BrainDen.com - Brain Teasers

# bonanova

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## Posts posted by bonanova

2498?

2. ### Typists problem

If two typists can type two pages in two minutes,

then one typist can type one page in one minute,

your math is off -- or your typist made a typo....

Two typists [together] can type at a pace of 1 page/minute.

and therefore, also, they can type 18 pages in 18 minutes

water?

4. ### Barbera

My name is Joe; my wife is Mary.

Our daughter is Amber.

Mary's mother is Barbera.

Who is Barbera's daughter?

Clue: it's Mary.

Who is my daughter?

Clue: it's Amber.

Who is her mother?

Clue: it's Mary.

Listen as Joe speaks:

Barbera's daughter is my daughter's mother.

The speaker could be Barbera's son-in-law.

5. ### Coins

"You will give me the gold coin." is close.

They can give you the gold coin [thus making the statement true and requiring them to do so.]

But they can also give you nothing [thus making the statement false and requiring them not to give you anything.] You have to add something to change the second outcome.

"You will give me the gold coin or you will give me nothing."

Now they can't give you nothing because that makes the statement true.

6. ### Lion and Unicorn II.

My head says any lying day; my heart says Monday.

What does the unicorn have to do with this?

7. ### Logic Problems at the Court III.

Use the shorthand notation IH IS IN GH GS GN for the 6 cases of

[innocent Guilty] [Honestant Swindlecant Normal].

Craft statements that distinguish you from the perpetrator [P] for 4 different scenarios:

[1] You are an IS. The court knows P is a GS. They don't know whether you are H or S.

[2] You are a GS. The court knows P is a GS. They don't know whether you are H or S.

[3] You are an IH. The court knows P is a GH. They don't know whether you are H or S.

[4] You are IH IS or IN [innocent, unspecified]. The court knows P is GH or GS [Guilty, but not Normal].

[1] Distinguish yourself [an IS] from P [a GS].

Create a statement that anyone but a GS can make.

It must be true for a GS [can't] or any H [can] and false for IS [can].

i.e. the statement must be false iff the speaker is IS.

"I am not an Innocent Swindlecant."

What the court can conclude is this:

If you were a GS [and fit P's known profile] this statement would be true and you couldn't make it.

If you were an IS it would be false, and that's ok.

If you were any flavor of H the statement would be true and you could make it; but no H is a GS.

Thus, only an innocent person can make this statement.

P cannot, and anyone else can. You could be anybody but P.

Go home.

[2] Distinguish yourself [a GS] from P [a GS].

You can't. You're cooked. Too bad.

There is no statement that distinguishes a GS from a GS.

Please report to the gas chamber.

[3] Distinguish yourself [an IH] from P [a GH].

Create a statement that anybody but a GH could make.

It must be true for an IH, false for a GH and false for any flavor of S.

i.e. the statement must be true iff the speaker is IH

"I am an innocent honestant."

You're free to go.

[4] Distinguish yourself [you are IH IN or IS] from P [a GH or GS].

Create a statement that any I can make, but neither a GH nor a GS can make.

Cases and Truth requirements.

IH - statement must be T

IN - doesn't matter: N's can say T or F as they choose.

IS - must be F

GH - must be F

GN - doesn't matter: such people do not exist. P is known not to be a GN.

GS - must be T

Taking the cases that do matter, the statement must be true iff the speaker is IH or GS.

"I am either an innocent honestant or a guilty swindlecant"

What the court can conclude is this:

If you were an IH the statement would be true and you can make it.

If you were an IN you could say anything you like, so you could say this.

If you were an IS, the statement would be false, and you could make it.

So you could be any flavor of Innocent.

Further,

If you were a GH the statement would be false and you could not make it.

If you were a GS the statement would be true and you could not make it.

You can't be a GN: P is known not to be N. GN's don't exist.

So you could not be any flavor of Guilty

All charges are dropped. You're free to go.

8. ### Honestants and Swindlecants VII.

The answer is - YES - head for the hills and bring your pick and shovel.

But I don't think anyone has given the correct analysis yet.

Here's mine:

First, note the statement that was made:

There is treasure only if I am an honest man.

Some have made the mistake of calling this logical equivalence.

It's not. A only if B is logically the same as if B then A.

Logical equivalence is more restrictive: A if and only if B.

The truth tables differ in the case of a false premise and a true conclusion:

"False implies Truth" is True for if; it's False for if and only if.

Thus, we can restate simply as if B then A:

If I am an honest man then there is treasure.

