Among the eight questions raised in the second mathematics test, faced at the 2026 Maturity by the students of the scientific high school, there is a famous fair game: Cover the spot, in Italian “the game of the red circle”.
The rules are very simple: there is a large red circle and five smaller discs, the player must drop the 5 small circles so as to completely cover the large one. Behind the very simple appearance, in reality the solution is not so immediate, because the dimensions of the circles are such that if you move even slightly from the only possible solution, the large circle is discovered.
The question of this year’s second Maturity test is a modified version of “Cover the Spot” in which you are asked to position 3 circles so as to entirely cover the surface of a square.
Cover the spot: the original game and the solution
The rules of Cover the Spot, one of the oldest and most profitable attractions at American and Canadian fairs, are elementary. A large circle, usually red, is painted on a table. The player receives five smaller blue discs and must drop them onto the circle until they cover it completely.
The diameter of the five discs, compared to that of the large circle, is calculated to the millimeter to leave zero margin for error. In fact, there is a single geometric configuration capable of covering the entire surface, every minimal movement of the first disk leaves a glimmer of red uncovered on the other side, and is therefore lost.
The solution everyone tries (and why it’s not enough)
Faced with the problem, almost everyone tries the same strategy, the one that seems more “orderly”: a disk in the center of the large circle and the other four arranged symmetrically around it, one for each cardinal side, like the arms of a cross.
It’s a reasonable intuition, but it’s wrong. This arrangement always leaves four “slices” of the red circle uncovered.
The real solution is part of the Five Disk Problem, a mathematical problem in which we ask ourselves what the minimum radius of the 5 small circles must be so that they can cover the large circle by arranging them symmetrically on a regular pentagon. The solution for “Cover the spot” starts from this idea, but varies the arrangement slightly, in order to optimize the spaces to the maximum even in an asymmetrical and irregular manner.
Below we show you how to arrange the circles one by one to solve the “Cover the spot” game.
The question of the second Maturity test: a variant of Cover the spot
Question number 1 of the second test of the 2026 Maturity exam tells of a small challenge between two students, Cecilia and Nicolò, who play a variant of Cover the spot: instead of a large circle to be covered with five discs, here there is a square with a side of √2 dm2 to be covered with three circles, each with a radius of 2/3 dm.
The question asks this: Cecilia positions the first of the three circles in a precise point, with the center on the diagonal of the square and at a distance from the vertex such that the edge of the circle passes exactly through that vertex. Without yet positioning the other two circles, Cecilia states only one thing: with this first circle she has already covered more than half the area of the square. Nicolò disagrees. The question of the question is simple: which of the two is right?
As visible in the image below, the square has an area of 2 dm2the circle of radius 2/3 dm has an area of 1.396 dm2. Putting the circle in that position, the area that “overhangs” (the one in green in the figure) is approximately 0.254 dm2 so the one that remains covered is 1.142 dm2. Since half of the area of the circle is exactly 1, in this way it is actually possible to cover more than half of the square. Cecilia is therefore right.
