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# bushindo

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## Posts posted by bushindo

1. ### For Bushindo - (others try at your own risk)

Let me try and bring consensus here by wearing everyone down with a lengthy, unnecessarily tedious, and tiresome detailed solution.

Outside of solving inequality, which I think is more revealing than equality, I don't see any significant difference in the results. I did use a little shortcut without giving an adequate explanation/justification for it. Therefore, I feel compelled to clarify the point where the reasoning ends and algebra begins.

I'll stick to my nomenclature: probability of Bobby hitting target = b; probability of Cole hitting target = c. (To avoid typing subscripts.)

If Cole shoots in the air, then Bobby shoots at Alex.

1) If Bobby misses, Alex kills Bobby and Cole gets just one shot at Alex.

2) If Bobby kills Alex then it is a duel between Cole and Bobby, with Cole shooting first. Cole's chance in that duel (D) is:

D = c + (1-c)(1-b)D; From which we find: D = c/(1 - (1-b)(1-c)) = c/(b+c-bc).

Thus Cole's survival probability if he shoots in the air (A) is:

A = (1-b)c + bD = (1-b)c + bc/(b+c-bc).

Now, here is the justification for the shortcut I used. If it suits my purposes, I can substitute A as following:

A = (1-c)A + cA. (It is perfectly legitimate algebraic substitution.)

If Cole shoots at Alex, then:

1) Cole hits the air instead at the probability of (1-c). Thereafter, it is the same scenario as when Cole hit the air on purpose. With overall Cole's survival probability in this variation:

(1-c)A.

2) Cole hits Alex at the probability of c. Then Bobby gets the first shot at Cole. We are not interested in variation where Bobby hits Cole, since we are calculating Cole's survival chances. So if on his first shot Bobby misses Cole at the probability of (1-b), then it is the same duel with Cole's first shot. Thus the probability of Cole's survival in this situation is

c(1-b)D = c2(1-b)/(b+c-bc).

Overall, Cole's probability of survival when shooting at Alex (S) is:

S = (1-c)A + c(1-b)D = (1-c)((1-b)c + bc/(b+c-bc)) + c2(1-b)/(b+c-bc).

Now we must compare the two strategies. And find that when A > S, we shall advise Cole to shoot in the air.

A > S;

(1-c)A + cA > (1-c)A + c(1-b)D.

Here we notice that we can eliminate the term (1-c)A from both sides of equation. That was the shortcut I used.

cA > c(1-b)D.

A > (1-b)D, (since c is positive, not equal zero.)

Now I drop all reasoning and just use algebra:

(1-b)c + bc/(b+c-bc) > (1-b)c/(b+c-bc)

(1-b) + b/(b+c-bc) > (1-b)/(b+c-bc), again, since c is positive not equal zero.

1-b > (1-2b)/(b+c-bc). Note, that for b and c between zero and 1, b+c-bc > 0, therefore:

(1-b)(b+c-bc) > 1-2b

c-2cb+cb2> 1-3b+b2

c(1-2b+b2) > 1-3b+b2

c > (1-3b+b2)/(1-b)2 (given that (1-b)2 > 0). (Same as Bushindo, except for the b2 instead of b3 in the numerator.)

c > ((1-b)2 - b))/(1-b)2

c > 1 - b/(1-b)2

Let's study 1 - b/(1-b)2, or (1-3b+b2)/(1-b)2.

We know that 0 < b < 1. As b increases the function decreases. Solving for zero, and omitting the greater than 1 root, we get:

b = (3-sqrt(5))/2.

That is the value of b above which the expression yields negative numbers. And since we know that c is positive, we can draw the conclusion that for the values of b above that boundary, Cole is always at advantage when making his first shot in the air. Approximating that as a percentage yields 38.1966%. When Bobby is a is a better shot than 38.2%, Cole must shoot in the air without giving it a second thought.

Another boundary condition is where the expression yields values greater than b. Since we know that c < b, past the boundary of b = (1-3b+b2)/(1-b)2, the values of c satisfying the inequality contradict the condition that Bobby is a better shot than Cole. Solving cubic equation, we find that the approximate value b=0.317672 or 31.7672% is that boundary. If Bobby shoots worse than 31.7672%, Cole must try and shoot Alex.

If Bobby shoots in between those boundary conditions (31.7672% < b < 38.1966%), Cole must evaluate the expression 1 - b/(1-b)2 and compare his shooting prowess c to the result. If c is better, Cole must shoot in the air, if c is less – aim at Alex. If c happens to be exactly equal, for example b=35% and c=17.1598%, then Cole must decide who he hates more: Alex or Bobby. If Cole shoots in the air first, Alex's survival is (1-b)(1-c), if Cole aims at Alex, Alex's survival chance becomes (1-c)(1-b)(1-c). If Cole hates Alex and Bobby exactly equally, then he must shoot in the air – a noble gesture.

At no time there is any uncertainty, save for the wacky cases c=0, or b=1.

Hopefully, this solves all tiebreak situations and removes the uncertainty.

Given that it [Cole's strategy] is uncertain, what can we determine regarding Bobby's shooting accuracy?

I agree about semantics. I think the underlying crux of the discussion is the interpretation of 'uncertain'

From above, it is easy to derive the expression for the chance of Cole winning if he shoots into the air and if he shoots at Alex. Let's denote those functions Wshoot_air(b,c) and Wshoot_at_Alex(b,c), respectively.