There are two cases: the speaker is a honestant [H] or a swindlecant .

[1] H - the speaker is an honest man

If the speaker is honest, the premise is true [fact] and the logical implication must be true [else he would be lying].

Therefore the conclusion is true: There is treasure.

[2] S - the speaker is lying.

If the speaker is lying, the premise if false [fact] and the logical implication must be false, also, [else he would be telling the truth.]

But, because a false premise validly implies every conclusion, such an implication is always true.

Thus we must conclude that the speaker could not have been a swindlecant:

one cannot invalidly conclude anything [tell a lie, as a swindlecant must do] starting from a false premise.

Since the speaker must have been a truth-teller, there must be a treasure.

9. ### Honestants and Swindlecants X.

Let's identify the four types [rich poor], [honestant, swindlecant]

of creatures as RH PH RS and PS.

There are two statements that must be crafted:

[1] A statement that only an RS can make, proving the speaker is a rich swindlecant.

[2] A statement that only an RH can make, proving the speaker is a rich honestant.

[1] A statement that only an RS can make.

Such a statement must be false if spoken by either type of honestant, eliminating RH and PH.

It must be true if spoken by a poor swindlecant, eliminating PS

It must be false if spoken by a rich swindlecant, making it OK for RS.

That is, the statement must be true if and only if the speaker is a poor swindlecant.

The simplest such statement is: "I am a Poor Swindlecant."

Only an RS can say that in character.

[2] A statement that only an RH can make.

Such a statement must be true if spoken by either type of swindlecant, eliminating RS and PS.

It must be false if spoken by a poor honestant, eliminating PH.

It must be true if spoken by a rich honestant, making it OK for RH.

That is, the statement must be false if and only if the speaker is a poor honestant.

The simplest such statement is: "I am not a Poor Honestant."

Only an RH can say that in character.

10. ### Honestants and Swindlecants IX.

...the identity would only be clear if one was honest and the other lied...

There are 3 cases [HH, SS and HS] and in each

case one or the other of the Aborigines could have

answered. So the gringo could have heard any of

6 answers. As Ploper points out, 5 of these answers

will be Yes. Only one answer could be No: the answer

given by the Swindlecant in the case HS.

So the answer is - the gringo heard "No" and that

made it clear a Swindlecant was speaking and that

there was one of each.

A "Yes" answer gives no information about either point.

11. ### Honestants and Swindlecants I.

Since A is going to say Honestant in both cases B is lying so he is Swindlecant and C is telling the truth he is an Honestant

This is correct.

The final bit of analysis is that we can't know what A is.

12. ### Lazy-bones Paradox

Thanks.

I love these things, too, and thoroughly enjoy this site.

- bn

13. ### Crocodile Sophism

Remember the question to answer is:

What statement shall the mother make to save her child?

First, let's ask: has the croc already decided what he'll do?

[1] Yes. [this is suggested by "if you don't guess his fate ..."]

The answer in this case is: Nothing the mother says can

affect whether the child is eaten or returned. It's a fallacy

to suggest there are Mother's words that will force the return

of her baby. Its fate is sealed; not open to negotiation.

Not a satisfying result for the puzzle solver or the mother.

Poor baby.

[2] No. [this is suggested by "If you guess [correctly] ... then I will ..."]

In this case, what the croc will do is determined by the

response of the mother. And the task is to construct a

reply which will logically bind the croc to return the child.

The best solution is reply that he will choose, and his choice

will be either to eat or to return. This is a clever approach

because it seems to cover all the bases and force the baby's

return.

But by assuring the baby's return, the mother seals its fate:

The croc merely says, "sorry, but you're wrong. I'll prove

you're wrong by eating the baby, and thus fulfill my word which is ...

If you don't guess his fate I'll eat him.

[Only because] she secured his return by her clever answer,

she wrongly guessed its fate.

The croc had dinner that night.

Another unsatisfactory result.

Personally, I like the strategy mentioned above of threatening

the croc with unbelievably heinous consequences if the baby is

not returned.

Always appeal to enlightened self interest.

14. ### A few sentences from life

6. Everyone who is not here, please indicate by raising your hand.

15. ### Is it possible to give what we don't have?

Depends on what is meant by "have."

[1] If have means "own," then yes. I can give someone

something that I do not own. e.g. if I stole it.

To give something, one only needs the ability to determine who

controls it. If I control it, I can pass its control to someone else.

So ...

[2] If have means "possess the control of" then no.

