Section 29.1

Measuring

Water jugs, burning ropes and hourglasses.

Toolkit

These puzzles exploit timing and capacity invariants rather than probability: track exact quantities (sand remaining, rope burnt, water held) through a sequence of discrete, reversible actions, and choose actions that create a known, reusable residual.

Key facts:

Burning a non-uniform rope/fuse end-to-end takes a fixed time $T$ regardless of burn-rate variation; lighting both ends simultaneously burns it out in exactly $T/2$ , since the two flame fronts’ combined progress always covers the whole rope.
An hourglass of total time $T$ can mark any instant by flipping it precisely when it empties, or by starting two glasses together and using the first one to empty as a fresh zero-point.
Two jugs of integer capacities $a,b$ (coprime) can produce any integer amount $0\le n\le a+b$ via fill/empty/pour moves, since the reachable quantities are exactly $\{xa-yb \bmod \text{capacities}\}$ ; by Bézout, $\gcd(a,b)=1$ guarantees reachability of 1 unit, and comparing the “fill $a$ , pour into $b$ ” cycle against its mirror image gives the minimal move count.
With a third vessel of known fixed content (e.g. a barrel), use it as an extra capacity: repeated fill-and-pour cycles between it and the jugs strand a target residual determined by solving $ka - \ell b \equiv r$ for small $k,\ell$ .

Problems