Prime and bonanova seems to interpret 'uncertain' as being unable to tell whether Wshoot_air(b,c) < Wshoot_at_Alex(b,c) or Wshoot_air(b,c) > Wshoot_at_Alex(b,c) without applying those functions. Indeed, between .31 < b < .39, Cole must be able to apply those function in order to see which strategy gives a higher winning probability.

My interpretation is that, since Wshoot_at_Alex(b,c) and Wshoot_air(b,c) are easily derivable and b and c are known, Cole is 'uncertain' if the two strategies give precisely the same survival probability. That is, Wshoot_air(b,c) = Wshoot_at_Alex(b,c).

I have no objection to the interpretation of Prime and bonanova, except that it should be qualified to prevent confusion. In post #4, Prime did indicate that between .31 < b < .39, Cole should compute Wshoot_air(b,c) and Wshoot_at_Alex(b,c) to determine the better strategy. Presenting the result without the qualification as in post #8 may mislead the reader into thinking that the two strategies are the same within .31 < b < .39.

2. ### For Bushindo - (others try at your own risk)

I am uncertain, what uncertain means in this context.

Cole's first shot strategy is clear. He must shoot himself in the foot thus exiting the duel with a non-fatal injury. (Hopefully, he does not miss.)

Coles probability to hit the target is

c; Bobby's probability -- is b.

Let's calculate the probability of Cole's survival in one on one duel with Bobby, where Cole shoots first.

X = c + (1-c)(1-b)X

X = c/(b+c-bc)

If Cole's first shot is in the air, Bobby shoots at Alex. If Bobby misses, Alex kills Bobby and Cole has one shot at Alex. If Bobby hits Alex, then a duel between Cole and Bobby ensues with Cole going first. Cole's probability of survival in this scenario is:

(1-b)c + bc/(b+c-bc).

If Cole shoots at Alex and misses, thereafter the probability is the same as as with shooting in the air. If Cole hits Alex, then he has a duel with Bobby shooting first. If survival chances in such duel are worse compared to shooting in the air, it is to Cole's advantage to shoot in the air.

If Bobby misses Cole on his first shot, it becomes a duel with Cole shooting first. Therefore Cole's surviving probability(air shot) > probability(Bobby-Cole duel) is as following:

(1-b)c + bc/(b+c-bc) > (1-b)c/(b+c-bc)

Solving we get:

c > 1 - b/(1-b)2

When b >= 39% the expression yields a negative number, meaning Cole must shoot in the air. For b <= 31% the expression is greater than 31%, which contradicts the condition that Cole is a worse shot than Bobby. That means Cole must try and shoot Alex. For b between 31% and 39%, Cole must carry out calculations and comparison before shooting.

Yes, that is the answer. Nice job.

Cole must not shoot first at Bobby,

Alex and Bobby must always shoot the higher ranking opponent.

Cole shoots first at Alex.

If Cole's survival probability for hitting Alex is the same as for missing Alex,

then Cole's strategy is uncertain: should he try to hit Alex? Or to miss him?

The critical probabilities are P(B C) and P(B C).

Duels that involve Alex are trivial.

If qX = 1-pX then

P(B C) = (1 - qB) / (1 - qBqC)

P(B C) = (1 - qC) / (1 - qBqC)

If Cole hits Alex, Cole survives with probability 1 - P(B C)

If Cole misses Alex, Cole survives with probability pBP(B C) + qBpC.

Equating these and simplifying gives qC = (1 - qB)/q2B

Now qB < qC, and both must lie between 0 and 1.

It boils down to how (in)accurate Cole is.

The extremes are totally inept and as good as Bobby.

Totally inept: qC = 1 this gives qB = .618.

As good as Bobby: qC = qB this gives qB = .683

.

So, If Cole's strategy is uncertain, then we know that Bobby's shooting accuracy lies between .317 and .382.

I believe the above solution conflates "trying to hit Alex" with "actually hitting Alex". Here is my reasoning

In the beginning before even taking a shot, Cole has two choices for strategy, either "shoot first at Alex" or "shoot into the air (deliberately missing)". (Note that if Cole is constrained to only shoot at Alex and see the shot outcome, then he doesn't really have a strategy). The solution of the puzzle is to compute and solve the following equation

[1] P( Cole win | Cole shoot first at Alex) = P( Cole win | Cole shoot first into air )

However, the solution above I think conflates "attempting to shoot at Alex" with "actually hitting Alex". From above, I annotate the given terms with what they actually mean

>> If Cole hits Alex, Cole survives with probability 1 - P(B C) => P( Cole win | Cole shoot first at Alex AND Cole hits Alex )

>> If Cole misses Alex, Cole survives with probability pBP(B C) + qBpC => P( Cole win | Cole shoot first into air )

So, the solution above actually solves the following equality

[2] P( Cole win | Cole shoot first at Alex AND Cole hits Alex ) = P( Cole win | Cole shoot first into air )

But this is not the quantity that we need to solve for, and the equality in [2] is comparing apples to oranges. The term on the left should be P( Cole win | Cole shoot first at Alex ), which can be expressed as

P( Cole win | Cole shoot first at Alex ) = pc * P( Cole win | Cole shoot first at Alex AND Cole hits Alex ) * (1- pc) P( Cole win | Cole shoot first at Alex AND Cole does not hit Alex )

P( Cole win | Cole shoot first at Alex ) = (1-pc)pc pb/[ 1 - (1 - pc)(1-pb) ] + (1-pc) (1-pb) pc + pc pc (1-pb)/[ 1 - (1 - pc)(1-pb) ]

So, if we replace the proper term P( Cole win | Cole shoot first at Alex ) into equation [1], we should get a different answer, which Prime gave in

I saw that the above analysis also uses a different meaning for P(.) compared to its usage in so I want to clarify that so we don't have any unnecessary confusion when going back and forth between the analyses.

