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This problem featured in the NRICH Secondary webinar in April 2022.
Iffy Logic
Age 14 to 18
Challenge Level 
Why do this problem ?
This problem is designed to help students think clearly about logical implication. It offers an opportunity to familiarise them with the arrows \Rightarrow and \Leftrightarrow that they may wish to use in their own mathematical writing.
Possible approach
This problem featured in the NRICH Secondary webinar in April 2022.
These printable cards may be useful.
When introducing the problem, please emphasise to students that n and m are positive integers.
Students could work on laptops and tablets, or using the printable cards. The interactivity allows them to check their answers.
It is valuable for students to work on this problem in pairs or small groups, so that they can talk about their logical statements and try to explain their answers verbally. Some cards can be matched in multiple ways, so working towards a complete solution will involve swapping statements and thinking clearly about whether an implication goes one or both ways.
When introducing the problem, please emphasise to students that n and m are positive integers.
Students could work on laptops and tablets, or using the printable cards. The interactivity allows them to check their answers.
It is valuable for students to work on this problem in pairs or small groups, so that they can talk about their logical statements and try to explain their answers verbally. Some cards can be matched in multiple ways, so working towards a complete solution will involve swapping statements and thinking clearly about whether an implication goes one or both ways.
Key questions
Are there any statements that match with only one other statement?
Which pairs can only be linked by a \Rightarrow arrow?
Which pairs can be linked by a \Leftrightarrow arrow?
Possible support
Rather than attempt to match all of the cards, students could try to make a selection of true statements.
Possible extension
Invite students to think about how the constraint that n and m are positive integers affects their answers.
Students could then move on to IFFY Triangles and Mind Your Ps and Qs
