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Why do this problem?

This problem provides a great opportunity to introduce a very useful way of representing numbers algebraically.

Possible approach

Present the learners with the problem displayed in the speech bubbles. Give the class some time to look for special numbers - once a few have been found, more will quickly follow. Once there is a consensus that a family of special numbers has been found, bring the class together and share what they think is going on. Ask them "Are you sure you've found all the special numbers? How could we be certain?" Suggest that an exhaustive search would be exhausting!
 
Introduce the class to the algebraic representation of two-digit numbers and work through the reasoning together to establish that there are only nine solutions.
 
Then present the suggestions for other types of special numbers, and challenge learners to predict, using algebra, which numbers will be special for each type.

Key questions

Can we use algebra to help us to understand what's going on?

Possible extension

Think of Two Numbers requires some of the same algebraic skills.
Legs Eleven is a very challenging extension.

Possible support

For students who find the leap into algebra too difficult, the problem can be used to offer an interesting numerical challenge.