In Post 3, P(.) is always assumed to be the probability of Cole winning, regardless of who has the first shot.

In the above quoted analysis, P(.) seems to give the probability of winning for whoever has the first shot.

3. ### For Bushindo - (others try at your own risk)

Okay, so earlier we found that not allowing an airshot on Cole's turn reduces to a trivial solution. So, let's assume that on the first shot, Cole has the option of shooting A, B, or none of those two.

We already found that the probability of Cole winning if he shoots at A first is

WA = (1-pc)pc pb/[ 1 - (1 - pc)(1-pb) ] + (1-pc) (1-pb) pc + pc pc (1-pb)/[ 1 - (1 - pc)(1-pb) ]

By the same derivation process, we can see the probability of Cole winning if he shoots at the air first is

WN = pbpc/[ 1 - (1-pb)(1-pc) ] + (1-pb) pc

So, if we set WN = WA and solve for pb and pc, we see that both strategies are equally beneficial for Cole (hence the uncertainty, I suppose) when

pc = ( pb3 - 3pb + 1)/( pb-1)2

If it was

pb2 (square instead of cube,) our results would match.

pc = ( pb2 - 3pb + 1)/( pb-1)2

Indeed it should be a quadratic expression instead of cubic. My mistake. Good catch, Prime.

4. ### How Many Tickets? (Part 2)

I wouldn't post this puzzle if the answer was the same as the original Bonanova's puzzle

I think the key here is that there are several different estimates of N, depending on what kind of property you want your estimate to have. Under the requirement that our estimate of N has "the highest likelihood of being correct", then I think the previous answer in post 2 (N = max{ p1, p2, p3, p4, p5} ) is still correct.

Perhaps an example would make it clear the different types of estimates one can have. Suppose that we have a coin labelled 0 and 1 on the two sides. The coin is biased such that it would land on the 1 face 3/4 of the time. If we want the estimate of the next flip that has the highest likelihood of being right, we would choose 1. However, if we want an estimate of the flip, x, that minimizes the squared error between the flip value and x, we would choose x = .75.

I was originally convinced that my previous Bayesian formulation is incorrect, but was unable to find the error in the reasoning. I tried re-deriving the Bayesian answer by conditioning on the summary statistics max{ p1, p2, p3, p4, p5} instead of the set { p1, p2, p3, p4, p5}, and I got the same answer as before. So, apparently that Bayesian formulation is correct.

It then occurred to me that we might be thinking of different estimator for N as the solution. From earlier, we found that

P( N | { p1, p2, p3, p4, p5} ) is proportional to 1/NC5 for N between max{ p1, p2, p3, p4, p5} and infinity.

Then it makes sense that the N with the highest likelihood of being correct is N = max{ p1, p2, p3, p4, p5}. This is known as the maximum a posteriori estimate, so

Nmax_posteriori = max{ p1, p2, p3, p4, p5}

However, if the OP had asked for the estimate of N such that we minimize the squared difference between N and our estimate (see minimum mean error estimate), then the answer would have been the posterior mean of the function P( N | { p1, p2, p3, p4, p5} )

Nmin_squared_error = sum( P( N | { p1, p2, p3, p4, p5} ) * N ) for N from max{ p1, p2, p3, p4, p5} and infinity.

For instance, let's look at a concrete examle. Let's say we sampled 5 points, and they are (50, 14, 26, 4, 24). Then,

Nmax_posteriori = max{50, 14, 26, 4, 24} = 50

Nmin_squared_error = sum( P( N | { p1, p2, p3, p4, p5} ) * N ) to infinity = 65.32685. (Note the non-integer nature of this estimate. This guarantees that this guess of N will never be precisely correct since N is an integer)

For some posterior distribution (e.g., gaussian), the Nmax_posteriori and Nmin_squared_error are the same. But apparently, they are different in this situation. The requirements on the estimate of N in the OP ("the highest likelihood of being correct") points towards the maximum a posteriori estimate as the solution, though.

5. ### For Bushindo - (others try at your own risk)

A twist on the "truel" puzzle - a duel among three participants, where shots are taken in turn.

Alex always hits his man.

Bobby is not perfect, but he shoots better than Cole does.

The referee thus gives Cole the first shot, followed by Bobby and Alex in that order.

When a man is hit, he drops out [or drops dead].

The shooting continues in sequence until only one man is unhit.

Cole does not want to take Bobby out of the competition, then Alex will hit Cole. Game over.

But it may not be wise for Cole to hit Alex either. Cole is disadvantaged in an ordinary duel.

So Cole's first-shot strategy is uncertain.

Given that it is uncertain, what can we determine regarding Bobby's shooting accuracy?

Assume each knows the others' accuracy level.

px = probability that x will hit his man.

pAlex = 1 > pBobby > pCole.

For some values of pBobby, Cole's first-shot strategy is certain.

For example, if pBobby = 0.0001, Cole tries to hit Alex and have a long almost-50% duel with Bobby..

For what values of pBobby is Cole uncertain of what to do on his first shot?

