Kent Haines describes the game of Integer Solitaire, which I find to be excellent for young kids learning arithmetic. I'm sure they will be motivated by this game to get a lot of practice.
Kent asks a question about his game, which I find very interesting, and so I am asking here, in the hopes that Math.SE might be able to answer.
The child draws 18 cards from an ordinary deck of cards, and then regards the cards to have values Ace = 1, 2, 3, …, Jack = 11, Queen = 12, King = 13, except that Black means a positive value and Red means a negative value.
Using 14 of the 18, the child seeks to find solutions of four equations:
For example, a successful solution would look like:
Question. Does every set of 18 cards admit a solution?
Kent Haines says, "I have no idea whether all combinations of 18 cards are solvable in this game. But I have played this game for five years with dozens of students, and I have yet to see a combination of 18 cards that is unsolvable."
Follow up Question. In the event that the answer is negative, what is the probability of having a winning set?
For the follow up question, it may be that an exact answer is out of reach, but bounds on the probability would be welcome.
Best Answer
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.