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Hundred Square

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

Thank you for your solutions to this problem. You found several different ways of tackling it.

Fergus, Nathan, Samuel and Jesse from Rutherglen Primary School drew another hundred square with the numbers on the opposite side of a printed hundred square. Jesse says:

On the back of $100$ was $91$, on the back of $58$ was $53$, on the back of $23$ was $28$ and on the back of $19$ was $12$.

Mikey from Archbishop of York CE Junior School wrote:

Having printed the page out I realised that if you turned it over you could see through the page. Looking where $100, 58, 23, 19$ would be meant you could read off the answers as $91, 53, 28, 12$. Or is this cheating?

I don't think this is cheating at all Mikey! Mikey then realised something else which was also spotted by "N" (he or she didn't give us a full first name):

If you draw a mirror line down the middle of the square you will be able to work out what number will be behind each number. You choose a number, then find its mirror on the other side of the line, this 'mirror' number will be the number on the reverse!

This is also a very handy method - well noticed. "N" sent an image with the "mirror line" drawn in:



100 square with mirror line drawn in

Well done to George from Bradon Forest, Mary from St Swithun's, Miss Grewcock's Class from St James CEVA Primary and Joshua from Queen Victoria Primary who also noticed this mirror line.

Devonshire Maths Club, Devonshire Primary School have found a pattern which they describe:

The tens in each pair don't change ie $58 - 53$, both $5$ tens.
The units in each pair add up to $11$.
$100$ & $91$ are different. $100$ has a nought in the tens column, and $90$ has a nine. In the units, $1 + 0$ doesn't = $11$.
$100 = 9$ tens & $10$ units. $91 = 9$ tens & $1$ unit. Now the tens stay the same, and the units add up to $11$.

Very well noticed. Luke from Queen Victoria Primary found another way:


If the number is $23$ then it is placed two places away from the end of the line.
Next you go two places back from the other end of the same line to get the correct answer of $28$. Like $19$ would be $12$ on the other side of the sheet.

Jake from the same school as Luke explained this a bit more generally:


To find the solution you need to first find out whether it's on the right or the left of the hundred square. Then see how many squares it is away from the end of the square. Then use the number of squares on the other side. Then you have the answer.

Sohpie and Anna from St Swithun's and Gabrielle from Hayesfield Girls School also tackled the problem in this way.