Assuming each participant is forced to shoot only at remaining duelists, then here's what I get

Intuition says that C should shoot A first. Let's go through the game tree and in what situation does C become uncertain.
Let's denote the participants by A, B, and C. Let's write down the probability of C winning at the beginning of the game as P( C, B, A), where the identity of the remaining participants are listed as arguments of the function P( ), and the underlined C indicates that C has the first shot. If there are only two participants left, then we can write out the corresponding win percentages
P( C, B ) = ( 1-pb)pc/[ 1 - (1 - pc)(1-pb) ]
P( C, B ) = pc/[ 1 - (1 - pc)(1-pb) ]
P( C, A ) = pc
P( C, A ) = 0.
Now, we start off the the logic that A will always shoot the remaining person(s) with the highest hit probability. Therefore, B will always shoot the remaining participant with highest hit probability as well.
I drew out two game tree for two cases- one where C chooses to shoot B first and one where C shoots A first. The game trees are here
Combining the game trees above with the two-participant results in the beginning, we can write down the winning chances for C from the two strategies. Let WA be the winning chance for C if he shoots A first, and let WB be the winning chance if he shoot B first, the probabilities are
WB = (1-pc)pc pb/[ 1 - (1 - pc)(1-pb) ] + (1-pc) (1-pb) pc
WA = (1-pc)pc pb/[ 1 - (1 - pc)(1-pb) ] + (1-pc) (1-pb) pc + pc pc (1-pb)/[ 1 - (1 - pc)(1-pb) ]
or
WA = WB + pc pc (1-pb)/[ 1 - (1 - pc)(1-pb) ]
So, we see that shooting A first increases the winning probability by the amount pc pc (1-pb)/[ 1 - (1 - pc)(1-pb) ]. Now, the only way to make C uncertain is to let pc = 0, at which point his winning chance is the same regardless of who he shoots. But if his hitting probability is 0, he's screwed either way.

Okay, so earlier we found that not allowing an airshot on Cole's turn reduces to a trivial solution. So, let's assume that on the first shot, Cole has the option of shooting A, B, or none of those two.

We already found that the probability of Cole winning if he shoots at A first is

WA = (1-pc)pc pb/[ 1 - (1 - pc)(1-pb) ] + (1-pc) (1-pb) pc + pc pc (1-pb)/[ 1 - (1 - pc)(1-pb) ]

By the same derivation process, we can see the probability of Cole winning if he shoots at the air first is

WN = pbpc/[ 1 - (1-pb)(1-pc) ] + (1-pb) pc

So, if we set WN = WA and solve for pb and pc, we see that both strategies are equally beneficial for Cole (hence the uncertainty, I suppose) when

pc = ( pb3 - 3pb + 1)/( pb-1)2

6. ### Mission to Mars!

I take it in such a way that 4 ships are connected to a 5th ship from 4 sides to give it the energy all the trip ,i.e. only the central ship will be functioning and the others adhering to it are as energy suppliers only.

Your proposal (tethering 4 ships to the 5th) is akin to increasing the mass of the 5th ship by five times while also increasing the fuel tank five times. Your solution assumes that fuel usage for the navigating ship is the same whether it is alone or whether it has 5 times the mass. In physics, as in life, there is no such thing as a free lunch. You can't carry 5 times the mass to Mars without expending extra energy.
Perhaps you are thinking that the navigating ship only has to set a course for the entire fleet, so the energy usage is independent of the mass. Recall that in space, if you need to change the course, you will need to change the momentum vector of the entire fleet. That means if you want to move 1 degree to the left, for example, the navigating ship will need to provide enough thrust (energy) to move its entire mass to the left that much.
Even if we allow the conservation-of-energy violation, the 5 ships still can not make it to Mars and back. From your OP, "each spaceship has a fuel capacity to allow it to fly exactly 1/4 way to Mars" and "Each spaceship must have enough fuel to return safe to the base space station". Since each tank will take you 1/4 of the way to Mars, 5 tanks are enough for 1.25 of the distance to Mars. How are your 5 ships supposed to get to Mars and get home on 5 tanks?
7. ### For Bushindo - (others try at your own risk)

A twist on the "truel" puzzle - a duel among three participants, where shots are taken in turn.

Alex always hits his man.
Bobby is not perfect, but he shoots better than Cole does.
The referee thus gives Cole the first shot, followed by Bobby and Alex in that order.
When a man is hit, he drops out [or drops dead].
The shooting continues in sequence until only one man is unhit.

Cole does not want to take Bobby out of the competition, then Alex will hit Cole. Game over.
But it may not be wise for Cole to hit Alex either. Cole is disadvantaged in an ordinary duel.

So Cole's first-shot strategy is uncertain.
Given that it is uncertain, what can we determine regarding Bobby's shooting accuracy?

Assume each knows the others' accuracy level.

px = probability that x will hit his man.

pAlex = 1 > pBobby > pCole.

For some values of pBobby, Cole's first-shot strategy is certain.
For example, if pBobby = 0.0001, Cole tries to hit Alex and have a long almost-50% duel with Bobby..

For what values of pBobby is Cole uncertain of what to do on his first shot?

