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Diagonal in a Spiral

Age 7 to 11
Challenge Level Yellow starYellow star
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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:
 


 


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?

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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

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