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# Numbered spoons

## Question

OK so there is this black velvet bag that contains a bunch of uniquely numbered balls spoons.

Every puzzle has numbered balls. Enough with the balls. This puzzle has spoons.

In order to get the needed information I ask my beautiful assistant here [pic load failed; sorry ]

to remove from the bag some quantity of spoons, add their numbers and write it on a sheet of paper.

She looks at you, smiles, and suggests that should be enough information for you.

You disagree and ask her to replace the spoons and repeat the process, which she does,

drawing the same number of spoons from the bag but this time the sum she writes is a different number.

You ask her to do it again. And again, ... until the sums begin to repeat themselves, and eventually

you are convinced that no more new numbers will appear.

She then erases any duplicates, keeping only one occurrence of each, and hands you the paper.

On it you see:

115 118 113 110 120 117 116 112 121 114

How many spoons are in the bag, and what are their numbers?

You think for only a moment and say that you need more information,

like how many spoons were drawn each time.

My smiling assistant says she can't tell you that; but she can tell you that the sum of the number

of spoons in the bag and the number of spoons drawn each time is an odd number.

You pull out a used envelope, turn it over [everyone does this], scribble something,

## Recommended Posts

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OK, five spoons numbered 54,56,58,59,62. Take two at a time.

How ? I got it a bit heuristically. Reasoning being.

1. number of Spoons + number picked each time is ODD. so one of these two numbers is even and the other odd.
2. There are in all 10 different combinations. If all sums were to be unique , you could achieve the 10 combinations by 5 C 2.
3. So the first attempt with 5 numbers - two at a time.
4. spread between the max and min sum is 11, so if this would imply generally a difference of 2 between numbers
5. Among the sums there are 4 odds and 6 even. having one odd number among 5 would make it convenient to achieve this
6. So make one aberration on the Arithmetic series by inseritng a difference of 1 somewhere, immediately followed by a jump of 3 in the series.

Generally above to start with, and a little bit of trial and error - QED.

The only point of OP, if at all, one may argue : "She then erases any duplicates...". The above solution has no duplicates. But may be the OP may be interpreted as " She then erases duplicates if any ..."

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Good going with the spoons!

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OK, five spoons numbered 54,56,58,59,62. Take two at a time.

How ? I got it a bit heuristically. Reasoning being.

1. number of Spoons + number picked each time is ODD. so one of these two numbers is even and the other odd.

2. There are in all 10 different combinations. If all sums were to be unique , you could achieve the 10 combinations by 5 C 2.

3. So the first attempt with 5 numbers - two at a time.

4. spread between the max and min sum is 11, so if this would imply generally a difference of 2 between numbers

5. Among the sums there are 4 odds and 6 even. having one odd number among 5 would make it convenient to achieve this

6. So make one aberration on the Arithmetic series by inseritng a difference of 1 somewhere, immediately followed by a jump of 3 in the series.

Generally above to start with, and a little bit of trial and error - QED.

The only point of OP, if at all, one may argue : "She then erases any duplicates...". The above solution has no duplicates. But may be the OP may be interpreted as " She then erases duplicates if any ..."

Very nice.!

The flavortext suggested that she kept drawing until you were satisfied no more NEW sums would be drawn.

So that is the sense in which it meant there were duplicates. The sums themselves are unique, but there are

only 10 pairs of spoons. If she draws eleven times, one of the sums would have to be repeated.

Repeated = duplicate.

I guess I could have made that clearer.

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115 118 113 110 120 117 116 112 121 114 are the spoons. The assistant draws one spoon each time.

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115 118 113 110 120 117 116 112 121 114 are the spoons. The assistant draws one spoon each time.

Very nice, and I thought you had me on that point.

But I read the OP again, and sure enough, it says ...

You disagree and ask her to replace the spoons and repeat the process,

But you get honorable mention anyway. Your solution works except for the red letter.

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I do agree it does say spoons.

115 118 113 110 120 117 116 112 121 114 are the spoons. The assistant draws one spoon each time.

Very nice, and I thought you had me on that point.

But I read the OP again, and sure enough, it says ...

You disagree and ask her to replace the spoons and repeat the process,

But you get honorable mention anyway. Your solution works except for the red letter.

However, if you didn't know how many spoons were taken, would you ask please replace the spoon?

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I do agree it does say spoons.

115 118 113 110 120 117 116 112 121 114 are the spoons. The assistant draws one spoon each time.

Very nice, and I thought you had me on that point.But I read the OP again, and sure enough, it says ... You disagree and ask her to replace the spoons and repeat the process, But you get honorable mention anyway. Your solution works except for the red letter.
However, if you didn't know how many spoons were taken, would you ask please replace the spoon?

The question of implication is an interesting one.

A previous Forum discussion centered on whether the statement "All my cars are Fords" implied that I had at least one car, and that it was a Ford. What do you think? My take was that it meant "For any object that is a car and that I own, that object is also a Ford." There may not be any such objects, but the statement nonetheless has meaning: If there is such an object then that object is a ford. The premise does not have to be true on a conditional.

The discussion continued until I said that I once saw an empty field that bore a sign "All trespassers will be prosecuted." The sign was meaningful even tho the field clearly was empty. That is, the sign did not imply the existence of at least one trespasser.

I'll concede your point as it applies to "some number of spoons." That number clearly could be one. Or zero, for that matter. But if she had drawn just one spoon, I could not say "replace the spoons". I would have to say "replace all the spoons that you drew." Well, that is the appropriate statement to have made, I guess. I didn't know the number; the OP simply presupposes that I knew is was plural.

So your point is better taken than I first thought. Let's say, though, that had she drawn only one spoon, then to be responsive to what I asked, she would have to have said "I can't do as you ask." or "I don't understand what you are referring to." or something other than just doing the replacement. In that sense, the OP (by careful reading, and way too much over-thinking) rules out the one-spoon case. Kind of.

In any case it was only a fortuitous accident that the OP covered the case of one spoon at all. I'm not clever enough to have done that by plan. I simply didn't think of the case.

Good solve!

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