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Impossible Sums

Age 14 to 18
Challenge Level


Why do this problem?

This problem, along with the rest of the problems in the Proof for All (st)ages feature, provides an excellent context for observing, conjecturing and thinking about proof, and for appreciating the power of algebra.


Possible approach

This problem featured in an NRICH Secondary webinar in January 2022.

These printable cards for sorting may be useful: 
Impossible Sums Proof 1
Impossible Sums Proof 2
Impossible Sums Proof 3
Impossible Sums Proof of Converse

Key question

Is there an obvious first line of the proof in each case?
Which line follows immediately from the previous line?


Possible support

Encourage students to work in pairs on the proof sorting exercise.
When collecting together the class's results for the numbers from $9$ to $15$, they can be arranged on the board in ways that will make it easier for patterns to emerge:
9 = 4+5 2+3+4    
10 =     1+2+3+4  
11 = 5+6      
12 =   3+4+5    
13 = 6+7      
14 =     2+3+4+5  
15 = 7+8 4+5+6   1+2+3+4+5


    Possible extension

    Students could be encouraged to work on the rest of the problems in the Proof for All (st)ages feature.