Diagonal in a spiral
Investigate the totals you get when adding numbers in threes on the diagonal of this pattern.
Age
7 to 11
Challenge level
Problem
This is the start of a spiral, starting with 1 and then moving clockwise.
For this challenge we're interested in the upper-left to lower-right diagonal, shown in green. Here is the starting example:
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Challenge 1
Find the numbers that would be in this green upper-left to lower-right diagonal for a spiral going up to 144 instead of just 16.
Challenge 2
The small example above shows 7, 1, 3, 13 as the diagonal.
Now that you have a bigger diagonal going beyond 100 you need to deal with all the numbers in that diagonal in order from upper-left to lower-right.
You'll need to add the diagonal numbers in threes in order as they move through the square.
In the small example it would be 7 + 1 + 3 = 11 and 1 + 3 + 13 = 17. So your new numbers would be 11 and 17.
The totals you get for each three will give you a new set of numbers to use for Challenge 3.
Challenge 3a
You now need to use the numbers you got from adding the diagonal up in threes.
Use these numbers to make a total that has a 2 as the ones digit. You can only use a number once in any addition.
Do this in as many different ways as possible.
Challenge 3b
Do the same as in Challenge 3a but now the ones digit has to be an 8.
How many different ways are possible?
Student Solutions
Well done to everybody who had a go at this task. By my calculations, there are over a hundred possible solutions for each of Challenge 3a and Challenge 3b, so it's not surprising that nobody has sent us a full solution yet! If anybody does have any more solutions that they'd like to share with us, please email
us.
Most children correctly worked out the diagonal numbers for the spiral going up to 144. Arlo from South Harringay Primary School in England sent in this picture of the spiral:
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Challenge 2 was slightly more difficult, and some children made mistakes with calculating the totals for each three numbers. The children from Stourport Primary Academy in the UK correctly found the ten totals and noticed a pattern between them:
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Well spotted! Have a look at Stourport Primary Academy's full solution to see each child's reasoning about the different challenges in this task.
Zac from the UK found 52 solutions for Challenge 3a and 84 solutions for Challenge 3b, but unfortunately some of Zac's numbers from Challenge 2 weren't quite right. Take a look at Zac's full solution to see their reasoning. How might Zac change some of these solutions to use the correct
numbers?
Ariel, Marlo and Robin from South Harringay found all of the totals in Challenge 2, but they also found a couple of extra totals that weren't made by adding groups of three numbers from the diagonal. They found these correct solutions for Challenge 3:
71 + 281 = 352
101 + 11 = 112
71 + 11 = 82
11 + 281 = 292
101 + 71 = 172
101 + 281 = 382
Thank you for sharing these with us! How can we be sure that this is all of the possible ways of completing Challenge 3 by only adding two numbers that end in the digit 1?
Shubhangee, the teacher of Viha, Abhiram, Eshann, Anirved, Arya, Rivaan, Miraya, Asma, Aprameya, Vibha, Rudraraj, Nithyashree, Adhrit, Kathir, Arnav, Arjun and Harshad from Ganit Kreeda, Vicharvatika, India, sent in their solutions to the task. They thought that there would be 75 solutions for Challenge 3a and 102 solutions for Challenge 3b. I agree with this
number of solutions for Challenge 3b but I think there are more solutions for Challenge 3a. Take a look at Ganit Kreeda's full solution to see their ideas.
Teachers' Resources
Diagonal in a Spiral
This is the start of a spiral, starting with 1 and then moving clockwise.
For this challenge we're interested in the upper-left to lower-right diagonal, shown in green. Here is the starting example:
Image
Challenge 1
Find the numbers that would be in this green upper-left to lower-right diagonal for a spiral going up to 144 instead of just 16.
Challenge 2
The small example above shows 7, 1, 3, 13 as the diagonal.
Now that you have a bigger diagonal going beyond 100 you need to deal with all the numbers in that diagonal in order from upper-left to lower-right.
You'll need to add the diagonal numbers in threes in order as they move through the square.
In the small example it would be 7 + 1 + 3 = 11 and 1 + 3 + 13 = 17. So your new numbers would be 11 and 17.
The totals you get for each three will give you a new set of numbers to use for Challenge 3.
Challenge 3a
You now need to use the numbers you got from adding the diagonal up in threes.
Use these numbers to make a total that has a 2 as the ones digit. You can only use a number once in any addition.
Do this in as many different ways as possible.
Challenge 3b
Do the same as in Challenge 3a but now the ones digit has to be an 8.
How many different ways are possible?
Why do this problem?
Moving through each of the challenges gives an opportunity for children to work investigatively. Children's curiosity will take them to many different places as they work on this challenge. This task is a good example that offers both a 'low threshold' starting point and at the same time offers a 'high ceiling' for those pupils who are inclined to explore much further.Possible approach
It will probably be best to have the whole group/class with you, extending together the spiral to 144.Then set challenge 2 by starting them off to ensure that as well as adding 111, 73 and 43 they'll have to add up the next three as 73, 43 and 21 (not just 21, 7, and 1), so that they have to produce ten totals altogether. (You may decide not to tell them that, but instead have some feedback from them when they have completed challenge 2).
Key questions
How are you getting your totals of three?What are you doing to try to get your totals to end in a 2 (or an 8)?
Tell me what you are doing.
Possible extension
Exploring the ten totals of threes, what can they find out?What about extending the spiral past 144?
Possible support
Some pupils may need a calculator to help with the many additions that they want to carry out.