Math buys you a smaller, easier-to-check rule, not automatic speed. That is the honest lesson of David Essien’s two C versions of Rock-Paper-Scissors. One spells out every matchup with branches. The other notices that the game is a cycle of three and encodes it as a 3×3 table or a single modular expression. The code gets shorter and easier to extend. The author’s own benchmarks show the speed ranking changing with compiler settings, and in a game that waits on a human, none of it is noticeable.
If you are a new CS student or a self-taught developer and you don’t quite buy that math matters day to day, this small example is a fair test case.
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The starting point: spelling out every case
The first implementation uses a switch that enumerates all nine pairings of two players’ choices. It works, and every case is easy to read in isolation. The cost is that correctness depends on you checking nine branches by eye, and a larger game multiplies the branches: Rock-Paper-Scissors-Lizard-Spock has 25 pairings.
A bug hiding in the starter loop
The article’s first example declares option without initializing it, then reads it in the loop condition before any input has been assigned. Reading an uninitialized local in C is undefined behavior, so the loop may start or skip depending on whatever happens to be in memory. Initialize it to a known value (or use a do { ... } while loop that reads input before testing). This is a correctness defect, not a harmless way to start a loop.
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How math helps: seeing the cycle
Number the moves Rock = 0, Paper = 1, Scissors = 2. Each pair then has exactly one of three outcomes: win, draw or loss. Because the game is a cycle, the outcome depends only on the relationship between the two numbers, not on nine unrelated facts.
Version 1: an outcome table
int rules_matrix[3][3] = {{0, -1, 1}, {1, 0, -1}, {-1, 1, 0}};The article encodes 1 as a win, 0 as a draw and -1 as a loss. Read it as rules_matrix[x][y], where the row x is the player whose outcome you want. Check row 0 (Rock): against Rock it is a draw (0), against Paper a loss (-1), against Scissors a win (1). Swap the meaning of row and column and every win becomes a loss, so document the convention next to the table.
Lookup is a single index, but only if both indices are within 0 to 2. Validate input before indexing; an out-of-range value reads outside the array.
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Version 2: a modular expression
The article also gives this formula for the same convention:
((x - y + 4) % 3) - 1Why it works: x - y is the distance around the cycle. The + 4 does two jobs. Adding 3 keeps the value non-negative (for inputs 0 to 2 the smallest value is 2), which sidesteps C’s negative-remainder behavior, and the extra 1 shifts the three residues 0, 1, 2 so that subtracting 1 gives -1, 0, 1. Spot checks against the matrix: (0,1) gives (3 % 3) – 1 = -1; (0,2) gives (2) – 1 = 1; (1,0) gives (5 % 3) – 1 = 1; (0,0) gives (4 % 3) – 1 = 0. All agree.
This only holds for inputs 0, 1, 2 under that ordering. It is not a general-purpose formula you can apply to arbitrary values without validation.
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Comparing the three approaches
| Axis | Switch | Table | Modular expression |
|---|---|---|---|
| Rule representation | Explicit cases | Indexed outcome data | Arithmetic |
| Easy to verify | Each case readable, nine to check | Nine numbers; row/column meaning must be known | Short, but needs the explanation above |
| Growing the gesture set | New branches for every pairing | Larger table, but you must still define every outcome | Only if the new game is also a clean cycle |
| Failure modes | Missing or wrong case | Out-of-range index; reversed convention | Out-of-range input; negative remainders if the offset is dropped |
The article describes extending the table and the option list instead of adding a case per rule. That is a real gain, but not free: a bigger table still needs correct rules, and the indexing scheme must still make sense. The modulo trick generalizes to odd-sized cycles where each move beats half the others, not to every rule set.
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What the benchmarks showed
All figures below are David Essien’s reported local results. The source does not state the machine or full compiler version, so treat them as one author’s measurements, not a claim about C, compilers or lookup tables in general.
Unoptimized build, switch versus table
Each function was called 100 million times, three runs each.
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| Version | Run times (s) | Mean of the three (my arithmetic) |
|---|---|---|
| Switch | 0.343620, 0.342246, 0.340185 | about 0.342 |
| Matrix | 0.275987, 0.272867, 0.272581 | about 0.274 |
The author summarizes this as roughly 0.7 ns per call, about 20%, and immediately notes that it is negligible in a game paced by a human.
With -O2
| Version | Run times (s) | Mean (my arithmetic) |
|---|---|---|
| Switch | 0.119248, 0.120751, 0.122600 | about 0.121 |
| Matrix | 0.133597, 0.128811, 0.132499 | about 0.132 |
The ranking flipped. In the author’s words: “The only thing I changed was adding the build flag, and switch went from consistently losing to consistently winning.”
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| Condition (ns per call) | Switch | Matrix | Modular |
|---|---|---|---|
| Forced inline | 1.293 | 1.339 | 1.261 |
| Real function calls | 2.261 | 1.697 | 1.793 |
The author says an AI helped write the harness for this comparison and does not try to explain the rankings beyond the evidence. Neither should you: the order changes with inlining, so no single winner emerges.
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So what does math actually buy you?
The author’s conclusion is that recognizing the modular structure gave a solution that was “simultaneously simpler to read, easier to extend, and measurably faster.” The first two hold up well. The third needs the caveat above: it was measurable in one unoptimized build and reversed or blurred in others. In a program that spends nearly all its time waiting for a keypress, a few nanoseconds per rules check change nothing the player can perceive.
The durable payoff is conceptual. Seeing the cycle turned nine special cases into one rule you can verify with four spot checks. When you want speed, measure under your real build flags and inlining conditions, and only after the code is correct, with initialized variables and validated indices.
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