### PROBLEM LINK:

#### Author:

Jaiwant Rawat

#### Tester:

Ankit Sultana

#### DIFFICULTY:

HARD

#### PREREQUISITES:

Dynamic Programming

#### EXPLANATION:

For solving it we will keep track of state as ( a , b ).

In the state ( a , b )

- a - denotes the sum
- b - denotes the sum formed from only using the first b numbers from the interval [ k1 , k2 ]

where k1 = minimum value of the start of all the intervals

k2 = maximum value of the end of all the intervals

### Example -

Taking the test case given in question

2

0 2

2 5

1 2

Here we have two intervals [ 0 , 2 ] & [ 2 , 5 ] and we have to find the sum from 1 to 3 .

so following the above approach we have

k1 = 0 ( min of ( 0 , 2 ) )

k2 = 5 ( max of ( 2 , 5 ) )

Now all the values that i can take will lie in this interval only and how many times i can take them is equal to the number of intervals in which that

value lies . Now according to the question k1 minimum value can be 0 and k2 maximum value can be 500 so i can have only 501 distinct numbers for consideration

in worst case

Now the state is ( a , b ) where

a is one of the number belonging to the interval [ N , N + x ]

b is one of the number belonging to interval in [ k1 , k2 ]

Now the transition of the states will be like ( a , b + 1 ) -> ( a , b ) if i am not using the b + 1 in my sum or ( a , b + 1 ) -> ( y , b ) where y

be one of depending on how many b i am considering for the sum .