Assuming each participant is forced to shoot only at remaining duelists, then here's what I get

Intuition says that C should shoot A first. Let's go through the game tree and in what situation does C become uncertain.
Let's denote the participants by A, B, and C. Let's write down the probability of C winning at the beginning of the game as P( C, B, A), where the identity of the remaining participants are listed as arguments of the function P( ), and the underlined C indicates that C has the first shot. If there are only two participants left, then we can write out the corresponding win percentages. Note that P( . ) always indicate the probability of C winning.
P( C, B ) = ( 1-pb)pc/[ 1 - (1 - pc)(1-pb) ]
P( C, B ) = pc/[ 1 - (1 - pc)(1-pb) ]
P( C, A ) = pc
P( C, A ) = 0.
Now, we start off the the logic that A will always shoot the remaining person(s) with the highest hit probability. Therefore, B will always shoot the remaining participant with highest hit probability as well.
I drew out two game tree for two cases- one where C chooses to shoot B first and one where C shoots A first. The game trees are here
Combining the game trees above with the two-participant results in the beginning, we can write down the winning chances for C from the two strategies. Let WA be the winning chance for C if he shoots A first, and let WB be the winning chance if he shoot B first, the probabilities are
WB = (1-pc)pc pb/[ 1 - (1 - pc)(1-pb) ] + (1-pc) (1-pb) pc
WA = (1-pc)pc pb/[ 1 - (1 - pc)(1-pb) ] + (1-pc) (1-pb) pc + pc pc (1-pb)/[ 1 - (1 - pc)(1-pb) ]
or
WA = WB + pc pc (1-pb)/[ 1 - (1 - pc)(1-pb) ]
So, we see that shooting A first increases the winning probability by the amount pc pc (1-pb)/[ 1 - (1 - pc)(1-pb) ]. Now, the only way to make C uncertain is to let pc = 0, at which point his winning chance is the same regardless of who he shoots. But if his hitting probability is 0, he's screwed either way.

8. ### For Bushindo - (others try at your own risk)

Probably not what you were looking for, but here's what I recommend Cole do

Cole should shoot into the air (i.e. not shoot Alex or Bobby) on his turn. Bobby and Alex will shoot one another in sequence. Cole then has the nice advantage of having the first shot on whoever survives between Alex and Bobby.

So, clarification please. Can any participant choose to shoot into the air instead of the other duelists?
9. ### How Many Tickets? (Part 2)

I wouldn't post this puzzle if the answer was the same as the original Bonanova's puzzle

You're right

I'm convinced that the answer, as you say, is not the same as bonanova's answer. My previous answer is incorrect, but I have yet to locate the faulty reasoning. Back to the drawing board.

10. ### How Many Tickets? (Part 2)

Here is a modified version of Bonanova's puzzle

Suppose n tickets numbered 1-n are in a box, and you draw five of them without putting them back in the box.

Your tickets have numbers p1, p2, p3, p4 and p5 on them.

You don't know how many tickets are in the box, but you are asked to estimate how many there are.

What estimate of n has the highest likelihood of being correct?

We wish to find P( N | p1, p2, p3, p4, p5). Let's assume that we have a prior distribution P(N). Also, the probabilities of the obtained number given N is P( p1, p2, p3, p4, p5 | N ) = 1/( NC5). So, here's the Bayesian derivations

So, if we have P(N), the derivation above is straightforward.

We are not given P(N), so let's assume that N is uniformly distributed between max( p1, p2, p3, p4, p5 ) and infinity. Unlike the previous bonanova puzzle, under this improper prior assumption, the denominator term in the equation above converges. Then P( N | p1, p2, p3, p4, p5) is proportional to 1/( NC5), which implies that we should pick N = max( p1, p2, p3, p4, p5 ).

11. ### How many tickets?

Bushindo, Thanks.

I learn so much from you.

Is there also a maximum likelihood estimate of the underlying assumptions

that would make the maximum likelihood estimate of n well defined?

Thanks for the kind words, bonanova, I also learned much from you over the years. The two wonderful examples (among many) that come quickest to mind are Hole in a Sphere and Maiden Escape, both of which taught me completely new ways of thinking.

The comment I made earlier is that the solution to this puzzle will depend on what kind of prior information (or assumption) we have about the probability distribution P(N), since the Bayesian derivation forces us to explicitly use P(N) in the calculation of the answer.

My interpretation of what you are asking is that is there a maximum likelihood estimate for the distribution P(N) such that P(N|p) is well-defined (and my impression is that this is a way to bypass the requirement to know exact values for P(N)). It is possible to assume the distribution P(N) as belonging to some well-defined probability class (exponential, gaussian, negative exponential, etc.) and then use maximum likelihood to find the best-fit curve within the assumed class. Of course, there is no free lunch. We just moved the prior assumption backwards a bit so that we make assumptions about the class of probability curves P(N) belongs to.

You may be asking is there a maximum likelihood estimate for estimating what class of probability function P(N) most likely is in, and that is a much tougher problem. If we have several candidate probability classes (again, exponential, binomial, gaussian, etc.) then we can easily compare between them and find the most likely probability class for P(N). The problem is that there are an infinite number of possible probability class (and most don't have nice tractable parametric forms), and finding the best class over this domain is not possible. So, if we restrict the possible probability classes to a finite number (note that we are making some assumptions here again), we can then solve the problem.

12. ### How many tickets?

I picked p in my draw. Lets estimate n = p. What would have been the likelihood that I got the max number ? : 1/p

If say n = p+1. What would have been the likelihood I picked p in my draw ? : 1/(p+1)

1/p > 1/(p+1).

So given that I picked p , the estimate of n that would give the best probability of me picking p is p itself.

So best estimate of n is p

>

Suppose n tickets numbered 1-n are in a box, and you draw one of them

You don't know how many tickets are in the box, but you are asked to estimate how many there are.

