A Fistful of Brain Teasers

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

Don Atkinson posted:

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

If the evidence suggests that the original decision was wrong, one should change it given the opportunity. Otherwise not (in the management example) or only if you are indecisive, or perceive some bias affecting things in the case of the game show. 

In this case there is no evidence that shows the original pick to be wrong. The best there seems to be is a conviction that the odds are 2/3 that car is behind the door that hasn’t been chosen or opened, so 1/3 that the original choice was correct. Unfortunately at this point it appears to me to be an unsubstantiated conviction by so eg people - which is not evidence...

 

Innocent Bystander posted:
Don Atkinson posted:

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

If the evidence suggests that the original decision was wrong, one should change it given the opportunity. Otherwise not (in the management example) or only if you are indecisive, or perceive some bias affecting things in the case of the game show. 

In this case there is no evidence that shows the original pick to be wrong. The best there seems to be is a conviction that the odds are 2/3 that car is behind the door that hasn’t been chosen or opened, so 1/3 that the original choice was correct. Unfortunately at this point it appears to me to be an unsubstantiated conviction by so eg people - which is not evidence...

 

That was a fairly common comment made by directors.

Innocent Bystander posted:
winkyincanada posted:
Don Atkinson posted:

Alright, here's an old one so I hope you haven't heard the answer already. If you haven't it is quite interesting. Enjoy!

Late one cold, rainy night in November after a few pints and a rather dubious curry you flop down onto your favourite listening couch and turn on the TV. Ah...that's better. But Have I Got News For You isn't enough to keep your eyes from closing and you soon fall into a deep, deep slumber and start to dream..

Congratulations! You have been selected as a participant in a very popular national TV gameshow. This is your first time on TV and on such a show and you are overwhelmed with excitement and nervousness.

You find the first rounds easy and before long the other contestants have been eliminated and you have made it to the final challenge. The audience go wild! What a night!

Now the final challenge begins. The lights dim and the cheers of the crowd die down. Ant & Dec show you three doors numbered 1, 2 and 3. They say that the Porsche is hidden behind one of the doors. You must choose which one. If you choose correctly you win the Porsche. If not, you win a goat. The tension mounts. The audience can't contain themselves: shouts begin..."door 1"..."no door 2"...etc. You hesitate. Ant urges you to make a choice. Finally you choose door number 3. A green light comes on over door 3. Oh no...did I choose wisely...

A hush comes over the theatre as Dec walks over to door number 1 and motions to open it. The atmosphere can be cut with a knife. Slowly, Dec opens door 1 to reveal a goat! Phew that was lucky. The crowd bursts into a relieved cheer. But soon Ant has gripped the handle of door 2 and the crowd go silent. You can hear a pin drop.

The tension is unbearable...

Suddenly Ant stops turning the handle of door 2 and turns towards you. He says he is in a good mood tonight and wants to help you as much as he can. He offers to do you a favour and allow you to change your selection. You can now stick with your original choice...door 3 or switch to door 2. It is up to you and you have just 10 seconds to decide. The audience erupts! "door 2, go for door 2"..."no, no stick with door 3"...

You have 5 seconds left. You are sweating. Everyone is hanging on your next words. Do you stick with door 3 or switch to door 2?

As I said, it's an old one, so you no doubt know the best thing to do and that's fine - just post your option. But..

...top marks go to the best explanation for your decision to open Door No............

Always switch. You have 2/3 chance of winning if you switch. If you stick with your first choice, your chance of winning remains at 1/3, the same as it was when you selected it. Think of it this way. After you have chosen the first time, your door has a 1/3 chance of winning. The other two doors, combined, have a 2/3 chance of containing the car. Removing one of those doors  effectively "compresses" that 2/3 probability into the other door. So switch!

(This game (The Monty Hall game) depends on the host knowing which door contains the car, and always purposefully choosing a goat-door. Obviously, if the host opens a door at random, they will sometimes reveal the car. It changes the premise of the problem a bit, but whenever the host-opened door reveals a goat, you should switch.)

However, when you make that second choice the odds at that point are 1/2 not 1/3 because one has been eliminated, so it is just a toss up: why change? So the question is simply, will you have the courage of your convictions, or swap? However if you don’t switch and don’t win you will kick yourself all the more for having chosen wrong twice, so maybe that is a reason for change, if a bad one - it is an evil thing to offer you just for greater entertainment. But then, that is what game shows are about!

if it was me I’d think the presenter is trying to save the cost of the Porsche (ok, the motorised goat) and tempting me to think that I should have chosen door 2.

