Twisty Logic

Age 16 to 18
Challenge Level

Why do this problem?

Logical paradoxes are an enduringly fascinating way to explore the limits of logical thinking, and their consideration provides a valuable mathematical workout. Students might be surprised to learn that it is possible so simply to create logical statements with no resolution True or False.

Possible approach

These problems make an excellent poster display or end of lesson filler; their intellectual challenge is appropriate at any stage of study. However, they could also be used particularly productively as a prequel to the study of logical operators in decision mathematics.

Give the questions to the students and ask them to think about two or three of them. Then ask the class to decide which ones are true and which ones are false.Whilst many might intuitively 'get it' that the statements are both true and false, explaining this clearly presents a different challenge. Get the class clearly to explain their reasoning to each other in pairs. Then suggest that those with particularly good explanations give their reasoning to the class.

Key questions

IF the statement were true, THEN what would be the implication?

IF the statement were false, THEN what would be the implication?

Possible support

To engage in this problem, students need to be confident with the logical concepts of IF (something) THEN (something else) and basic notions of True and False. 

Possible extension

Could students make up similar questions of their own?

Could students change the questions slightly so that they are either true or false?