What is a key feature of recreational mathematics problems?

Concept: Recreational mathematics is classified by how accessible a problem's statement is, not by the branch of mathematics it belongs to and not by the training its solver holds. A problem joins this family when it can be posed in ordinary language using only elementary ideas, so that a non-specialist grasps at once what is being asked. Nothing in that definition caps how hard the problem is to settle; the difficulty simply moves out of understanding the question and into answering it.

Application: Read the four described features against that definition. The Collatz problem is the standard illustration of the family: take any positive whole number, halve it when it is even, otherwise triple it and add one, and ask whether the process must always reach 1. A primary-school child can carry out the rule, yet the question has stood open since 1937. The four-colour problem was posed in 1852 in a single sentence about colouring a map and waited until 1976 for a proof. The bridges of Königsberg were posed as a walking route, and Euler settled them in 1736 only by inventing what became graph theory. In each case the statement costs one line, while the answer has cost a new branch of mathematics, more than a century of work, or a wait that is still running. That gap between the cost of asking and the cost of answering is exactly the feature described as easy to ask but difficult to answer.

Cross-check: The three remaining descriptions each break the definition at a different point.

• Immediately obvious solutions would describe a drill exercise: once the statement is read the work is over, so nothing is left to explore and the puzzle character disappears.

• Always involving number theory would confine the family to one branch, yet tangram dissections are geometry, magic and Latin squares are combinatorics, knights-and-knaves puzzles are logic, and the Königsberg bridges are graph theory.

• Requiring advanced mathematical training reverses the accessibility that defines the family; a problem whose statement can only be read after years of specialist study belongs to technical research mathematics, not to this tradition, even though research-level difficulty by itself is no bar.

The defining feature is therefore the gap between the cost of asking and the cost of answering: recreational mathematics problems are easy to ask but difficult to answer.