Seven Answers to a Shakuntala Devi's Puzzle

In the Shakuntala Devi's book entitled "Puzzles to Puzzle You"; puzzle no. 104 is "At a certain festivity a rich man decided to distribute free rice to deserving people. He had altogether 100 Kg rice and he wanted to distribute the grain to 100 people in such manner that each old person received 3 Kg, each young person 2 kg and each child ½ Kg. How many old persons, young persons and children were there?"

Solution: 

Suppose,  

 i) no. of old persons     = X

ii) no. of young persons = Y

iii) no. of children           = Z

Then            X + Y + Z   = 100     -------   (1) (Given)

 i.e.                           Z   = 100 - X - Y   ---- (2)

We have, 3X + 2Y + ½ (100 - X - Y) = 100 Kg     (Given)      

i.e. 6X + 4Y + 100 - X - Y = 200 

i.e. 5X + 3Y = 100   ---- (3)  

It seems, it is impossible to get the values of X and Y as there is only one equation and two variables. But one inference of the puzzle that X and Y are whole numbers helps to get the solution.

Let us assume the extreme case that Y = 0; then 5X = 100; i.e. X = 20 and Z = 80.

On observation of equation (3), it is concluded that if we subtract successively multiples of 3 (i.e. 3, 6, 9, 12, 15 and 18) from the value of X obtained above, and add respectively multiples of 5 (i.e. 5, 10, 15, 20, 25 and 30) to the value of Y assumed above; the equation (3) remains true to certain limit. The data can be tabulated as follows:-

S.No.   5X      + 3Y       = 100; Z (= 100 - X - Y)

1)       5 * 20 + 3 * 0   = 100; 80 

2)       5 * 17 + 3 * 5   = 100; 78

3)       5 * 14 + 3 * 10 = 100; 76 

4)       5 * 11 + 3 * 15 = 100; 74 

5)       5 * 8   + 3 * 20 = 100; 72 

6)       5 * 5   + 3 * 25 = 100; 70 

7)       5 * 2   + 3 * 30 = 100; 68 

Thus, all the possible answers of the puzzle are seven as follows.

S.No.   X;  Y; Z 

1)       20;  0; 80 

2)       17;  5; 78

3)       14; 10; 76 

4)       11; 15; 74 

5)         8; 20; 72 

6)         5; 25; 70 

7)         2; 30; 68 

Please note: (i) Our personal record shows that the values 14, 10, 76 were obtained by my son Lalit [now Dr. Lalit Patil, M. Tech., IIT, Mumbai, Ph. D.  Michigan.], in the year 1990, when he was a 10th standard student. But I do not know how he got it. (ii) 5, 25, 70 are given in the book, "Puzzles to Puzzle You" by Shakuntala Devi, Orient Paperbacks, Delhi Publication , 1989. (iii) 11, 15, 74 are given on internet in “PLZ solve this Aptitude Question – ChetanaSforum”