Problem A: Trains

Trains are cool. You can get pretty much anywhere in Europe on a train, including little towns. In Canada, train service is not quite as good. Sometimes you have to change trains many times, and it's often hard to know the quickest route and where to switch. Depending on the time of day, the route may even pass through completely different cities. Fortunately, you can write a program to help you. Given the schedules of trains running between different places, your task is to find all the shortest connections between two places. A shortest connection is one for which there is no other connection that would allow you to leave later and arrive at the same time or earlier, or leave at the same time and arrive earlier. Time to change trains need not be considered, as it is already built into the schedule.

The first line of input contains a positive integer N, the number of test cases. The first line of each test case contains T <= 20, the number of train routes. Each train route is described by one or more lines containing:

• S <= 20 The number of stations on the route including origin and terminus.
• hh:mm The starting time of the route (in 24-hour clock notation; that is, in the range 00:00 to 23:59). You may assume that a train leaves the origin every day at this time. A station name is a string of no more than 40 alphabetic characters.
• A list of the names of the S stations, separated by the travel time between adjacent stations. Travel time is given in hours and minutes.

Finally, each test case gives the name of an origin and destination for which a schedule is to be generated. You may assume there is at least one route from the specified origin to the specified destination. The output consists of a list of shortest connection departure and travel times from the origin to the destination, ordered by departure time. Each combination of departure and travel time should be listed only once, even if there are multiple routings yielding the same connection times. The departure time should be given as hh:mm (that is, exactly 2 digits each). Travel time on the other hand should be given as h:mm, hh:mm, hhh:mm, etc., as appropriate to avoid unnecessary leading 0's. Leave an empty line between the output for successive test cases.

For example, the first route in the sample input begins in Windsor at 08:00 a.m., arrives in London at 09:55, in Kitchener at 11:30, Guelph at 12:25, Toronto at 13:30 and Montreal at 18:20. To get from Waterloo to Toronto we can leave at 07:00 and travel direct for a total time of 1:45. Or we can leave at 08:00 and travel to Kitchener, then to Guelph and Toronto, arriving at 13:30 for a travel time of 5:30. Or we can depart Waterloo at 09:00 for Hamilton, Niagara, and Toronto, arriving at 14:00. Finally, we can depart Waterloo at 23:00 and arrive in Toronto at 07:05 the next morning, for a travel time of 8:05.

Problem B: Tic Tac Toe

Tic Tac Toe is a child's game played on a 3 by 3 grid. One player, X, starts by placing an X at an unoccupied grid position. Then the other player, O, places an O at an unoccupied grid position. Play alternates between X and O until the grid is filled or one player's symbols occupy an entire line (vertical, horizontal, or diagonal) in the grid.

We will denote the initial empty Tic Tac Toe grid with nine dots. Whenever X or O plays we fill in an X or an O in the appropriate position. The example below illustrates each grid configuration from the beginning to the end of a game in which X wins.

...  X..  X.O  X.O  X.O  X.O  X.O  X.O
...  ...  ...  ...  .O.  .O.  OO.  OO.
...  ...  ...  ..X  ..X  X.X  X.X  XXX

Your job is to read a grid and to determine whether or not it could possibly be part of a valid Tic Tac Toe game. That is, is there a series of plays that can yield this grid somewhere between the start and end of the game?

The first line of input contains N, the number of test cases. 4N-1 lines follow, specifying N grid configurations separated by empty lines. For each case print "yes" or "no" on a line by itself, indicating whether or not the configuration could be part of a Tic Tac Toe game.

Problem C: Square

Given a set of sticks of various lengths, is it possible to join them end-to-end to form a square?

The first line of input contains N, the number of test cases. Each test case begins with an integer 4 <= M <= 20, the number of sticks. M integers follow; each gives the length of a stick - an integer between 1 and 10,000.

For each case, output a line containing "yes" if is is possible to form a square; otherwise output "no".

Problem D: Blocks

Donald wishes to send a gift to his new nephew, Fooey. Donald is a bit of a traditionalist, so he has chosen to send a set of N classic baby blocks. Each block is a cube, 1 inch by 1 inch by 1 inch. Donald wants to stack the blocks together into a rectangular solid and wrap them all up in brown paper for shipping. How much brown paper does Donald need?

The first line of input contains C, the number of test cases. For each case there is an additional line containing N, the number of blocks to be shipped. N does not exceed 1000. Your program should produce one line of output per case, giving the minimal area of paper (in square inches) needed to wrap the blocks when they are stacked together.

Problem E: Faucet Flow

A faucet is pouring water into a long, thin aquarium with various vertical dividers (walls) in it. The aquarium is initially empty, and its bottom is perfectly level. How long will it take for water to spill over its left- or right-most divider? The faucet is above location x=0, and the dividers are located at x=-1, -3, -5, ..., leftx and 1, 3, 5, ..., rightx. The dividers are attached perpendicular to the floor and sides of the aquarium, and have various heights. The aquarium's length is greater than rightx-leftx, its walls are higher than the highest divider, and its width is 1 unit everywhere. Water pours from the faucet at a rate of 1 cubic unit per second. [You may assume that water is an ideal liquid: it always flows downhill and if it cannot flow downhill it spreads at an equal rate in all horizontal directions.]