[ Arithmetic. Mental arithmetic, etc.]

In the turning shop, parts are machined from steel blanks, one part from one blank. The chips left after processing three blanks can be melted down to make exactly one blank. How many parts in total can be made from 9 blanks? And from 14? How many blanks need to be taken to get 40 parts?
The answer "From 9 blanks, 9 parts can be made" is incorrect. Why?

## Solution

From 9 blanks, on the first stage, 9 parts can be obtained, and from the remaining chips, 3 blanks can be made; on the second stage, 3 parts, and from the remaining chips, 1 blank can be made; on the third stage, 1 part and chips for 1/3 of a blank will remain. In total, from 9 blanks, 13 parts can be made.

From 14 blanks, we will get: 1) 12 parts, 2 blanks, and chips for 4 blanks; 2) 6 parts and chips for 2 blanks; 3) 2 parts and chips for 2/3 of a blank. Thus, from 14 blanks, 20 parts can be made.

The last question about 40 parts is solved by trial and error. If 20 parts were obtained from 14 blanks, it is natural to assume that 40 parts will be obtained from 28 blanks. Let's check this assumption.

From 28 blanks, we will get: 1) 27 parts, 1 blank, and chips for 9 blanks; 2) 9 parts, 1 blank, and chips for 3 blanks; 3) 3 parts, 2 blanks, and chips for 1 blank; 4) 2 parts and chips for 2/3 of a blank. In total, 41 parts and chips for 2/3 of a blank.

We got many parts, but not too many. If we now check what will happen if we take 27 blanks, we will find that in this case, 40 parts and chips for 1/3 of a blank will be obtained. Notice that every time we turn 3 blanks into 3 parts and melt the chips to make 1 new blank, the number of blanks decreases by 2, and the number of parts increases by 3, i.e., 2 blanks are as if turned into 3 parts. And such a transformation is possible as long as more than 1 or 2 blanks remain.

Thus, if the number of blanks is even, we will gradually turn all of them, except the last 2, into parts. From the last 2, we will make 2 more parts, and chips for 2/3 of a blank will remain, which we will not be able to turn into a whole part (although the weight of metal in them is sufficient). This means that from (2n + 2) blanks, (2n × (3/2) + 2) = (3n + 2) parts will be made.

If the number of blanks is odd, we will turn all of them, except the last one, into parts. From the last blank, we will get 1 more part and chips for 1/3 of a blank, i.e., from (2n + 1) blanks, (2n × (3/2) + 1) = (3n + 1) parts will be made.

From this, it follows that, no matter how many blanks we take, the number of parts we get cannot be a multiple of 3.

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