Logic Puzzle Forums Advanced Techniques using "Either-Or" clues
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#1
08-08-2013, 05:29 PM
 BillsBayou Member Join Date: Mar 2013 Location: New Orleans Posts: 77
Advanced Techniques using "Either-Or" clues

I've expanded upon my analysis of the "Either-Or" type clues. My earlier post is HERE.

From that discussion we see how to substitute a single value with a pair of potential values, do some eliminations, and come up with something very useful. I'll call that one "Either-Or Advanced Technique PART ONE".

Either-Or Advanced Technique PART TWO:

I've since found that substitutions can be made in the OPPOSITE direction. In an Either-Or type of clue, Multiple values can be replaced with a Single value.

Here are the clues that clued me in:
1) The peacock butterfly was either the specimen from Samoa or the butterfly that sold for \$105.
2) Of the butterfly that sold for \$105 and the specimen from Samoa, one was won by Victor and the other was the swallowtail butterfly.

Both clues use the values "butterfly that sold for \$105 or Samoa". Thus we can substitute the given text in one clue with the equivalency from the other clue.

That is to say, either clue can be rewritten as:
The peacock butterfly is either the one won by Victor or the swallowtail butterfly.

Either-Or Advanced Technique PART THREE:

In PART ONE, I eliminated a contradiction after the substitution. In PART THREE, we eliminate potential values after the substitution using a tautology.

Here are the clues:
1) Vestor is either the planet with a diameter of 714,000 mi. or the exoplanet orbiting star PLC 120.
2) Of Vestor and the planet with a diameter of 650,000 mi., one orbits star PLC 120 and the other is 47 light years from earth.

*** WARNING!!! *** WARNING!!! *** WARNING!!! *** WARNING!!! *** WARNING!!! ***
***
*** THE FOLLOWING LOGIC IS NO LONGER RELIABLE. I APPOLOGIZE. ***
***
*** WARNING!!! *** WARNING!!! *** WARNING!!! *** WARNING!!! *** WARNING!!! ***

Here is how to combine these clues:

Start with these two facts from the clues:
- Vestor is 714,000 or PLC120
- Vestor is PLC120 or 47-light-years

Replace one "Vestor" with the equivalency from the other clue and you get:
- 714,000 or PLC120 is PLC120 or 47-light-years.

Spell that out as you would see it in a Puzzle Baron clue:
- Of the planet with a diameter of 714,000 miles and the planet that orbits star PLC120, one planet orbits star PLC120, and the other is 47-light-years from Earth.

The tautology is that the planet that orbits star PLC120 is the planet that orbits PLC120. Once eliminated, we have the planet with a diameter of 714,000 miles is 47-light-years from earth. For that, we get a green dot!

I hope I explained these two techniques well enough.

Last edited by BillsBayou; 11-19-2013 at 01:36 PM.
#2
08-08-2013, 05:47 PM
 BillsBayou Member Join Date: Mar 2013 Location: New Orleans Posts: 77

If you want to rot your brain, here is everything we know from PART THREE:

1) Vestor is 714,000 or PLC120
2) Vestor is PLC120 or 47-light-years
3) 650,000 is PLC120 or 47-light-years
4) PLC120 is Vestor or 650,000
5) 47-light-years is Vestor or 650,000
6) 714,000 is not PLC120
7) Vestor is not 650,000
8) PLC120 is not 47-light-years

Here are the areas for further study:

1 + 4: Vestor is 714,000 or Vestor or 650,000
(What do I do with that second Vestor? Can I eliminate it and get "Vestor is 714,000, or 650,000"? If so, from #7, above, we know we can eliminate 650,000 from the potential values and get "Vestor is 714,000". Which it is, but is the approach correct? The counter example would be that Vestor is not 714,000, and eliminating the second "Vestor" is a big mistake.)

2 + 4: Vestor is Vestor or 650,000 or 47-light-years
(From #7, above, I know that Vestor cannot be 650,000. So "Vestor is Vestor or 47-light-years". If I drop the second "Vestor" I get something that is actually the answer to the puzzle.)

3 + 4: 650,000 is Vestor or 650,000 or 47-light-years
(Since 3 and 4 come from the same clue, we can see why incest is banned in most countries. Don't do this.)

5 + 1: 47-light-years is 714,000 or PLC120 or 650,000
(Gets me nowhere.)
#3
11-18-2013, 12:46 PM
 Grigor Junior Member Join Date: Dec 2009 Posts: 4

From those two clues we do not know that 714,000 is 47 light years, because Vestor could be PLC 120 making 47 light years be 650,00.

So the tautology elimination is not a valid technique.
#4
11-20-2013, 08:12 AM
 BillsBayou Member Join Date: Mar 2013 Location: New Orleans Posts: 77

Using "What If?", I ran two boards at 4x5 and determined that you are correct. The tautology does not work. This is not a valid solving technique.

One thing stands out in my solution sets for the two clues given. 47 Light Years can only be 650,000 or 714,000. Somewhere in these two clues is the answer to this. There has to be a way of determining this without doing a "What If?" technique.