No. The probabilities are now 1/3 - 2/3, not 50:50. The likelihood that you were right the first time doesn't change. If there were a thousand doors, one car, and 999 goats, and the host opened 998 of them showing goats, would the chance that you picked the correct door the first time now be 50%? No, it stays at 0.1% and the likelihood that the car is behind the other door is 99.9%.

The answer is always to switch.

I don’t pretend that my explanation is any better than the explanation given by others but I set it out below nonetheless. I have already alluded to some aspects of it anyway.

Doors are labelled A, B and C

A car lies behind one door and a goat behind each of the other two

 

Given the choice of zero doors, the probability of getting the car is 0

Given a choice of only one door, the probability of getting the car is 1/3

Given a choice of all three doors, the probability of getting the car is 1 (it’s a dead cert !)

Given the choice of any two doors, the probability of getting the car is 2/3

 

That last option is crucial and………..it’s yours for the taking !

 

Instead of choosing C and sticking with it (1 in 3 chance)…..

…………..choose A and B

 

Yes, you heard right…..choose A and B

All you need now is to persuade the Show-Host to open both doors A and B

….and you then stand a 2/3 chance of winning that car !

 

And what’s so good about this is, you already know how to do it !

 

Youtell the Show-Host that you have “Chosen” Door C.

He then voluntarily opens either A or B (let’s say A)

You then tell him to open the other one ie B

No time to think about it at work, but I sussed it for myself cycling home (see, cycling is good for the brain, too!)

actually a very simple explanation:

At the time of picking, there is a 1/3 chance of it being behind the door I pick, so 2/3 chance that it will be behind  one or other of the other two doors? Rule out one of those two, and the 2/3 chance is now resting on just the one door, that I didn’t pick, my priginal choice still being 1/3.

THe evidence is now there, so I change my mind and swap!

Innocent Bystander posted:

No time to think about it at work, but I sussed it for myself cycling home(see, cycling is good for the brain, too!)

actually a very simple explanation:

At the time of picking, there is a 1/3 chance of it being behind the door I pick, so 2/3 chance that it will be behind  one or other of the other two doors? Rule out one of those two, and the 2/3 chance is now resting on just the one door, that I didn’t pick, my priginal choice still being 1/3.

THe evidence is now there, so I change my mind and swap!

I'm not yet convinced of that.

For sure it's true for the sort of off-road recreational cycling that I do (see my photos of Canada).

However, when people commute-cycle on our current road system, either knowing the risk involved or totally ignorant of the risk, I am inclined to think the opposite is true.

But that's for another thread

Don Atkinson posted:
Innocent Bystander posted:
Don Atkinson posted:

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

If the evidence suggests that the original decision was wrong, one should change it given the opportunity. Otherwise not (in the management example) or only if you are indecisive, or perceive some bias affecting things in the case of the game show. 

In this case there is no evidence that shows the original pick to be wrong. The best there seems to be is a conviction that the odds are 2/3 that car is behind the door that hasn’t been chosen or opened, so 1/3 that the original choice was correct. Unfortunately at this point it appears to me to be an unsubstantiated conviction by so eg people - which is not evidence...

 

That was a fairly common comment made by directors.

And why should anyone reverse their decision without a reason to believe it is better than the original decisiom? You saying it is better doesnt make it so: what is needed is a weighing up of allavailable evidence, balance of probability of the new information being correct, and assessment of pros and cons of change. Given the facts as I reasoned in my explanation it is clear cut, but someone else saying without doing anything to show why their proposal is better is not a reason for making the change, though it can be a reason to consider it and make additional investigations or considerations.

Innocent Bystander posted:
Don Atkinson posted:
Innocent Bystander posted:
Don Atkinson posted:

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

If the evidence suggests that the original decision was wrong, one should change it given the opportunity. Otherwise not (in the management example) or only if you are indecisive, or perceive some bias affecting things in the case of the game show. 

In this case there is no evidence that shows the original pick to be wrong. The best there seems to be is a conviction that the odds are 2/3 that car is behind the door that hasn’t been chosen or opened, so 1/3 that the original choice was correct. Unfortunately at this point it appears to me to be an unsubstantiated conviction by so eg people - which is not evidence...

 

That was a fairly common comment made by directors.

