Is this math game always winnable?

Unsatisfyingly, a counterexample is (all black):

$$(5,5,6,6,7,7,8,8,9,9,10,10,J,J,Q,Q,K,K)$$

which does not satisfy the last two equations, since

$$\_+\_+\_ \ge 5+5+6 =16>13 = K$$

Extending this result, we need at least $22$ cards to guarantee a solvable $14$-tuple since we have the $21$-card counterexample

$$(3,4,4,5,5, \dots , K, K)$$

where $3+4+4+5+5+6 = 27 > 26 = 2K$, so the last two equations cannot both be satisfied. I do not know whether a counterexample to $22$ cards exists at this moment.


28 card counterexample:

$$ black: K, K, J, J, 9, 9, 7, 7, 5, 5, 3, 3, A, A $$ $$ red: K, K, J, J, 9, 9, 7, 7, 5, 5, 3, 3, A, A $$

cannot satisfy __ + __ = __ because 2 odds make an even (whether added or subtracted), and there are no evens in the set.

Edit: it's 28 cards, not 26.

The 29th counterexample is easy: with only one additional even, it isn't enough to satisfy both of the top two equations. So, 2 evens are needed to be added.

Tags:

Combinatorics

Card Games

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