As stated, the paradox arises from the different antecedents of

"with sorrow." Sorrow is the consequence of giving, not a possession

before the act. But the language permits that interpretation by

its form.

Cute.

• 1
16. ### Fortune Cookie

When a denial implicitly asserts a truth you have a paradox.

Kind of an empty one, tho -- as was pointed out, nothing was predicted!

17. ### The Surprise Execution Paradox

Putting it another way, the more certain the man is
[on Friday night, should he live that long] that he
cannot be hanged on Saturday [the last possible
day for the hanging] the more surprised he is
when it happens. It's possible because of the
prisoner's unwavering belief in the truth of the judge's
statements.

As with other semantic paradoxes, the assumption
that what is being asserted is true comes into play.
If the prisoner didn't believe what the judge said
was true, he might [expectantly] dread a Saturday
hanging, thereby precluding it!

The moral:
Never believe a judge. It might cost your head.

18. ### Double Liar Paradox (Jourdain's paradox)

I don't quite understand the fascination with 'paradoxes' of this sort, which basically come down to which of the two statements are true, if any.

......

Can someone tell me what I am missing?

It's the fact that one statement can be a contradiction.

[1] "I am lying."

Spreading that over two statements does not change the

nature of the paradox:

[2] "I am telling the truth."

[3] "The previous statement is a lie."

Here, one can simply eliminate statement [2],

which carries no information, and change [3] into

[4] "This statement is a lie."

which is equivalent to statement [1].

To my mind the paradox arises from an explicit assertion of something's falseness

using a vehicle [declarative sentence] which implicitly asserts its truth.

19. ### Lazy-bones Paradox

Destiny and free will become paradoxical if and only if

[1] they both determine the outcome of events.
[2] they apply to the same event at the same time.

Clearly, two independent forces cannot each have their way
in the same matter [e.g. going to the doctor] at the same
time [today, at 10:00, the time of my appointment.]

Either [if Destiny has its way] I become simply an observer,
capable only of telling someone after the fact that I did [or didn't]
keep my appointment or I choose to keep my appointment
and am forced [lamely] into the supposition that [my then unknown]
Destiny must have been that I do so. Destiny and free will cannot
both be the deciding factor and be independent of each other.

What does that mean? One conclusion is that one or the other
simply does not exist. [At least] one of them is simply an illusion:
one that people discuss, but that owns no real affective power.

Usually at this point we cling to our free will and throw Destiny
out the window. But experience suggests that particular outcomes
seem to happen despite the choices of others to the contrary:
one might assert that Hitler was destined to fail, despite protracted
and horrendous [free] choices made, by many, to obtain his success.
This is the seemingly favorite way to invoke Destiny - something that
happens over an extended period of time, contrary to many free
choices [or other inanimate obstacles] to the contrary.

This is the point of [2] at the beginning of my post. Free choices and
Destiny can coexist if they to not apply to the same matter at the
same time. It might be my Destiny, born out of my underlying desire for
health, to recover from an illness even though I decide not to keep today's
appointment with the doctor. Eventually I will seek and get the help I need.

In this sense, free will applies to the microscopic decisions and strategies
I employ moment by moment, and can logically co-exist with a Destiny
which sees eventual outcomes -- outcomes that are outside my ability
directly to create. Coexistence is non paradoxical -- both can determine
certain outcomes -- if they each operate in their respective, disjoint arenas.

Finally, if one believes in an overpowering Destiny that applies to every
event in every arena
, and if, in the face of that prospect, one simply gives
up an active role in life, becoming only its spectator, it can be accurately
said that one has so chosen.

A paradox of a different type.

20. ### Double Liar Paradox (Jourdain's paradox)

What underlies paradoxes of this type is the syntactical rule

that a declarative sentence is by its nature an assertion of

some particular truth. To use a presumed assertion of

truth to deny that same truth is paradoxical: One cannot

convey usable knowledge by asserting a denial. Nor can one

meaningfully deny a truth: the coin has two paradoxical

sides:

[1] "I am asserting a falsehood." or "I am lying."

[2] "I am not asserting something that is true." or "I am not telling the truth."

Putting it another way, it's physically possible to speak the

words, "I am lying." But when one undertakes a linear

analysis of what has happened when the words are spoken,

one is drawn into the syntactical analogy of a Moebius Strip:

a piece of paper having a physical connection of its two sides.

The circular reasoning forced on the mind by a linear

analysis of such statements creates a pleasantly frustrating

tease, and the desire for consistency and meaning leaves

one in a disturbingly uncomfortable state.

