Imagine that round the back of where you live is a piece of ground that your family want to change. Instead of grass or plants growing there you want to have a patio. To prepare to change it the ground is dug up and levelled, it is then measured and found to be a square $10$ by $10$ [this could be feet, metres or yards it does not really matter for this challenge]. You then go to a large store that sells tiles to go on the ground. To your surprise you find that they sell tiles in ten different sizes; $1$ by $1$, $2$ by $2$, $3$ by $3$, etc. up to $10$ by $10$. You decide to go back home for a while and work a few ideas out.

You start by drawing a $10$ by $10$ square; and then you think about what sizes of tiles to use to cover it.

If you use $1$ by $1$ tiles then you find [or know already!] that you will need $100$ of them. So your shopping list would look like :-

$100$ tiles @ $1$ by $1$ > Total number of tiles $100$

If you use $10$ by $10$ tiles then you find [or know already!] that you will need 1 of them. So your shopping list would look like :-

$1$ tiles @ $10$ by $10$ > Total number of tiles $1$

If you use $2$ by $2$ tiles then you find [or know already!] that you will need $25$ of them. So your shopping list would look like :-

$25$ tiles @ $2$ by $2$ > Total number of tiles $25$

You could of course use a mixture of different sized tiles.

If you use $12$ tiles $2$ by $2$ and $4$ tiles $3$ by $3$ and $1$ tile $4$ by $4$ your shopping list would look like :-

$12$ tiles @ $2$ by $2$
$4$ tiles @ $3$ by $3$
$1$ tile @ $4$ by $4$ > Total number of tiles $17$

So you could, with these examples use $1, 17, 25$ or indeed $100$ tiles.

Your challenge starts by asking you to try to find lots of different numbers of tiles that could be used to cover this $10$ by $10$ patio.

You could think about the artistic appearance of it as well if you like. However all the patio must be covered and no cutting any of them to make different sizes or shapes ... they're actually very, very hard to cut anyway.

You could now think about these kinds of questions and perhaps you would think of other questions of your own to pursue.

1. Are there some numbers of tiles used that actually are made up of different mixtures of tiles? [For example in our one that used $17$, could you get to use just $17$ tiles but they require a different shopping list from the one above?]
2. Are there some numbers between $1$ and $100$ that cannot be used as the total number of tiles required?
3. Is there some system for getting new examples that use different numbers of tiles?