BdMO National 2013: Junior 6

[BdMO]

You have $1, 2, 3, 4, 5, 6, 7, 8, 9$ kg weights in your home. You have only one piece of each of the weights. You also have a balance. If you put a weight on the left side of that balance, its weight becomes twice. Every time you choose three weights to put on the balance in such a manner that the balance remains in equilibrium. On every turn, the total weight the balance can carry gets reduced by $3$ kg. At the first time the balance may carry at most $20$ kg in total. In how many ways can you keep weights on balance?

[Kiriti]

এইটার উত্তর mathematically কিভাবে বের করবো ?? গুনে গুনে না ।

[Labib]

If you are looking for some hints,

If you think logically, you will realize that you do not need to use brute force for this problem. (In fact, you cannot. There are $2*9*7*4=504$ choices for the 3 weights.)
case 1: You put 1 weight on the left.
Let, $w_L, w_{R_1}$ and $w_{R_2}$ be the three weights you chose at first. Here, $w_L$ is the weight you put on the left.
According to the statement,
$2w_L = w_{R_1} + w_{R_2}$
and,
$w_L + w_{R_1} + w_{R_2} \leq 20$
$\Rightarrow w_L \leq 6$

case 2: You put 2 weight on the left.
Let, $w_R, w_{L_1}$ and $w_{L_2}$ be the three weights you chose at first. Here, $w_R$ is the weight you put on the right.
$2(w_{L_1} + w_{L_2}) = w_R$
and,
$w_R + w_{L_1} + w_{L_2} \leq 20$
$\Rightarrow w_{L_1} + w_{L_2} \leq 6$

Use these conditions to your favour. Think about the possible values of the next $w_L$ and $w_R$.

*edited to add a missed out case. (thanks to Mahi)

[*Mahi*]

According to the statement,
\[2w_L = w_{R_1} + w_{R_2}\]

The question allows $2(w_{L_1}+w_{L_2}) = w_R$ too. Did you count those?

[Labib]

According to the statement,
\[2w_L = w_{R_1} + w_{R_2}\]

The question allows $2(w_{L_1}+w_{L_2}) = w_R$ too. Did you count those?

Missed out on those I guess.
I'll fix it anyway.

[atiab jobayer]

Vai Eku sohoj basai lekhen

[Labib]

I'm not sure if I can make it sound any easier than this Atiab.
I'd leave it to someone else who could come up with an easier solution.
Meanwhile, I'd give you a heads up. Please do not use "Banglish" in the forum. (I believe it is prohibited)
If you wish to right bangla, please use Avro/some other software.

[atiab jobayer]

I want to know what is the final answer

[asif e elahi]

I think the answer is 1.I solved it by brute force.

[Fatin Farhan]

আমারও ১ আসছিল। গুনে গুনে করসিলাম