It must be 2000

Here are many ideas for you to investigate - all linked with the number 2000.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



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:-

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It must be 2000

 

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

Image
It must be 2000

 

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:

Image
It must be 2000

 

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.

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It must be 2000



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
$2$ $110110000$
$3$ $2202002$
$4$ $133100$
$5$ $31000$
$6$ $13133$
$7$ $5555$
$8$ $3720$
$9$ $2662$
and
$10$ $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. #set var="roll-text" value="Up to you!" --><!-- #set var="roll-text" value="" --></p> <p>Talking of $2000$ dots.</p> <p>This is what they look like in a $40$ by $50$ rectangle.</p> <p><img alt="" src="/content/00/01/bbprob1/dotsma.gif" /></p> <br /> <br clear="all" /> <br /> <br /> <br /> <br /> <p>You might like to print this off and explore as I did.</p> <p>I just drew a line and then explored the number on each side of that line.</p> <p><img alt="" src="/content/00/01/bbprob1/dotsma2.gif" /><br /> <br clear="all" /> <br /> <br /> <br /> <br /> <!-- #set var="roll-text" value="Dotty!" --><!-- #set var="roll-text" value="" --></p> <p>I then thought that squares might be better to look at:-</p> <p><img alt="" src="/content/00/01/bbprob1/squas.gif" /><br /> <br clear="all" /> <br /> <br /> <br /> <br /> In another investigation on this site, we looked at <a href="/public/viewer.php?obj_id=52">tiling a patio</a> with square tiles. So what if we have to cover an area of $2000$ squares in a $40$ by $50$ arrangement with squares?</p> <p>You could have:-</p> <p><img alt="" src="/content/00/01/bbprob1/squ1.gif" /></p> <br /> <br clear="all" /> <br /> <br /> <br /> <br /> <p>That used one $40$ by $40$ and four $10$ by $10$ tiles.</p> <p><img alt="" src="/content/00/01/bbprob1/squ2.gif" /></p> <br /> <br clear="all" /> <br /> <br /> <br /> <br /> <p>This one used one $30$ by $30$, two $20$ by $20$ and three $10$ by $10$ tiles.</p> <p><img alt="" src="/content/00/01/bbprob1/squ3.gif" /></p> <br /> <br clear="all" /> <br /> <br /> <br /> <br /> <p>This third one used two $25$s, three $15$s and three $5$s. {It's a bit easier to write that way!}</p> <p>Lastly I made</p> <p><img alt="" src="/content/00/01/bbprob1/squ4.gif" /></p> <br /> <br clear="all" /> <br /> <br /> <br /> <br /> <p>This used one $25$, one $20$, one $15$, four $10$s and fourteen $5$s.</p> <p>You could print off the $40$ by $50$ squares and try some of your own.</p> <p>BUT</p> <p>Be on the look out for patterns</p> <p>Be on the look out for relationships between your results. <!-- #set var="roll-text" value="Have a go!" --><!-- #set var="roll-text" value="" --></p> <p>Lastly I just had to look at triangles.</p> <p>Have a good look at these :-</p> <p><img alt="" src="/content/00/01/bbprob1/tris1.gif" /><br /> These four triangles are the $19$th, $20$th, $39$th and $40$th triangular numbers and their total is ... $2000$!!!!</p> <p>Well isn't that great?!</p> <p>Have a go a looking at combining triangular numbers of your own that add up to $2000$. <!-- #set var="roll-text" value="Have a go!" --><!-- #set var="roll-text" value=""

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

So what if the triangles were made differently?

Like this one:-

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It must be 2000






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

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It must be 2000



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 $2000$!