Constraints, guarantees and flows

Decide what an entry represents

Before doing arithmetic, name the objects in the problem and what one mark on paper will mean. For a shift puzzle, a mark might pair one person with one evening. For a guarantee, a mark might be a parcel assigned to one of several lockers. For a flow, a mark could represent one litre passing a junction in a minute. The words “at least”, “exactly” and “per minute” change what operations are legitimate. Write these qualifiers alongside your sketch so that a neat arrangement on paper still answers the original question.

For a one-to-one assignment draw people down the side of a grid and time slots across its top. Cross out forbidden pairings and tick forced ones. Whenever a row has just one possible slot, tick it and cross out that slot in every other row. A column with a single possible person can be equally informative if every slot needs filling. If people can cover several slots, crossing out an entire row after one assignment would discard legal possibilities. Check the capacity rule before you use the grid.

Recognise what a guarantee demands

“Can happen” asks for one feasible arrangement. “Must happen” asks for an argument covering them all. Suppose objects are divided among boxes and you want a guarantee that one box contains three objects. Deliberately avoid that outcome by allowing each box at most two. With four boxes you can place eight objects while still avoiding a triple. A ninth forces one box beyond two. This is the pigeonhole principle (principio dei cassetti). The balanced arrangement proves that eight is insufficient, and the capacity count proves that nine suffices.

A target can name a particular category instead. To guarantee a green token when drawing without replacement, a deliberately unlucky draw takes every non-green token first. Here the limiting arrangement is concentrated in the wrong categories, whereas in the box example it spreads the objects around. Describe the failure condition in words each time. Then count the largest arrangement compatible with that description. Check also that the inventory really contains enough objects for the proposed guarantee.

Add contributions in compatible units

If two pumps fill a tank together, convert each completion time into a rate. A pump that fills one tank in six minutes contributes one sixth of a tank per minute. Express the other pump in the same unit, add the fractions and ask how long that combined rate takes to deliver one whole tank. A drain takes water away, so its contribution has a negative sign. Write the direction next to each quantity before adding. If a pump runs only during part of the interval, split the job into phases and calculate the amount moved in each.

Routes can have limits too. Two independent pipes in parallel can both carry water to the destination at the same time. Two pipes in series carry the same stream one after the other. The narrower capacity limits a steady flow through both. At a junction with no storage, inflow must match outflow. Label each link separately and make sure that two seemingly different routes do not both claim the full capacity of a shared link. A drawing with arrows is enough to check where a shared segment begins.

Verify that your arrangement is attainable

When you have a candidate schedule, read each original clue against the completed grid. When you have a candidate maximum flow, describe an actual allocation of that many units along the routes. Showing that a limit cannot be exceeded only gives an upper bound. Showing a feasible allocation as large as that upper bound establishes the maximum. For a minimum guarantee the corresponding pair of checks is an arrangement just below your answer that fails, and a reason no arrangement at your answer can fail.