Challenge Level

Here are some dominoes taken out of the full set:

Sort them into two groups - one group with an odd number of
spots and one group with an even number of spots.

Do you have any dominoes left over? Why or why not?

Now put the dominoes into pairs. The number of spots on each
pair of dominoes must make a total of $5$.

How many pairs can you make?

Which dominoes are left over?

Can you pair them up in any different ways so that each pair
adds to $5$?

Which dominoes are left over now?

Are there any dominoes which are always left over?

Can you explain why?