Why Threes

A cube that only knows one number

3927 is 2048 rebuilt in three dimensions and in a different arithmetic. The board is a 3×3×3 cube, twenty-seven cells, and you shove the whole cube along any of six directions. When three tiles of equal value collide, they don't double; they triple. A three becomes a nine, a nine becomes a twenty-seven, and onward: 81, 243, 729. The name itself counts up the first three tiles, 3·9·27, and everything about the design is threes all the way down: three per axis, three to a merge, three as the growth factor.⁴

Its ancestor is stubbornly binary. Classic 2048 deals in powers of two, 2, 4, 8, 16…, and merges tiles in pairs. Two is the number of the coin flip, the transistor, the bit. So the natural question a curious player asks is whether swapping two for three is mere novelty, or whether three was hiding a real advantage all along. It turns out mathematics has an opinion, and the opinion is surprisingly firm.

The cost of counting

Imagine you must build a machine that stores numbers. Each digit position costs you something, a wheel, a wire, a tube, and that cost scales with the base, because a base-b digit needs b distinguishable states. But a larger base also means you need fewer digits to reach any given number. There's a tension here, and it has a name: radix economy.

Formally, the economy of representing a number N in base b is the base times the digit count, E(b,N) = b ⌊log_b N + 1⌋.¹ A big base buys you short numbers but expensive digits; a small base is cheap per digit but sprawls. Somewhere between the two extremes sits a sweet spot, and it is not where most of our technology landed.

The optimal base is irrational

For large N the economy behaves like (b ÷ ln b) · ln N, so the efficiency of a base rests entirely on the factor b ÷ ln b. Minimize that with a little calculus and the answer is exact and strange: the most economical base of all is e, roughly 2.71828.¹ You cannot build a computer with 2.718 states per digit, of course. But you can ask which integer sits closest to that ideal, and the winner, beating both two and ten, is three.

The gap is not academic. Base three lands within half a percent of the theoretical optimum, while binary pays a six-percent surcharge and decimal a crushing sixty. A concrete way to feel it: to write every number from 0 up to 999,999, binary needs 20 digits, decimal needs 6, but ternary needs only 13, and 13×3 is a smaller total than 20×2.¹ Three genuinely packs more counting into less structure.

Nature was asked to name the most efficient way to count, and it answered with an irrational number. The nearest whole number to that answer is three.

The machine that believed it

This is not just a curiosity of the textbooks. In 1958, at Moscow State University, a team led by Nikolay Brusentsov and the mathematician Sergei Sobolev built the Setun, the most famous ternary computer ever made.³ Where the West was committing to bits, Setun counted in trits, and it used a particularly beautiful scheme: balanced ternary, whose three digit values are not 0, 1, 2 but −1, 0, +1.²

The symmetry pays off immediately. A number carries its own sign, no separate sign bit, no two's-complement contortions, and to negate any value you simply flip every digit. Rounding and truncation become the same operation. Donald Knuth judged it "perhaps the prettiest number system of all."² Brusentsov, lacking transistors, coaxed three stable states out of paired ferrite cores; roughly fifty Setuns were built, most sent to teach mathematics across the Soviet universities.³ The machine worked, and it was cheap, and it was quietly ahead of its neighbors.

Is threes the more elegant game?

Here is where a careful writer must resist himself. Radix economy is a claim about storing and processing numbers under specific cost assumptions, and even mathematicians note those assumptions are only approximate, a fuller accounting often re-justifies binary, which is why the world runs on it.¹ A merge puzzle is not a memory bank; nobody is minimizing tube counts on a coffee break. So 3927 is not "more efficient" in any load-bearing engineering sense, and it would be a cheat to say so.

And yet. The game inherits something true. Growth by tripling reaches large numbers in fewer, more consequential moves; each merge does more work, and a 3×3×3 cube is the smallest board that lets three-way fusion breathe in every direction.⁴ The design didn't stumble onto three, it landed on the integer that the mathematics of counting has quietly preferred for a century, the one Setun's engineers chose on purpose. Two is the number we defaulted to. Three is the number we'd have chosen if efficiency, and not the coin flip, had the final say. That the tiles happen to spell 3, 9, 27 is a small joke the universe was already telling.