1) Vestor is either the planet with a diameter of 714,000 mi. or the exoplanet orbiting star PLC 120.
2) Of Vestor and the planet with a diameter of 650,000 mi., one orbits star PLC 120 and the other is 47 light years from earth.

Translating to a nameless approach:

1) A1 is either B1 or C1
2) Of A1 and B2 one is C1 and the other is D1

Somewhere in there is this: D1 is B1 or B2

I'm determined to find out how to summarize this in a logical expression.
#5
11-20-2013, 06:41 PM
 zenobia43 Senior Member Join Date: Mar 2012 Posts: 175

I think the difficulty stems from the fact that the Englishized clues don't state the missing logical terms (thankfully).

Ready for some tedium? If not, start another puzzle and have some more fun .

Let's take your clue 2 first:

"2) Of Vestor and the planet with a diameter of 650,000 mi., one orbits star PLC 120 and the other is 47 light years from earth."

If we isolate the relationship involving 47 light years, the clue is really:

47 = (Vestor AND (NOT 650)) OR ((NOT Vestor) AND 650)

This is just saying that if 47 is Vestor, 47 cannot also be 650, and if 47 is 650, it cannot also be Vestor. I have added the missing NOT terms.

Now we have our fully specified equation for 47.

We would like to replace Vestor and its inverse in this equation with the information we get from your clue 1. Let's find out what Vestor is, and then express its inverse.

Your clue 1:

"1) Vestor is either the planet with a diameter of 714,000 mi. or the exoplanet orbiting star PLC 120."

Written more completely with the missing NOT terms:

Vestor = (714 AND (NOT PLC)) OR ((NOT 714) AND PLC)

I.e. Vestor must be one and only one of the two alternatives given.

Now here's the tricky part.

When expressing Vestor in terms of two other logical variables that can be true or false (true means Vestor is equal to that element, false means Vestor is not equal to that element), there are four combinations. That is the entire universe of possibilities.

Clue 1 states that only two of the combinations are possible.

Then the inverse must be the remaining two combinations. The inverse includes "Vestor is both 714 and PLC" and "Vestor is neither 714 nor PLC."

Now we know what "NOT Vestor" (Vestor's inverse) is.

(NOT Vestor) = (714 AND PLC) OR ((NOT 714) AND (NOT PLC))

So let's get back to our equation for 47 and do some substituting.

Let's do the first term which is the easiest.

47 = Vestor AND (NOT 650) ...

Substituting for Vestor ...

47 = ((714 AND (NOT PLC)) OR ((NOT 714) AND PLC)) AND (NOT 650) ...

We know from your clue 2 that 47 cannot be PLC, so using some boolean algebra, this reduces to:

47 = (714 AND (NOT PLC)) AND (NOT 650) ...

Rearranging terms ...

47 = 714 AND (NOT 650) AND (NOT PLC) ...

Now let's take the second term of the equation for 47.

47 = ... OR ((NOT Vestor) AND 650)

Substituting for (NOT Vestor) ...

47 = ... OR (((714 AND PLC) OR ((NOT 714) AND (NOT PLC))) AND 650)

Again - since 47 cannot be PLC, we can reduce the above to:

47 = ... OR (NOT 714) AND (NOT PLC) AND 650

Rearranging terms ...

47 = ... OR (NOT 714) AND 650 AND (NOT PLC)

Now putting the two modified terms in the original equation together and bringing the common term, (NOT PLC), outside, we get ...

47 = (NOT PLC) AND ((714 AND (NOT 650)) OR ((NOT 714) AND 650))

We know that 47 is NOT PLC. So the NOT PLC term is true, and we can eliminate it.

Or as you put it more simply, 47 is either 714 or 650.

I have the uneasy feeling that I just proved that 1 = 2 by dividing by zero.

I glossed over some definitions, and I took great liberties by using boolean algebra to prove your thesis.

I'd much rather use the grid and the technique indicated by 2manynames in one of the original posts (06-09-2013, 08:40 AM) to solve the nominal logic puzzle. I usually screw something up if I try to use boolean algebra.

Last edited by zenobia43; 11-20-2013 at 08:52 PM.
#6
11-25-2013, 01:11 PM
 BillsBayou Member Join Date: Mar 2013 Location: New Orleans Posts: 77

I actually read this. You expanded the clue into a very simplified form and worked it out rather well. Since the negation of an XOR is a biconditional, I can't fault your logic. Thanks.

Here's how I worked it out from my list of things I get from the two clues:

1) Vestor is 714,000 or PLC120
...
5) 47-light-years is Vestor or 650,000
...
8) PLC120 is not 47-light-years

Substituting #1 into #5 I get:
47 is 714,000 or PLC120 or 650,000

Applying #8 eliminates PLC120:
47 is 714,000 or 650,000

Mine is simple in English, but yours is more of a proof of why mine works. Given my screw-up with the tautology approach, I'll take all the help I can get.

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