Your ticket has the number p on it.

What estimate of n has the highest likelihood of being correct?

It must be p only because then the Probabability of picking out p = 1/p, for n>p, the probability decreases.

You are both correct. bona_gold_star.gif

nakulendu posted first, so I marked his answer as the solution.

The solutions above are making some assumptions about the properties of underlying distribution of N, which may or may not be warranted depending on your interpretation of the OP.

Implicit within the explanations above is the assumption that P(N) = P(N+1). The explanation goes on to say that since P(p|N) decreases as N increases (without bound, presumably), then it makes sense that p is the most optimal choice.

From an earlier post, the full Bayesian expression for P(N|p) is

Note that if we assume that that N is uniformly distributed between p and infinity (which is an improper prior distribution), then the denominator of the term above becomes the harmonic series, which does not converge. If we assume that N is uniformly distributed between p and some large finite M, then the solution is warranted.

So, choosing N=p is the correct solution IF P(N) is uniformly distributed AND P(N) -> 0 as N approaches infinity.

13. ### How many tickets?

Suppose n tickets numbered 1-n are in a box, and you draw one of them

You don't know how many tickets are in the box, but you are asked to estimate how many there are.

Your ticket has the number p on it.

What estimate of n has the highest likelihood of being correct?

It seems like this problem is dependent upon what is a reasonable a priori distribution for N.

Let P(N) be the prior distribution for the number N. Let P(p|N) be the conditional probability for p given N; we know that P(p|N) = 1/N. We wish to compute the conditional probability P(N|p), which can be expanded using Bayes theorem

So, if we are given a well-defined probability function P(N), we can easily compute the number N that gives that maximum probability P(N|p).

The problem is that we are not given what P(N) is. One reasonable choice would be to assume that P(N) is the improper uniform distribution from p to infinity (every number has an equal chance of occuring). However, if we plug that improper distribution into the equation above, we would be able to cancel out P(N), but we'd end up with a harmonic series in the denominator, which does not converge.

Oh wait, the answer is much simpler than that. Turns out this is the easiest bonanova puzzle I have ever seen =)

Since "n tickets numbered 1-n are in a box", if we get number p = (1-n), then it is easy to see that n = 1 - p.

14. ### How many tickets?

Suppose n tickets numbered 1-n are in a box, and you draw one of them

You don't know how many tickets are in the box, but you are asked to estimate how many there are.

Your ticket has the number p on it.

What estimate of n has the highest likelihood of being correct?

It seems like this problem is dependent upon what is a reasonable a priori distribution for N.

Let P(N) be the prior distribution for the number N. Let P(p|N) be the conditional probability for p given N; we know that P(p|N) = 1/N. We wish to compute the conditional probability P(N|p), which can be expanded using Bayes theorem

So, if we are given a well-defined probability function P(N), we can easily compute the number N that gives that maximum probability P(N|p).

The problem is that we are not given what P(N) is. One reasonable choice would be to assume that P(N) is the improper uniform distribution from p to infinity (every number has an equal chance of occuring). However, if we plug that improper distribution into the equation above, we would be able to cancel out P(N), but we'd end up with a harmonic series in the denominator, which does not converge.

15. ### More divisibility

Indeed, this number is a bit smaller than the one I found.

Perhaps, it is easier to evaluate as 410000...000214: "41", 97 zeros, and "214" (which is, clearly, divisible by 7) multiplied by 100 "1"s.

But where is the proof that it is the smallest number that meets the conditions?

I can't prove that it is the smallest number. The solution that you posed made me realize that my solution is flawed. Originally, I thought I had a divisibility rule for 100 ones, which allowed me to compute the smallest integer within those rules. The number you came up with made me realize that the divisibility rule I had only worked on a subset of the possible divisors.

So, there might be smaller solutions out there. I made a mistake in an earlier post. I should have called that number "the smallest number that I could find" instead of calling it the solution.

16. ### More divisibility

I found this interesting 102-digit number: 490000...000077 (“49”, followed by 98 zeros, followed by “77”.) If you multiply that number by 111...111 (one hundred 1-s), the resulting 201-digit number:

54444...4447555...55547 (“5”, followed by 99 “4-s”, followed by “7”, followed by 98 “5-s”, followed by “47”) meets the conditions set forth in the OP, except I cannot guaranty it is the smallest number that does. (I did not find any simple combination of "1-s" and "0-s", which multiplied by all "7-s" would yeild "4-s", "5-s", and "7-s", while avoiding "8-s".)

On a side note. While playing with these numbers I found a simple yet curious divisibility rule, of which I was not aware before. I should try and construct a problem based on it.

Excellent work so far. Sorry for not responding earlier. I was on a vacation and didn't have access to a computer

The number above indeed satisfies the conditions giving in the OP, but there is a smaller number. The number I have in mind is a 201-digit number, but the 201-th (leftmost) digit is a 4.

I'm interested in seeing a puzzle based on the divisibility rule you found. For what it is worth, I also based this puzzle on a divisibility rule as well.

Solution to the puzzle

If we multiply 100 7's with the following number

``` 58 571 428 571 428 571 428 571 428 571 428 571 428 571 428
571 428 571 428 571 428 571 428 571 428 571 428 571 428 571
428 571 428 602
```

We would get the following 201-digit number

```455 555 555 555 555 555 555 555 555 555 555 555 555 555 555
555 555 555 555 555 555 555 555 555 555 555 555 555 555 555
555 555 555 747 777 777 777 777 777 777 777 777 777 777 777
777 777 777 777 777 777 777 777 777 777 777 777 777 777 777
777 777 777 777 777 777 754
```

If you look closely at the number above, you might be able to see the divisibility rule used in the construction of this puzzle.

