Deal or no deal?

Logic Level 2

Raj, an artifacts dealer, has to sell artifacts to the other corner of the city. He has three bags, with each bag containing 30 artifacts which totals to 90 artifacts. He is carrying all the three bags with him.

Now in his way are 30 checkpoints. Each checkpoint being a few kilometers away from each other. He has to pass through all 30 checkpoints. While crossing through a checkpoint he will have to give one artifact from each bag, otherwise he will not be allowed to pass through the checkpoint.

There is no other way except the way in which the 30 checkpoint exist.

After passing through all the checkpoints, how many artifacts is he left with?

Stipulation and Assumptions: If required the bags can be disposed off. Transfer of artifacts from one bag to another is allowed. No other ways are possible to save the artifacts.

30 0 9 33 3 25

1 solution

Abhay Tiwari
Apr 20, 2016

What Raj will try to do first is that, to get rid of the bags one by one as soon as he can. For this he, after passing through each check point will transfer artifacts from one bag to other bag, like

After passing through $first$ checkpoint. Raj will be left with $\boxed{29}$ artifacts in each bag. He will transfer two artifacts from one bag to the other two bags, which makes $\boxed{30}$ artifacts in two bag, and $\boxed{27}$ in the other.

Like this 3 artifacts gets subtracted from one bag after every checkpoint.

That means, after the $tenth$ checkpoint one bag will become empty which he can dispose. Raj will be left with only two bags now(each with 30 artifacts). He will repeat the process again and after the $25^{th}$ checkpoint he will be left with one bag only with 30 artifacts.

After crossing the $30^{th}$ checkpoint he is left with $\boxed{25}$ artifacts.

If Raj disposes a bag, how will he get through checkpoint? He has to give an artifact from each bag (there is no stipulation that he must have the bag in his possession). If we assume that he need only give artifacts from bags being carried, why doesn't he simply sew the bags together to make one bag, such that he will keep 60 artifacts? Or put a bag inside of a bag inside of a bag, so that one artifact from each bag is one artifact because the sets are equal? Or better yet, simply dump all the artifacts out of the bags and find some other sort of container that isn't subject to the checkpoint rules?

Shawn Franchi - 5 years, 1 month ago

ha ha, you have some sense of humor. But the possible thing here is that he can just dump the bags off. The other things that you mentioned are possible only when I mention some hint in the Question and Yes, he has to give an artifact from $each$ bag, I did not mention that he has to give an artifact from $all the three$ bags, I said from $each$ bag. This has to be noticed in the question.

Abhay Tiwari - 5 years, 1 month ago

I confess I still don't understand. "Each" is a determiner used to refer to every one of two or more people or things, regarded and identified separately. Thus, in mathematical logic, the statement would be represented as $\forall x \in A$ , where $A =\{ Bag_{ i }: i \in ℕ \wedge 1 \le i \le 3\}, \quad P(x)$ , defined as $P(x) \quad : \quad$ An artifact from $x$ must be presented.

This is simply translating the stipulation "he will have to give one artifact from each bag" from English into mathematical logic. The only way I could see to make the intended solution make sense would be to stipulate that rather than saying "he will have to give one artifact from each bag", have it reworded so as to say something to the effect of "At each and every checkpoint, if it is the case that Raj is in possession of at least one nonempty bag, each such bag shall have one artifact removed." It might furthermore be restricted with such caveats such as stipulating that the bags are finite with a limit of thirty artifacts and that each artifact must be in one and only one bag. For examples of how to make syntactically correct problems of this sort, I strongly recommend perusal of the publications of Peter Winkler. If you do not have access to any of his puzzle books, you can look him up through his position as William Morrill Professor of Mathematics and Computer Science at Dartmouth College. His brief paper "Seven Puzzles You Think You Must Not Have Heard Correctly" he created for the Seventh Gathering for Gardner is available in PDF publicly via his university webpage.

Shawn Franchi - 5 years, 1 month ago

@Shawn Franchi – Shawn, thanks a lot for the valuable information, I have made the correction please check. And if you don't mind can you please check this and this also, please do tell me if some correction is required. And I read the PDF, I have come across some of those puzzles, but they are really nice. Please have a look at this also :). Thanks again.

Abhay Tiwari - 5 years, 1 month ago

Yeah, bag in bag in bag let me keep 60. If bag A was in bag B was in bag C, then everything in bag A would also be in bags B and C, it would suffice to fulfill the required condition, and the guards should be satisfied with receiving just 1 artifact from the innermost bag A.

Saya Suka - 2 years ago

Deal or no deal?

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