#P13036. [GCJ 2021 #2] Matrygons
[GCJ 2021 #2] Matrygons
题目描述
A matryoshka is a type of doll that originated in Russia over a century ago. Their defining characteristic is that they consist of a set of dolls, all of a different size, with smaller dolls fitting nicely inside larger dolls.
In this problem, we work with matrygons, which are sets of regular convex polygons that follow a similar nesting pattern. A matrygon consists of a set of regular convex polygons with positive area such that, for all , the vertices of overlap with a proper subset of the vertices of ( has strictly less vertices than ).
For example, the following pictures illustrate two matrygons. The first one contains 3 regular convex polygons: a regular icositetragon (24 sides), a regular hexagon (6 sides), and an equilateral triangle (3 sides). The second one contains 2 regular convex polygons: a regular icosidigon (22 sides) and a regular hendecagon (11 sides). Each of these matrygons has 33 total sides among all polygons in it.
Given a fixed total number of sides , calculate the largest number of polygons that can be part of a matrygon such that the total number of sides among all polygons in it is exactly .
输入格式
The first line of the input gives the number of test cases, . lines follow. Each line represents a test case and contains a single integer , the target total number of sides.
输出格式
For each test case, output one line containing Case #x: y
, where is the test case number (starting from 1) and is the maximum number of polygons in a matrygon such that the total number of sides among all polygons in it is exactly .
3
33
15
41
Case #1: 3
Case #2: 2
Case #3: 1
提示
Sample Explanation
The first matrygon pictured in the problem statement is an optimal solution for Sample Case #1.
In Sample Case #2, we can get to two polygons by fitting a regular pentagon (5 sides) inside a regular decagon (10 sides).
In Sample Case #3, there is no way to create a matrygon with multiple regular polygons, so our only option is to use a single regular tetracontahenagon (41 sides).
Limits
- .
Test Set 1 (7 Pts, Visible Verdict)
- Time limit: 20 seconds.
- .
Test Set 2 (13 Pts, Visible Verdict)
- Time limit: 40 seconds.
- .