Things that cannot be done — page 12
Discrete
A search for room that never ends
The textbook way to find the most that a network can carry is to find any route that still has room, push as much along it as it will take, and repeat until no route has room left. With whole-number capacities that must stop. Give one road a capacity of 0.618…, the golden ratio's reciprocal, and choose the routes badly, and it never stops: every push is positive, the flow converges, and it converges to 4.236 when the network can carry 5. The same choice made carelessly in floating point does stop — because of rounding, at the wrong answer, convinced it has finished.
Logic
The plane cut by a list
The continuum hypothesis looks like a statement about sizes: no infinity sits strictly between the whole numbers and the line. Wacław Sierpiński found in 1919 that it is equally a statement about the plane. It is true exactly when the plane splits into two pieces, one meeting every vertical line in only countably many points and the other meeting every horizontal line in only countably many. The split is drawn by listing the line's points in a particular order, and with the rationals in place of the reals it can be drawn exactly — a set that, counted column by column, has no area, and counted row by row, has all of it.
Number
How close two sums of square roots come
Two different sums of square roots of whole numbers are never equal: square roots of different squarefree numbers are independent, so no cancellation can make them agree. But they can come very close. Among sums of two square roots of numbers up to 320, √88 + √289 and √171 + √177 agree to eight decimal places. How many digits must be computed, in the worst case, to tell two such sums apart is not known — the best proof allows a number of digits doubly exponential in the length of the sums, the examples suggest only a modest polynomial, and the gap between them is why nobody knows whether the shortest route through points in the plane can be checked quickly.