Double Digit
Choose two digits and arrange them to make two double-digit numbers, for example:
If you choose $1$ and $2$,
you can make $12$ and $21$
Now add your double-digit numbers.
Now add your single-digit numbers.
Divide your double-digit answer by your single-digit answer.
Try lots of examples. What happens? Can you explain it?
What happens if you choose zero as one of the digits?
Try to explain why.
How does it work if you choose the same digits, for example $3$ and $3$?
What happens if you use negative numbers?
Now choose three digits and arrange them to make six different triple-digit numbers.
Repeat the steps above: add the triple-digit numbers, add the single digits then divide the triple-digit answer by the single-digit answer.
Do you get the same results?
If you're feeling very organised, try more digits and see what happens.
This problem has been adapted from the book "Numbers in Your Head" by John Spooner, published by BEAM Education. This book is out of print but can still be found on Amazon.
Everyone said correctly that the answer will always be $11$. Many of you also managed to justify this using algebra. The first to do sowas Stephen from Singapore International School. His solution went along the following lines:
Let single numbers $x$ and $y$ represent our digits. Then our two digit numbers will be $10x + y$ and $10y + x$.
For example, if $x=1$ and $y=2$, our two digit numbers are $12 = 10\times 1 + 2$ and $21 = 10\times 2 + 1$. Now $12 + 21 = 10\times 1 + 2 + 10\times 2 + 1 = 11\times 1 + 11\times 2$. So $$\frac{12+21}{1+2} = \frac{11\times 1 + 11\times 2}{1+2} = \frac{11\times 3}{3} = 11$$ In general, the sum of our two digit numbers will be $(10x + y) + (10y + x) = 11x +11y$ and this sum divided by $x+y$ will be 11.
Patrick from Woodbridge School extended the problem to three digits in the following way:
Suppose we pick three digits, say $2$, $3$ and $7$. Then we construct the six three digit numbers from our digits - in our case $237$, $273$, $327$, $372$, $723$ and $732$. If we add these numbers together and divide by $2+3+7$ we get $222$. Try this with your own set of digits. Can you explain what is happening?
Patrick then extended the problem in the same way to four digits. Investigate further.
To facilitate the ease of obtaining accurate results it would be useful to have calculators available for the children, particularly for the later extension ideas.
Children tackling this investigation will be:
- Using knowledge of place value
- Applying understanding of addition and division
- Choosing appropriate calculation strategies
- Working systematically to sort and organise data
- Recognising, explaining, generalising and predicting patterns
- Explaining methods of reasoning
It may be worth working through a few examples of the double-digit problem with the whole class so that the children get a feel for the procedure. Having done this, they are bound to start making their own predictions.
Before leaving them to investigate for themselves, it may be useful to talk about how they are going to record their results. This becomes particularly important when they come to tackle the three-digit extension suggested at the end. For those who reach this stage, a discussion about how to write down the six different triple-digit numbers systematically may prove valuable.
As usual, encourage children to talk to each other about their theories, helping them to express these clearly. Bring the class together as appropriate to share their findings and possible explanations.
The pupils themselves will come up with further variations to investigate so you can take on board their suggestions too.