November 29, 2025
Pigeons vs averages: FIGHT!
The undeserved status of the pigeon-hole principle (EWD 1094)
Math legend dunks on “pigeonhole”; commenters squawk over pigeons vs averages
TLDR: Dijkstra says the pigeonhole trick is overhyped and should be framed as “the maximum is at least the average.” Commenters split between loving the clean math and defending the bird metaphor as easier for beginners, sparking a side debate on proof styles and teaching clarity.
In a spicy blast from computer-science icon Edsger Dijkstra, the essay claims the famous “pigeonhole principle” gets too much awe and should be rephrased as the simple idea that the maximum is at least the average. He calls the pigeon/box metaphor “noise” and says the generalized version is stronger. Cue the comments section turning into a coop fight. One camp cheered the clean, math-first framing—“ditch the bird cages, keep the numbers”—while others accused Dijkstra of professional blindness, saying regular people grasp pigeons way faster than averages. One reader even admitted, “I thought this was parody,” arguing the “bad” pigeon version is actually easier to understand.
Then came the philosophy squad: a bold take declared the pigeonhole principle “just the Law of Excluded Middle with extra slots”—that’s the idea that something is either in one category or another—adding that proofs by contradiction aren’t chic. Another thread rallied behind constructive proofs, urging folks to cut double negatives and make arguments feel more hands-on, even if reductio ad absurdum (proving the opposite is absurd) helps. Meanwhile, jokesters asked if pigeons were unionizing, memed “Average vs Max: Cage Match,” and turned Dijkstra’s German vibe—“Schubfach prinzip”—into a posh branding exercise. The German Soccer Lotto cameo? Commenters called it the bonus level of bird math.
Key Points
- •The article criticizes traditional pigeon-hole formulations as overspecific and metaphor-heavy, obscuring the principle’s arithmetic nature.
- •It proposes a generalized formulation: in a finite set of real numbers, the maximum is at least the average and the minimum is at most the average.
- •The article highlights logical equivalence between the traditional statement and its contrapositive, cautioning against misleading visual interpretations.
- •It advises against reading inequalities as conceptually different when they state the same relationship (e.g., average ≤ maximum vs. maximum ≥ average).
- •The generalized formulation is presented as strictly stronger and more broadly applicable, with a mention of the German Soccer Lotto problem as an example.