The chess clock

A chess clock is really two clocks in one box. Each player has a button on their side, and only one of the two clocks runs at a time.

Two clocks in one box; each player has a button; only $1$ of the $2$ clocks runs at a time.

Buttons: $2$; clock running: $1$.

When you make a move, you press your button. Your clock stops and your opponent's clock starts. So the clock does not measure the game; it measures how long each player thinks.

Move $\Rightarrow$ press: your clock stops, your opponent's starts. The box measures each player's thinking time, not the game.

Move: clock $A$ stops, clock $B$ starts.

Say each player gets ten minutes, and every move adds five seconds back. A player who thinks for twenty seconds on every move loses fifteen seconds each time. Ten minutes is six hundred seconds, so that player's time runs out after forty moves.

Each player: $10$ min $=600$ s, plus $5$ s per move. Thinking $20$ s per move: net $5-20=-15$ s, so after $600÷15=40$ moves the time is gone.

Time $T$, bonus $i$, thinking $c$ (s): $T=10\times 60=600,i=5,c=20$. Net $i-c=-15$, and $600÷15=40$ moves.

A player who moves in five seconds loses nothing at all and could play for ever. The clock rewards steady thinking more than hurrying.

Thinking $5$ s per move: $5-5=0$, nothing lost. Steady beats hurried.

$c=5$ gives $i-c=0$.