17. ### Mission to Mars!

Yes...thats right...Thanks

If the fuel is used for guidance and anti-explosibility measures only, and not for propulsion; then all five spaceships could be flung from the moon towards Mars starting at the same time. All the ships would be tethered in such a way that adjusting the course of one would alter the course of the fleet as a whole. For the first quarter of the trip, Spaceship #1 does all of the course correction. For the second quarter, Spaceship #2; and so on until landing on Mars. Once on Mars, Spaceship # 5 redistributes it's fuel equally between all of the spaceships to power their lasers and defeat the Martians.

Dear wolfgang, the solution that you consider correct suffers from two problems: 1) It violates common understanding of thermodynamics and 2) even if we allow such violation, it does not satisfy the conditions in your original post. Here is why

From your original post, each ship needs to make constant trajectory corrections since "without fuel the spaceship will miss its direction". The solution that you accept states that 4 space-ships can "piggy-back" on the course correction ability of 1 ships. That is, the ships would be tethered in such a way that adjusting the course of 1 ship would alter the course of the whole.

Of course, a simple understanding of physics would indicate that we don't really save any fuel if we allow 1 ship to change the course for all 5. If a single ships needs to make a minor course correction, it would need to provide enough thrust (energy) to move its mass however much required. If a single ship is tethered to 4 other ships, Newton's equation (F=ma) implies that the navigating ship would need 5 times the energy to alter the current course. So, in reality, tethering 5 ships together would make the single navigating ship burn fuel 5 times as fast as it changes course.

That said, even if we allow such violation, how do you suppose to get the ships home. From your OP, "each spaceship has a fuel capacity to allow it to fly exactly 1/4 way to Mars" and "Each spaceship must have enough fuel to return safe to the base space station". Even allowing for the thermodynamics violation, all your 5 ships would end up on Mars with 1 tank left, which is only good enough for 1/4 of the way back to the moon base.

While I understand that lateral and out-of-box thinking are valued around here, in general such solutions have to be internally consistent (and avoid violating explicit conditions set forth in the original post). Also, in this forum, we obey the law of thermodynamics.

18. ### Mission to Mars!

Dear Bushindo....you should notice that I have concidered the moon as our base station(!) and not the earth.So you should think a little bit (laterally),and try to reduce the number to 5 ships only!...note : all of them will reach mars

So I guess the lateral solution of this puzzle is contingent upon the properties of this 'fuel'. My previous solution assumes that "the fuel spent is proportional to distance travelled". Since that doesn't seem to be the case, I'd like some clarification about the conditions of the puzzle.

1) My impression is that the fuel is used to make constant minor course corrections during the trip (per OP: "without fuel the spaceship will miss its direction") and to maintain the life-support and ship computer (per OP: "… and may explode"). Is this correct? If that is not correct, what is the fuel being used for?

2) Consider the following two scenarios

* A single ship flies from base (x=0) to the one-fourth point (x=1/4)

* A ship flies from the halfway point (x=1/2) to the three-fourth point (x=3/4)

Does the ship use the same amount of fuel (1 tank) in both scenarios?

3) Are the ships supposed to land on the surface of Mars?

19. ### Checker lines

Here is the amended equation that should return the desired outcome of the larger of the two input values, m or n.

This assumes a few things:

1. Given m and n, the largest value presented is the minimum number of checkers needed to be placed on the board.
2. The factorial of a negative number will be equal to the negative value of the factorial of that number's positive counterpart. -z! = -(z!)
3. The factorial of 0 equals 1.
4. I haven't messed up anywhere.

*Items 2 and 3 are supported by WolframAlpha ( 0! = 1 ; -5! = -120 ; -6! = -720 ; -7! = -5040 ; etc... ), but don't seem to be uniformly accepted elsewhere.

Comments on point 2. and 3.

The factorial of 0 is indeed 1 by convention.

As for the assumption (-z)! = -(z!) for a positive z, it is not true. Check out the gamma function, which is the extension of the factorial function to real and complex numbers. The gamma function is undefined at non-positive integers. I suspect Wolfram Alpha is interpreting your testing inputs as -1 * (z!). For instance, -1*(5!) = -120 ; -1*(6!) = -720 ; -1*(7!) = -5040.

20. ### Mission to Mars!

Well...9 ships are too much...try to reduce the number

I don't think the number of ships can be reduced below 9. Here is why

Let's say that the moon is at location (x = 0), and Mars is at (x=1). Each ship can travel a distance of (1/4) on a full tank.

The moon base has unlimited fuel, so if we only have 1 space-ship, the max range we can reach (and return safely) is (1/4)*(1/2) = 1/8.

We can imagine that each spaceship we use can extend the fuel capacity of the base by a distance up to (but not including) 1/8. For instance, at a lesser distance of (1/9), a single spaceship can take as many trips as required to shuttle back and forth to deposit fuel at location (x=1/9). Each trip the ship can deliver 1/36 of a tank, but we can take as many trip as necessary. Note that the single ship *cannot* extend the fuel-supply to the precise distance of (1/8), as it would be not able to deposit any fuel at (x=1/8) and return home safely.

