Two merge games, one question: can either be played flawlessly? The answer runs through the corner of a board, an opponent who never arrives, and a number seventeen exponents tall.
Start with the game everyone knows. 2048, four directions, a 4×4 grid, tiles that double when they touch, is, strictly, unsolved. No one has written an algorithm that plays it perfectly. What we have are very strong approximations. An expectimax search, peering eight or so moves ahead and grading each board by a handful of hand-tuned heuristics, open cells, big tiles pinned to an edge, smoothness, reaches the 32768 tile in better than a third of its games,1 and the strongest public engine touches the 65536 tile a few percent of the time.2 Generalized to an m×n board, merely deciding whether a target tile is reachable is NP-hard.3 Strong is not the same as solved.
How high can a single tile climb? On sixteen cells the answer is a small, elegant piece of counting. Lay the board out as a descending staircase, 65536, 32768, 16384, on down to a lone 2. Each tile is exactly one power of two below its neighbour, so the sixteen distinct powers 21 through 216 fill the board perfectly, and 65536 = 216 sits at the summit: one exponent per cell. That is the ceiling if the game only ever hands you 2s. But 2048 spawns a 4 one time in ten, and a single well-timed 4 smuggles in a seventeenth exponent, lifting the true maximum to 131072 = 217, seventeen tiles' worth of powers packed into sixteen squares.4 No human has built it; a few AIs have brushed it.
Why do these boards reward tucking your largest tile into a corner? A tile in the middle can be shoved four ways, and it keeps getting divorced from the tiles it wants to marry. A corner tile touches two walls; it only moves when you push toward walls it already hugs, so it sits still while everything organizes around it. Chain the rest into a monotone snake, highest in the corner, folding back and forth in descending order, and one swipe can set off a cascade of merges.1 This is the heuristic that dominates casual human play, and it is very nearly what the AIs rediscover when you let them tune their own weights.
A merge game with random spawns has no opponent, only weather. To "solve" it is to beat the dice on average, not to beat a mind.
Now tilt the board into three dimensions. 3927 is 2048's cubic cousin: a 3×3×3 lattice of 27 cells, six shift directions instead of four, and tiles that fuse in threes, 3 to 9 to 27 to 81, base three where 2048 is base two.5 Does corner-stacking survive the extra dimension? A cube has eight corners, and a corner cell now touches three faces rather than two, it should be steadier still, pinned by three walls at once, though the six directions give the board more ways to shrug your structure loose. The snake becomes a folded path threading all three layers. As far as I can find, no one has worked out whether the analogy actually holds, reasoning, not measurement.
And the ceiling? The one-exponent-per-cell rule would suggest 327 ≈ 7.6 trillion as a loose upper bound. But the analogy frays badly. 2048's extra exponent came from a lucky 4; 3927 only ever spawns the smallest tile, a bare 3, so there is no bonus. Worse, a triple-merge needs three tiles aligned in a line, and every row, column and pillar on a 3×3×3 cube is exactly three cells long, so every merge consumes an entire line. That constraint bites far harder than anything in the flat game and almost certainly drags the real maximum well below 327. What that true number is, I have found computed nowhere. (Clearly-labelled reasoning; the mechanics above are measured from the game's design documents.)
Here is the subtlety that makes "perfect play" slippery. A merge game with random spawns is a single-player stochastic game, a solitaire against a die, not a duel. Nothing chooses the worst spawn to ruin you; there is only the RNG, indifferent. So the right notion of optimal play is expectimax: maximize the expected outcome over the distribution of spawns. It is emphatically not minimax, minimax presumes an adversary, and if you actually let one place every tile ("evil 2048"), the game becomes a crueler thing you can be forced to lose. Because the dice can in principle deal any sequence, a strategy that guarantees a given tile may simply not exist. So the honest answer to "is there a perfect strategy?" is that for a stochastic game the best you can even define is a best-on-average one, and computing it exactly is out of reach for 20486 and wide open for 3927.