TWENDS - Two Ends


In the two-player game “Two Ends”, an even number of cards is laid out in a row. On each card, face up, is written a positive integer. Players take turns removing a card from either end of the row and placing the card in their pile. The player whose cards add up to the highest number wins the game. Now one strategy is to simply pick the card at the end that is the largest — we’ll call this the greedy strategy. However, this is not always optimal, as the following example shows: (The first player would win if she would first pick the 3 instead of the 4.)

3 2 10 4

You are to determine exactly how bad the greedy strategy is for different games when the second player uses it but the first player is free to use any strategy she wishes.

Input

There will be multiple test cases. Each test case will be contained on one line. Each line will start with an even integer n followed by n positive integers. A value of n = 0 indicates end of input. You may assume that n is no more than 1000. Furthermore, you may assume that the sum of the numbers in the list does not exceed 1,000,000.

Output

For each test case you should print one line of output of the form:

In game m, the greedy strategy might lose by as many as p points.

where m is the number of the game (starting at game 1) and p is the maximum possible difference between the first player’s score and second player’s score when the second player uses the greedy strategy. When employing the greedy strategy, always take the larger end. If there is a tie, remove the left end.

Example

Input:
4 3 2 10 4
8 1 2 3 4 5 6 7 8
8 2 2 1 5 3 8 7 3
0

Output:
In game 1, the greedy strategy might lose by as many as 7 points.
In game 2, the greedy strategy might lose by as many as 4 points.
In game 3, the greedy strategy might lose by as many as 5 points.

hide comments
Surendra : 2015-08-10 21:14:03

give any other test cases please .. All the test cases shows correct output . but in it shows wrong output

anshal dwivedi: 2015-08-07 18:21:26

AC in one go.. !..Nice DP for beginners like me..!

ram: 2015-08-02 15:04:32

Easy DP!!
My 50th on SPOJ :) :) !!

PRIBAN91: 2015-07-22 06:26:54

Very nice problem!

mayank: 2015-06-30 13:56:15

Height of stupidity, thinking that I memoized, but did not, and then clueless about TLEs :P -_-

ASHU: 2015-06-22 21:23:45

Is removing left end means
4 3 2 10 4
After removing left end
4 2 10 4
Or
4 10 4

Nitesh Tiwari: 2015-06-10 14:01:24

Isn't the result for the first test case should be 9, 16- (23-16) =9 ,Hows's it 7.
Also for the last test case I'm getting 9 instead of 4. Someone please varify how's that, 7 and 4.
Thanks.

:.Mohib.:: 2015-06-08 07:47:14

Learned a lot.... nice que!!

Rushikesh Potdar: 2015-05-30 16:42:58

is complexity is less than ( queries*O(N^2))

Last edit: 2015-05-30 16:44:27
forma: 2015-04-14 05:22:44

Could anyone explain why in a case of tie we should always remove the left end ?
why not the right!


Added by:Camilo Andrés Varela León
Date:2007-07-26
Time limit:0.100s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ERL JS-RHINO NODEJS PERL6 VB.NET
Resource:East Central North America 2005