• 1
21. ### Pouring water VI.

Alternatively you can use the following solution:

1. Fill C with B leaving 4 litres in B

B starts with 3 liters.

Filling C [takes 1 liter] leaves 2 [not 4] liters in B.

22. ### Weighing IV.

Each weighing has three outcomes:

Left side is [lighter than] [equal to] [heavier than] Right side.

Three weighings can thus discern among [3]x[3]x[3]= 27 cases.

We have only 24 cases:

one of 12 balls is heavier or lighter than the rest.

So we can solve the problem, so long as ...

[1] The first weighing reduces the cases to no more than 9.

[2] The second weighing reduces the cases to no more than 3.

[3] The third weighing then distinguishes among 3 or fewer cases.

First weighing:

Set aside four balls. Why?

Because, if the first weighing balances, we have 8 [fewer than 9] cases:

One of the 4 excluded balls is heavier or lighter.

[1] Weigh 1 2 3 4 5 6 7 8

Outcome 1[a] 1 2 3 4 balances 5 6 7 8.

Since only one ball is odd, 1 2 3 4 5 6 7 8 must all be normal.

We have 8 cases: 9 10 11 or 12 is H or L

[shorthand: H=heavier; L=lighter]

[2] Weigh 1 2 3 9 10 11 [we know 1 2 3 are normal]

If this balances, 9 10 11 are also normal, and 12 is H or L.

[3] Weigh 12 [any other ball]

If 12 rises, then 12L; if 12 falls, then 12H.

If [2] 9 10 11 rises, we have 3 cases: 9L, 10L or 11L

[3] Weigh 9 10.

If [3] balances, then 11L;

If 10 rises, then 10L; If 9 rises, then 9L

If [2] 9 10 11 falls, we have 3 cases: 9H, 10H or 11H

[3] Weigh 9 10.

If [3] balances, then 11H;

If 10 falls, then 10H; If 9 falls, then 9H

end of Outcome 1[a]: balance.

Outcome 1: 1 2 3 4 falls, and 5 6 7 8 rises.

This gives 8 cases: 1H, 2H, 3H, 4H, 5L, 6L, 7L, or 8L

The second weighing in this case becomes tricky.

Remember each of its three outcomes can lead to no

more than 3 cases for the third weighing to resolve.

Again, we exclude a number of balls and involve the others.

We exclude any three of the balls. Why? Because if the

other [included] balls balance, we have exactly 3 cases.

Without loss of generality we exclude balls 1 2 3.

Since that leaves an odd number of balls, 4 5 6 7 8,

we need to use one of the normal balls.

Finally we choose which three to weigh against the others.

And here's the only hard part of this problem.

We must mix some of the possibly light balls with some

of the possibly heavy balls. Otherwise, one of the

outcomes of the second weighing will leave us with more

than 3 cases, and the third weighing will not resolve this.

[2] weigh 4[H] 5[L] 6[L] 1[normal] 7[L] 8[L]

in parentheses I've indicated the POSSIBLE cases

that we have determined:

4[H] means 4 is heavier if it's not normal.

Outcome 2[a]: 4 5 6 balances 1 7 8

These balls are all normal.

We have 3 cases: 1H 2H or 3H.

[3] weigh 1 2

If [3] balances, then 1 and 2 are normal, and 3H

if 2 falls, then 2H

if 1 falls, then 1H

Outcome 2b: 4 5 6 falls, and 1 7 8 rises.

We have 3 cases: 4H 7L or 8L

[3] weigh 7 8

if [3] balances, then 7 and 8 are normal, and 4H

if 7 rises, then 7L

if 8 rises, then 8L

Outcome 2[c]: 4 5 6 rises, and 1 7 8 falls.

We have 2 cases: 5L or 6L.

[3] weigh 1[normal] 5[L]

If 5 rises, then 5L

If balance, then 5 is normal, and 6L

Now we can go back and take the remaining case

Outcome 1[c]: 1 2 3 4 rises, and 5 6 7 8 falls.

to distinguish among the remaining 8 cases:

1L 2L 3L 4L 5H 6H 7H and 8H

exactly as we analyzed Outcome 1b.

Simply substitute H for L and v.v.

Problem solved.

23. ### Barbera

The puzzle asks: Who am I to Barbara - NOT who is Barbara to me. Hence the correct answer is ... Daughter. Unless of course I am a male. Then the correct answer is not in the choices listed, it's Son-in-Law.

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