One-Button Flyer

Click, tap or press space to flap. Every level is generated inside what the bird can physically reach, so any run you fail was flyable. Everything runs in your browser; nothing is uploaded.

Climbing is linear, falling is quadratic

Between one pipe and the next the bird has a fixed number of seconds to change height, and the two directions are not alike. Tapping every frame holds it at the impulse speed, so the climb grows in proportion to the time. Doing nothing lets gravity accelerate, so the drop grows with the square of it.

So there is a crossover. Below it the bird can rise further than it can drop; above it, falling pulls ahead and keeps pulling ahead. At the standard settings that crossover is 0.428 seconds — and every realistic pipe spacing is past it:

Every row drops further than it climbs, by up to 5.8×. That is the asymmetry a level designer has to live with: a gap can be placed far below the last one and still be reachable, but the same distance above it may simply be out of range.

The textbook formula is not the one this game obeys

The obvious closed forms are impulse × t for the climb and ½ × gravity × t² for the drop. Run them against the actual game loop and the climb one is wrong by a constant 6.98% at every duration — which is the signature of a modelling error rather than a rounding one.

The cause is the order of two lines. The loop applies the tap and then adds that frame's gravity, so a bird tapping on every single frame rises at impulse − gravity/fps — 400 rather than 430. The shortfall is exactly one frame of gravity, which is why it is the same fraction at every duration.

The forms this game actually obeys are (impulse − gravity/fps) × t for the climb and ½ × gravity × t × (t + 1/fps) for the drop. Checked against the loop across five different gravity, impulse and frame-rate settings, they agree to floating-point precision — worst disagreement 1.5e-12 pixels.

This is not pedantry — it decides whether levels are possible. A generator using the textbook climb places gaps up to 36 pixels beyond where the bird can get: levels that look reasonable, pass a glance, and cannot be flown. The generator here uses the discrete forms, and every level it makes was checked to be inside reach before you saw it.

The same correction moves the crossover: the continuous formula puts it at 0.478s, the discrete one at 0.428s — 50 milliseconds apart, and the loop lands on the discrete answer to within a single frame.