# QOJ

Memory Limit: 1024 MB Total points: 45

### Problem

Your task in this problem is to find out the minimum number of stones needed to place on an N-by-M rectangular grid (N horizontal line segments and M vertical line segments) to enclose at least K intersection points. An intersection point is enclosed if either of the following conditions is true:

1. A stone is placed at the point.
2. Starting from the point, we cannot trace a path along grid lines to reach an empty point on the grid border through empty intersection points only.

For example, to enclose 8 points on a 4x5 grid, we need at least 6 stones. One of many valid stone layouts is shown below. Enclosed points are marked with an "x".

### Input

The first line of the input gives the number of test cases, T. T lines follow. Each test case is a line of three integers: N M K.

### Output

For each test case, output one line containing "Case #x: y", where x is the test case number (starting from 1) and y is the minimum number of stones needed.

### Limits

Memory limit: 1 GB.

1 ≤ T ≤ 100.

1 ≤ N.

1 ≤ M.

1 ≤ KN × M.

#### Small dataset (15 Points)

Time limit: 60 10 seconds.

N × M ≤ 20.

#### Large dataset (30 Points)

Time limit: 120 20 seconds.

N × M ≤ 1000.

### Sample

2
4 5 8
3 5 11

Case #1: 6
Case #2: 8