Elevenses
Problem
In the grid below, look for pairs of numbers that add up to a multiple of 11.
9  46  79  13 
64  90  2  97 
25  31  20  22 
4  52  55  7 
Are there any numbers that can only have one partner?
Are there any numbers that could have more than one partner?
What is special about numbers which have the same set of partners?
Can you find every possible pair?
How can you be sure you haven't missed any?
You may have solved the problem by looking at how close each number is to a multiple of 11...
Here is another grid.
This time, look for pairs that add up to a multiple of 13.
11  54  93  15 
76  106  2  115 
29  37  24  26 
4  62  65  9 
How can you use your insights from the first problem to be sure you have found all the possible pairings?
Thank you to Susanne Mallett from Comberton Village College for introducing us to this problem.
Click here for a poster of this problem.
Getting Started
What can you combine with 9?
What can you combine with 20?
What can you combine with 31?
What is special about these numbers?
What is special about their partners?
Student Solutions
Megan from the Thomas Deacon Academy used a spreadsheet to find all 28 pairs of numbers that add up to a multiple of 11. This is how she did it:
To work out the first box I started with 9, adding the rest of the numbers, and then moved on to 46, but didn't do 46 add 9 since it had already been done. Ithen carried this out through out the table but making sure I had not done any in front of the number I was working on as it would have already been done.
Then I highlighted all the answers that are in the 11 times table.
In the attached document are my notes.
Alex from St. Anne's School noticed something special about the numbers in some of the pairs:
There were over 25 different pairs of numbers wich totalled a multiple of 11.
We noticed that the numbers we added to 9, 20 and 31 were all the same:
9+46=55
9+79=88
9+13=22
9+90=99
9+2=11
20+46=66
20+79=99
20+13=33
20+90=110
20+2=22
31+46=77
31+79=110
31+13=44
31+90=121
31+2=33
The difference between 9 and 20 is 11 and the difference between 20 and 31 is 11.
When we added 11 to 31 and made 42. We added 46, 79, 13, 90 and 2 to this number and found that each result was a multiple of 11.
Jack and Zaim from London sent us this very clear explanation of why this happens.
Curtis from Shatin College used a similar strategy :
I divided all of the numbers by 11 and wrote down their remainders. Then I wrote a chart of them, in the same spot. After that, I checked in the remainder box for any pairs that added up to 11. Finally I transfered the numbers back on to the provided grid, and came up with 28 solutions for both the 11's and the 13's.
Adil from Valentines High School discovered the same property of the numbers that could be paired:
 The numbers that are of the form 11X+2 or 11X2 will pair up to give a multiple of 11.
 Obviously pairs of multiples of 11 will add to be a multiple of 11.
 Finally single digit numbers will pair with another number of the form 11X minus the single digit number itself.
Well done to you all.
Teachers' Resources
Why do this problem?
On first inspection this appears to be an opportunity to practice mental calculation strategies, but it soon becomes apparent that this context offers an opportunity to think about the structure of numbers, and multiples in particular.
Possible approach
This printable worksheet may be useful: Elevenses.
Give out the grid and allow a little time for the students to find a couple of pairs that add to a multiple of 11. Collect suggestions and display on the board.
Set the challenge  how many can they find? Can anyone find them all?
When they are well into the problem, stop them and ask "What have you noticed about the pairings?" Collect ideas and note them on the board. If no one has suggested it, draw attention to the pairings involving 9, 20 and 31.
"What is special about these numbers? What is special about their 'partners'?"
Suggest that they return to the problem and use this insight to find out how many pairings are possible.
Can this be done without listing them?
When appropriate, bring the class together and draw out ideas that lead to an efficient strategy.
Offer the follow up grid to consolidate the strategy.
Key questions
What is special about the numbers in each pairing?
Are there some numbers that can only be used once? Why?
Are there some numbers that can be used many times? Why?
Possible support
The grid below could be used to ask students to find pairs that add to a multiple of 10.
The key questions are useful prompts to focus students on the structure of the numbers rather than multiple calculations. This could be useful preparation before going on to the main problem.
8  42  72  12 
58  82  2  88 
23  28  18 
20

4  47  50  6 
This image may be useful to show the students that the sum of two multiples of 11 is a multiple of 11, and the sum of two numbers in the form 11n+2, 11n2 is a multiple of 11 as well.
Possible extension
Both grids contain less than 30 possible pairings. Can you produce a grid of numbers that has more than 30? What is the maximum number of possible pairings in a 4x4 grid?
Legs Eleven may provide an interesting followup challenge.