MIXTURES - Mixtures

Harry Potter has n mixtures in front of him, arranged in a row. Each mixture has one of 100 different colors (colors have numbers from 0 to 99).

He wants to mix all these mixtures together. At each step, he is going to take two mixtures that stand next to each other and mix them together, and put the resulting mixture in their place.

When mixing two mixtures of colors a and b, the resulting mixture will have the color (a+b) mod 100.

Also, there will be some smoke in the process. The amount of smoke generated when mixing two mixtures of colors a and b is a*b.

Find out what is the minimum amount of smoke that Harry can get when mixing all the mixtures together.


There will be a number of test cases in the input.

The first line of each test case will contain n, the number of mixtures, 1 <= n <= 100.

The second line will contain n integers between 0 and 99 - the initial colors of the mixtures.


For each test case, output the minimum amount of smoke.


18 19
40 60 20


In the second test case, there are two possibilities:

  • first mix 40 and 60 (smoke: 2400), getting 0, then mix 0 and 20 (smoke: 0); total amount of smoke is 2400
  • first mix 60 and 20 (smoke: 1200), getting 80, then mix 40 and 80 (smoke: 3200); total amount of smoke is 4400

The first scenario is a much better way to proceed.

hide comments
kushal_singh: 2023-02-24 11:38:37

great question!!!

fizz_05: 2023-01-20 06:25:17

awesome problem.
learn a lot.
notice. if you try to solve this problem. you must have deep knoweldge of MCM.
the question is how MCM works.
thank you very much the auther.

old_monk1: 2022-07-31 21:32:43

For C++ users getting RTE, don't use -> ios_base::sync_with_stdio(false) as well as cin.tie(NULL)

cloverr20: 2022-04-07 20:58:59

@tommy_trash dude your comment saved me, thanks a lot!!

siba_tripathy: 2021-12-25 19:44:12

how to input no of testcases is not clearly mentioned in question. use while(cin>>n);

yasser1110: 2021-07-16 11:29:04

Maybe its more useful to think of this problem as a divide and conquer recursion problem.

Waseem Ahmed: 2021-07-04 04:32:15

Got an AC only after
1) Thoroughly reading user comments.
2) Following their advice (learnt Matrix Chain Multiplication thoroughly before attempting to solve this problem. And then learnt DP - https://www.quora.com/Are-there-any-good-resources-or-tutorials-for-dynamic-programming-DP-besides-the-TopCoder-tutorial/answer/Michal-Danilák)
3) Solving the problem for a few test cases using pen and paper to understand the problem
4) Modifying the code given on https://www.geeksforgeeks.org/ slightly to fit this problem

I solved the problem without using DP first (got an expected TLE) and then modified it using DP to get an AC.

Lovely problem!!! Learnt a lot.

Last edit: 2022-06-01 07:21:32
atultyagi: 2021-05-27 23:48:36

Modified version of Matrix Chain multiplication, good problem but keep in mind ,number of test cases are not given use while(cin >> n )

Last edit: 2021-05-27 23:49:05
anurag_20: 2021-02-21 06:46:19

Awesome use of Matrix chain multiplication concept

shubham7811: 2021-01-23 23:50:01

ikd why it is showing worng answer
solved by matrix chain multiplication still wrong answer

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