lastly I just had to look at triangles.

Have a good look at these :-

These four triangles are the $19$th, $20$th, $39$th and $40$th triangular numbers and their total is ... $2000$!!!!

Well isn't that great?!

Have a go a looking at combining triangular numbers of your own that add up to $2000$.

We're always asking "I wonder what would happen if I ...?"

So what if the triangles were made differently?

Like this one:-

Then, for my final contribution, and eagerly waiting to receive your contributions, I offer you this splendid picture:-

This has four large triangles, the $20$th in their sequence each with an area of $400$.

There are $16$ small triangles, the $5$th in their sequence each with an area of $25$.

So you see the total area is $2000$!