Strange Cows - 1

Level 2

A farmer has $9$ cows and $3$ sons. All the cows in the farmer's farm are marked from $1$ to $9$ . Each cow gives that many Litres of milk (i.e the marked number). For e.g. Cow $6$ will give $6$ liters of milk each day. In the farmer's will, he mentions that each of his three sons should get an equal number of cows such that each of his sons receives the same amount of milk each day (i.e The sum of milk obtained from the cow should be equal).

What is the least sum of squares of milk amongst three brothers (i.e. if the son has $a, b, c$ cows, then what is $a^2 + b^2 + c^2$ ) ?

The answer is 83.

1 solution

Mahdi Raza
Apr 13, 2020

Sum of milk

$\begin{aligned} \text{Total milk } &= 1 + 2 + 3 \ldots + 8 + 9 \\ &= \frac{9(9+1)}{2} \\ &= 45 \\ \\ \text{Sum of Milk to one son } &= \frac{45}{3} \implies \boxed{15} \end{aligned}$

Using Magic square

Options to choose are: $1, 2, 3, 4, 5, 6, 7, 8, 9$ . Thus, a magic square can be made with dimensions $3 \times 3$ . This magic square can fulfill our conditions such that the sum of rows is equal. Surprisingly there is only 1 unique magic square of order 3. Thus we only need to account for two possibilities (row and coloumns).

$\begin{array} {|c|c|c|c|}\hline \\4&9&2 \\ \hline \\ 3&5&7 \\ \hline \\ 8&1&6 \\ \hline \end{array}$

We get two sets of possibilities: $\bigg( [4,9,2], [3,5,7],[8,1,6]\bigg)$ and $\bigg( [4,3,8], [9,5,1],[2,7,6]\bigg)$ accounting for row sum and coloumn sum respectively

$\begin{aligned} [4,9,2] &\implies 4^2 + 9^2 + 2^2 = 101 \\ [3,5,7] &\implies 3^2 + 5^2 + 7^2 = \color{#D61F06}{\boxed{83}} \\ [8,1,6] &\implies 8^2 + 1^2 + 6^2 = 101 \end{aligned}$

$\begin{aligned} [4,3.8] &\implies 4^2 +3^2 +8^2 = 89 \\ [9,5,1] &\implies 9^2 + 5^2 + 1^2 = 107 \\ [2,7,6] &\implies 2^2 + 7^2 + 6^2 = 89 \end{aligned}$

Maybe u wrote the sum wrong 😅 it's 83 right. Yeah magic square! Actually I also did a mistake I took sum of the squares of diagonal 4,5,6 which gives 77 but then u can't actually split cows properly then I realized that u have to go through the cols or rows😂.

Eswar Charan Bonda - 1 year, 1 month ago

How do you construct the magic square?

Prakash Arora - 1 year ago

There's an algorithm for it. decent one here

Mahdi Raza - 1 year ago

Thanks. I tried constructing 4X4 magic square. Got stuck at the condition i=n,j=-1. The video only mentions the condition i=-1,j=n. What do we do here?