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Thursday, January 17, 2008

Interesting problems

1.Here`s conversation between liars and truthtellers:
B to A:You are a liar.
C to B:It`s you who is a liar.
D to C:Both of them are liars.
D to C:And you are a liar too.
Can you tell who is who?
2.There are 2 ropes.It is known that each of them burns out completely in exactly 1 hour.
How one can measure 45 minutes using these ropes and a box of matches, if pace of burning is not uniform?
These 2 problems are very good examples of good problems.


mathmom said...

D must be a liar, because if he were a truthteller, all of A, B, and C must liars, but from the first statement we see that exactly 1 of A or B must be a liar.

Since D is a Liar, the last thing he said, that C is a liar, is not true so C is a Truthteller.

Since C is a trutheller, when he said B is a liar that was true, so B is a Liar.

Since B is a liar, when he said that A was a liar, that was untrue, so A is a truthteller.

Checking the statements
1) Liar B says Truthteller A is a Liar, which is a lie, so that's ok

2) Truthteller C says Liar B is a Liar, which is true, so that's ok

3) Liar D says both Truthteller A and Liar B are liars. This is a lie (only one of them is a liar, not both) so that's ok

4) Liar D says that Truthteller C is a liar, which is a lie, so that's ok.

The rope one has me stumped, as that type of puzzle often does. ;-)

Filip Jekic said...

Yes, nice solution.
D to C:And you are a liar >>too<<.
Rope is very nice problem, try measuring 30 minutes first.

mathmom said...

How about this. We assume that each match in the (large) box burns completely in a constant amount of time that is the same for each match. Then we light one rope, and also light a match. When the match burns out, we light another one, and so on until the rope burns out. Count the burnt out matches (not including the one used to light the rope). Burn 3/4 of that number of matches to measure 45 minutes.

Probably not what you had in mind, but I can't think of anything useful to do with 2 lengths of rope, each of which burns the same amount of time, non-uniformly, other than measuring multiples of that amount of time.

Filip Jekic said...

Try lighting 1 rope on both sides, and at the same time other rope on 1 side.Then, exactly when first rope burns out - in half an hour - light other rope from other side.You have measured 45 minutes when second rope burns out.
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Light A,B and C at the same time, then D when AB burns out.
When you are solving this kind of problems think about what you can do with objects (in this case, ropes), and try everything.
Sorry for my English.I`m from Serbia.

mathmom said...

Very clever. :) I still think my solution falls under the description of "think about what you can do with objects, and try everything." The box of matches was explicitly included in the problem description ;-)

Filip Jekic said...

How did you make that categories menu on your blog?

mathmom said...

I used a wordpress widget to create the categories menu on my blog. I don't know how to do it on blogspot. You could try asking Dave over at Mathnotations