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Russian Cubes

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted in order 'blue then red then blue then red then blue then red'. Having finished one cube, they begin to paint the next one. Prove that the girl can choose the faces she paints so as to make the second cube the same as the first.

Iffy Logic

Stage: 4 Short Challenge Level: Challenge Level:1

Mathematical logic and thinking are grounded in a clear understanding of how the truths of various mathematical statements are linked together.

For example, for any number $x$ the expressions $x> 1$ and $x^2> 1$ are both mathematical statements which might be true or might be false. However, we always know that $x^2> 1$ IF $x> 1$, whereas it is not always the case that $x> 1$ IF $x^2> 1$ (consider $x=-2$, for example). Thus:

It is correct to write $\quad\quad x^2> 1$ IF $x> 1$

It is incorrect to write $\quad\quad x> 1$ IF $x^2> 1$

Test out your logical thinking with these statements where n and m are positive integers, assuming any obvious properties about numbers (full screen version ).
 
 
 
 
 
 
 
 
 
 
 

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You can view and print out the cards here.

Are there multiple solutions? If not, how do you know?

How would the logic change if $n$ and $m$ were not necessarily positive or not necessarily integers?

Extension: Note that this activity does not prove that the statements are true. How might you go about proving that certain combinations are correct? How might you go about proving that certain combinations are incorrect?