Why does Greenland look enormous on some maps? And why can the shortest route between two cities appear curved? Turning a round planet into a flat surface creates an impossible problem: something has to be distorted. This chapter explores the mathematics of map projections, from Mercator to maps designed to preserve area rather than shape. Distances, directions and sizes change depending on what each projection chooses to protect. A flat map cannot preserve every geometric truth at once. Before asking which map is more accurate, there is a more important question to answer: accurate for what?

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You find a pretty good option, but one dangerous question appears: what if the next one is better? This chapter explores the mathematics of knowing when to stop searching. Through the secretary problem, we discover why choosing too early can be a mistake, but waiting too long can be just as costly. The famous approximate 37 percent rule offers an elegant solution under very specific assumptions, although real life quickly complicates them. Apartments, jobs and other choices reveal that searching longer does not always mean choosing better. Sometimes the real mathematical problem is not finding options, but knowing when you have seen enough.

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You press the elevator button, wait a few seconds, and press it again just in case. Why does it seem to arrive instantly when you do not need it and take forever when you are in a hurry? This chapter explores the mathematics of waiting and a counterintuitive idea: when arrival intervals are irregular, we are more likely to show up during the longer ones. Elevators, buses and trains reveal how the same average frequency can hide very different waiting experiences. Sometimes we are not unusually unlucky. We have simply arrived at random in a system where long intervals have more chances to find us.

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