So, if a single space-ship can extend the fuel-supply line by 0 < d < 1/8, then we would need minimum of 8 ships to extend the fuel-supply line past (x=7/8). We then need 1 more ship to go to Mars and return. That makes 8+1 = 9 ships.

21. ### Mission to Mars!

I want the fewest number of ships required , and at least one of them should launch with enough energy keeping it working.

If you want the number of unique ships required, then

You would need 9 different ships. Let's label the ships 1-9. Let the distance to Mars be 1 unit. Let the base space station be located at (x=0) and Mars be at (x=1). The idea is as follows

1) Let ships 2-9 go to the location (x = 1/9) and stay there. Ship 1 would then shuttle back and forth between the starting point (x = 0) and (x = 1/9) to deliver gas to the ships 2-9 until all 8 ships are full of gas. Each trip ship 1 can deliver 1/36 of a tank.

2) Let ships 3-9 go to the point (x=2/9). Ship 2 now shuttles back and forth between (x=1/9) and (x=2/9) to deliver gas to ships 3-9 until all are full of gas. Each trip ship 2 will have to wait at (x=1/9) for ship 1 to deliver gas to it, and then it can deliver (1/36) of a tank at (x=2/9).

3) Ship 4-9 will then proceed to point 3/9, and then ship 3 shuttles back and forth as above. Eventually ship 9 will be able to reach Mars and go back.

22. ### Meeting of the dragon minds

i got a bigger number

Dragons have to meet for a brainstorm in a convention center. The delegates have to be selected to provide the maximum efficiency of the brainstorm session. A dragon has any amount of heads, and for any N, any amount of N-headed dragons is available if needed. The problem is that the size of the convention center is limited so no more than 1000 heads can fit into the assembly hall. The intellectual power of a dragon pack is the product of head numbers of dragons in the pack. How should an optimum pack look like (the total number of dragons, the head number distribution)?

I'm getting the max of

Intellectual power is 3^333

Ha, silly me

Intellectual power is 3^(332) * 4

23. ### Meeting of the dragon minds

Dragons have to meet for a brainstorm in a convention center. The delegates have to be selected to provide the maximum efficiency of the brainstorm session. A dragon has any amount of heads, and for any N, any amount of N-headed dragons is available if needed. The problem is that the size of the convention center is limited so no more than 1000 heads can fit into the assembly hall. The intellectual power of a dragon pack is the product of head numbers of dragons in the pack. How should an optimum pack look like (the total number of dragons, the head number distribution)?

I'm getting the max of

Intellectual power is 3^333

24. ### Mission to Mars!

The thing about the space travel is that it doesn't require fuel to cover the distance. The fuel is required to break free of the gravity and to establish the direction and speed. Once you're far enough from any large celestial bodies you can cover arbitrarily large distances without any fuel consumption. So the fuel consumption is not proportionate to the distance, but is proportionate to the masses of the celestial bodies you're launching from or you're in gravitational field of.

Anyway, assuming that the fuel is consumed at a constant rate for the distance travelled I have a couple of questions:

1) Does the spaceship that reaches Mars also have to return to the station?

2) Can the spaceships that return to the base be used again, and if so, then do we count the number of spaceships used or do we count the number of spaceship launches?

k-man makes a good point. For the sake of the puzzle, let's assume that the fuel spent is proportional to distance travelled. Let's also assume that the ship that gets to Mars will need to return home.

It's not clear whether the OP wants the minimum unique number of ships required or the minimum unique number of ship launches.

Let's assume that we want the minimum unique number of ships (a ship can launch more than once). I'm getting an answer of 9 ships required if we allow a ship to turn off its engine and wait for fuel; see below for an outline of the solution. This will take a *long* time, though.

Let's assume that we want the minimum number of ship launches (also the one that takes the least time to get to Mars). Here are where things get interesting.

Let the distance to Mars be 12 unit. On a full tank a ship can travel 3 units. We define a function s(d) as the number of unique launches required to put 1/3 gas tank at distance d (and have all the ships return home to base). It is easy to see that

s(1) = 1

s(2) = 3

s(3) = 9

And in general,

s(N) = 2 * ( s( 1) + s( 2 ) + … s( N - 1 ) ) + 1 = 3(N - 1 )

The above is derived by assuming for distance N, there is a special ship designated to reach distance N. At every integer distance less than N on the trip out and on the trip back, it gets (1/3) of a tank from a supporting network on other ships.

So, if we want to put (1/3) gas tank on Mars, then we would need 311 launches. Since we don't want to put any gas on Mars, we would only need 2*310.

25. ### Trailer Parks

Why are trailers rectangular And flower pots circular in shape? (There is a mathematical explanation)

Okay, I'll bite =)

I would guess that it is dependent on the intended function. Flower pots are intended to hold both soil and plants. The soil comes in contact with both the bottom of the pot and the sides, and thus the flower pot needs to be circular in order to distribute the horizontal force from the soil and plants evenly. A rectangular flower pot would have weak-points at the corners where the sides join, and thus would not be as optimal a design.

Trailers are rectangular because they are intended to carry their load on the floor or bottom of the enclosure. The walls of the trailers are not primarily intended for load-bearing, and thus they do not need to be circular to distribute the weight.

As a further argument, consider gas trucks that carry their liquid load in cylindrical tanks. Since the liquid gas applies pressure on both the floor and the sides of the tanks, the tanks are designed as a cylinder (actually more like a sphere elongated on 1 axis- see attached image) to prevent structural faults that would arise from uneven force distribution.

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