# square numbers challenge

25 Feb 2014

Using the numbers 1 to 17 exactly once, make a list where every pair of terms sum to a square number. For example

each pair of numbers here sums to a square number:

$5+11=16$,

$11+14=25$,

$14+2=16$,

$2+7=9$.

## useful information

• the pairs of numbers can sum to the same square number, e.g. $5+11 = 16$ and $14 + 2 = 16$.
• each from 1 to 17 must be used once.
• it is possible to create a list for the first 17 numbers, but no longer list.

Until recently I had given this task to more able classes at KS3, but I was pleasantly surprised by the resilience of a bottom set year 7 class where 3 students produced a complete list.

## differentiation

There are a number of times where there will be a choice as to which square number can be created. Reassure students that any work they do doesn’t go to waste, even if it doesn’t create a complete solution. They will create short chains of numbers that can be reused later.

Model the process by writing a list from 1 to 17. Then ask a student to pick a random number between 1 and 17 and cross it off the list. Choose the next number so it sums to a square number and cross it off the list. Continue you until students “get it”, they soon get the hang of it.

If they get to a dead end, then suggest they start from 17. This definitely produces a solution.