# Twenty weights and a balance

On the table there are twenty weights with masses $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $11$, $12$, $13$, $14$, $15$, $16$, $17$, $18$, $19$ and $90$ grams, and a two-armed balance with two pans.

• Nine of the weights are made of bronze, nine are made of silver, one is made of gold, and one is made of iron.
• The iron weight is $90$ grams.
• If you put all bronze weights and the iron weight into one pan, and all silver weights into the other pan, then the balance is in perfect equilibrium.

Question: How many grams is the gold weight?

gold weight is

10 grams

because:

• if you remove 90 g from the last series (11 ,12 ... 19) it matches first series (1,2,....9)

Agawa001 has the right answer, but here's a fuller explanation.

Let $B_1,\dots,B_9,S_1,\dots,S_9,G$ be the weights of the bronze, silver, and gold weights respectively in grams (these numbers are 1 to 19 in some order).

We know the sum of all $B_i$ is at least $1+2+3+\dots+9=45$ (the sum of the nine lightest weights) while the sum of all $S_i$ is at most $11+12+13+\dots+19=135$ (the sum of the nine heaviest weights). So $\Sigma_i S_i\leq135\leq\Sigma_i B_i+90$. Since the left and right quantities in these inequalities are equal, we must have equality all the way through, which means the $B_i$ are $1,\dots,9$ in some order and the $S_i$ are $11,\dots,19$ in some order. The only possiblity left for $G$ is 10.

Sum up the iron weight (of 90 grams) with the nine smallest weights, i.e., 1 through 9.

This will give you 135 grams. So one side of the balance will have a weight of 135 grams.

So we need 135 grams in the second pan to balance both sides.

Now, sum 10 through 18. This will give you 126 grams. Then we still need 9 grams to balance.

Let's remove the 10 from this pan, and add the 19-gram weight instead of it. So, 126-10+19=135.

Both sides are balanced, and the 10-gram weight is left over.

So the gold weight is 10 grams.

10 grams

Iron is 90g

The two plates are:

90 + bronze(1,2,3,4,5,6,7,8,9) = 135g

silver(11,12,13,14,15,16,17,18,19) = 135 g