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The Collatz conjecture is the simplest open problem in mathematics. You can explain it to all your non-mathematical friends, and even to small children who have just learned to divide by 2. It doesn't require understanding divisibility, just evenness.

The lack of connections between this conjecture and existing mathematical theories (as complained of in some other answers) is not an inadequacy of this conjecture, but of our theories.

This problem has led directly to theoretical work by Conway showing that very similar questions are formally undecidable, certainly a surprising result.

The problem also relates directly to chaotic cellular automata. If you look at a number in base 6, you will see that multiplying by 3 and dividing by 2 are the same operation (differing only by a factor of 6, i.e. the location of the decimal point), and the operation is local: each new digit only depends on two of the previous step's digits. Using a 7th state for cells that are not part of the number, a very simple cellular automaton is obtained where each cell only needs to look at one neighbor to compute its next value. (Wolfram Mathworld has some nonsense about a CA implementation being difficult due to carries, but there are no carries when you add 1, because after multiplying by 3 the last digit is either 0 (becomes a non-digit because number was even so we should divide by 6) or 3 (becomes 4), so there are never any carries.)

It is easy to prove that this CA is chaotic: If you change the interior digits in any way, the region of affected digits always grows linearly with time (by $\log_6 3$ digits per step). This prevents any engineering of the digit patterns, which are quickly randomized. If the final digit behaves randomly, then the conjecture is true. Clearly any progress on the Collatz conjecture would immediately have consequences for symbolic dynamics.

Emil Post's tag systems (which he created in 1920 expressly for studying the foundations of mathematics) have been studied for many decades, and they have been the foundation of the smallest universal Turing machines (as well as other universal systems) since 1961. In 2007, Liesbeth De Mol discovered that the Collatz problem can be encoded as the following tag system:

$\begin{eqnarray} \hspace{2cm} \alpha & \longrightarrow & c \, y \\ \hspace{2cm} c & \longrightarrow & \alpha \\ \hspace{2cm} y & \longrightarrow & \alpha \alpha \alpha \\ \end{eqnarray}$

In two passes, this tag system processes the word $\alpha^{n}$ into either $\alpha^{n/2}$ or $\alpha^{(3n+1)/2}$ depending on the parity of $n$. Larger tag systems are known to be universal, and any progress on the 3x+1 problem will be followed with close attention by this field.

In short the Collatz problem is simple enough that anyone can understand it, and yet relates not just to number theory (as described in other answers) but to issues of decidability, chaos, and the foundations of mathematics and of computation. That's about as good as it gets for a problem even a small child can understand.