Puzzle | 100 Cows And Milk

A milkman has 100 cows numbered from 1 to 100. Every cow gives milk according to its number, i.e i'th cow gives i litres of milk. Milkman has 10 sons, and he wants to divide his cows among his 10 sons so that every son gets an equal amount of milk.

The task is to help him know about the division of these cows among the sons.

Check if you were right - full answer with solution below.

Solution: 

As we know that the i'th cow gives i litres of milk.

So first, let us count the total litres of milk which is obtained from these cows. This can be solved by using the arithmetic progression sum.

We know that the sum of n numbers starting from 1 is always sum = n*(n+1)/2.

So the total quantity of milk obtained from 100 cows is total milk = 100(100+1)/2 = 5050 litres.

As the problem says that the division of the cows should be done in a manner that every son gets an amount of milk.

So every son should get 5050/10 = 505 litres of milk.

Now our main problem is to divide 1 to 100 numbers in such a way that every son should get 10 cows whose numbers sum up to 505.

Let us reduce the complexity of this puzzle using the example below:

Now, let us suppose you have 10 numbers and you have to divide them among 5 people so that each one gets an equal sum of these numbers.

So these 10 numbers sum up to 55, now each of the 5 people will get 11 as a sum, so it's obvious that every person will get two numbers whose sum is 11. So the division would be like this:

First_person ---> 1, 10
Second_person ---> 2, 9
Third_person ---> 3, 8
Fourth_person ---> 4, 7
Fifth_person ---> 5, 6 so each person will get 11 .

Now coming back to the problem, we have 100 numbers we have to divide this in 10 groups so that each son get 505 litre of milk. So,

First_son ---> 1, 2, 3, 4, 5, 96, 97, 98, 99, 100
Second_son ---> 6, 7, 8, 9, 10, 91, 92, 93, 94, 95
Third_son ---> 11, 12, 13, 14, 15, 86, 87, 88, 89, 90
Fourth_son ---> 16, 17, 18, 19, 20, 81, 82, 83, 84, 85
Fifth_son ---> 21, 22, 23, 24, 25, 76, 77, 78, 79, 80
Sixth_son ---> 26, 27, 28, 29, 30, 71, 72, 73, 74, 75
Seventh_son ---> 31, 32, 33, 34, 35, 66, 67, 68, 69, 70
Eighth_son ---> 36, 37, 38, 39, 40, 61, 62, 63, 64, 65
Ninth_son ---> 41, 42, 43, 44, 45, 56, 57, 58, 59, 60
Tenth_son ---> 46, 47, 48, 49, 50, 51, 52, 53, 54, 55

as the problem said each son should get equal amount of milk.