What can be found out about the maximum Set-less pile of
Do not be frustrated if you cannot determine an exact answer for this
question. A reasonable approach to this problem is to try to establish
limits on the possible answer. What would you need to do to show that
the maximum Set-less pile is at least some number n? What would you
need to do to show that it is no bigger than some upper limit? If you can
narrow down the possibilities, that is a worthy accomplishment.
One general approach is to start with a simpler problem: a smaller deck
where cards have two or three attributes instead of four (or fewer values
per attribute). Find the maximum Set-less pile in those cases and try to
extend your methods as you add attributes.
Another approach: Imagine dividing the whole deck of Set into two piles:
the first one, which is Set-less, and the second one, which contains all of
the rest of the cards. Any two cards of the first pile make a Set with exactly
one card of the deck. For the first pile to be Set-less, this card must be in
the second pile. As you try to construct Set-less piles, it is helpful to think
about the number of cards being forced into the complementary pile. How
can you choose cards for your Set-less pile such that the second pile does
not grow too large?
Alternatively, you could try to think geometrically about the problem.
How can the different attributes and values be represented visually?