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Learning Opportunities

This puzzle can be solved using the following concepts. Practice using these concepts and improve your skills.



Two rooms are connected by a service duct allowing floor-sweeping bots to move between the rooms.
The duct is very narrow, just wide enough for one bot to pass through.

The bots do not have visual, lazer, radar or anything to detect obstacles. Their only sensory is touching. If they meet in the duct, they will bump into each other. Collision will cause each of them to turn around to continue moving in the opposition direction.

Bob finds that it is very interesting to watch the bots bumping and bouncing back and forth. He prepared a longer transparent duct and put more bots into it, wishing to watch a sequence of bumping and bouncing "bot-show". He designed some starting locations for the bots and then turned them all ON at the same time.

- assume the bots move at a constant speed of 1 meter/second,
- assume the bots have a diameter small enough to be negligible in calculation of travel distance,
- assume no bot will go from the rooms into the duct during "show-time",
- assume Bob is free to arrange every bot to face it either Left or Right before starting the show...

What is the longest possible time for the bots to run around in the duct until all bots exit?


In a duct of length 10m two bots move face-to-face

    >       <
+-+-+-+-+-+-+-+-+-+-+ init state
0 1 2 3 4 5 6 7 8 9 10

X after 2 sec
+-+-+-+-+-+-+-+-+-+-+ bump at 4, bounce
0 1 2 3 4 5 6 7 8 9 10

< > after 4 sec
+-+-+-+-+-+-+-+-+-+-+ the L bot is exiting
0 1 2 3 4 5 6 7 8 9 10

> after 2 sec
+-+-+-+-+-+-+-+-+-+-+ the R bot is exiting
0 1 2 3 4 5 6 7 8 9 10

Total bot running time: 8 sec
Line 1: An integer L for the length of the service duct, in meter
Line 2: An integer N for the number of bots
Line 3: N integers indicating the initial locations of the bots measured in meter from one end of the duct. Numbers are not arranged in special order.
Line 1: An integer, the maximum number of seconds for the bots to continue running in the duct before they all exit.

If the answer is not an integer, round it to become the nearest integer.
2 ≤ L ≤ 1000
1 ≤ N ≤ 100
2 6

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