And why should anyone reverse their decision without a reason to believe it is better than the original decisiom? You saying it is better doesnt make it so: what is needed is a weighing up of allavailable evidence, balance of probability of the new information being correct, and assessment of pros and cons of change. Given the facts as I reasoned in my explanation it is clear cut, but someone else saying without doing anything to show why their proposal is better is not a reason for making the change, though it can be a reason to consider it and make additional investigations or considerations

I used the Car/Goat puzzle to encourage directors to seek the advice of experts. (see one of my earlier posts)

This puzzle demonstrates that a "gut" feeling can lead to inappropriate decisions. The concept is difficult to grasp. Don't make decisions until you understand the issues and likely outcomes. Accept the evidence of your experts. If you're not sure, ask two or three of them.

When I and others (in other words two or three experts) said it was better to switch, you knew that we had justifiable evidence. We simply hadn't yet revealed it. Even when revealed, many directors don't understand the evidence in front of them and have to trust their experts.

JRHardee posted:

FWIW, I'm pretty sure that Monty Hall and the "Let's Make a Deal" show were long retired when this puzzle became a thing. Tapping into my childhood memories, the opportunity to change your mind was NOT a part of the show.

I never saw this Monty Hall show and I only became aware of this "puzzle" some 15 years or so ago.

It's more than likely that a bit of "folk law" has developed around the show and the puzzle such that have become synonymous (is that the right word ?)

I hope we all enjoyed this one, I think it's a real Teaser.

Don Atkinson posted:
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I hope we all enjoyed this one, I think it's a real Teaser.

Indeed. 

Incidentally, I wonder how many of those who answered correctly had come across it before and learntbthe correct answer - or am I the only Dumbo who couldn’t see it? If so, no matter - it is always good to learn.

This will be especially useful if I ever find myself having to choose which path to cycle down, knowing one is nice and smooth and has no motorists or pedestrians, one joins a narrow road with speeding cars and pedestrians with dogs, and the third is a road with a cycle lane, but cyclists are taxed heavily for using it - when just after I have chosen one of them and am about to set off someone erects a sign on one of the others saying “Cycle route: toll £5 a mile”.

Target Board JPEG

Arrows !!

Aaron, Bertie and Chuck have created their own Target Board.

Seven circles on a beige background. Each circle has its own points value as does the beige background which is so large, it simply can’t be missed !

Each player gets to fire six arrows at the Target Board.

Aaron made 25 with his first three shots. It was a coincidence that the third shot scored 3.

Chuck hadn’t scored at all, when Bertie picked up 2 points.

At the end of the tournament, 50 had been scored only once, 20, 10 and 1 were scored three times, and 25, 5, 3 and 2 were each scored twice.

To their utter amazement, despite all hitting different combinations of circles and background, all three participants wound up with the same score !

Who shot the Bull’s-Eye 50 ?

The Baron's Treasure

The Anglo-Austrian Baron Von Dane-Fraydenegg kept his gold treasure (fairly got by moderating various fora) in a treasure house. In each room of the treasure house were as many chests as there were rooms. In each chest there were as many gold coins, as there were chests in that room. All the gold coins were of the same size and value.

When he died, the Baron's will was that his surgeon-barber should receive one chest of gold coins. The remaining coins were to be divided equally between the Baron's three sons .

The three sons were proud and fierce men who would definitely resort to bloodshed if the coins could not be divided equally.

The question is simple: -

a) Was there blood shed.
b) Was there no blood shed.
c) Is there no way of knowing whether there was blood shed or not.

The proof is the real requirement, not simply a one in three guess.

There wil be no bloodshed (unless there was only 1 room and the sons turned on the surgeon barber).

r = no of rooms and no of coins per chest

r^2 = total no of chests

r^3 = total no of gold coins

barber receives 1 chest = r coins

balance of coins = r^3 - r = r(r^2 -1)

Obviously if the no. of rooms is a multiple of 3 then r(r^2 -1) is divisible by 3 otherwise, if r is not divisible by 3 then (r^2 -1) is divisible by 3.

Either way the product of r(r^2 - 1) must be divisible by 3 so that the coins can be divided equally between the 3 sons.

 

sjbabbey posted:

There wil be no bloodshed (unless there was only 1 room and the sons turned on the surgeon barber).

r = no of rooms and no of coins per chest

r^2 = total no of chests

r^3 = total no of gold coins

barber receives 1 chest = r coins

balance of coins = r^3 - r = r(r^2 -1)

Obviously if the no. of rooms is a multiple of 3 then r(r^2 -1) is divisible by 3 otherwise, if r is not divisible by 3 then (r^2 -1) is divisible by 3.

Either way the product of r(r^2 - 1) must be divisible by 3 so that the coins can be divided equally between the 3 sons.

 

Neat solution sjb.

I normally expand the expression (r² - 1) to give (r-1)(r+1)

which together with "r" gives the number of coins as the product of r(r-1)(r+1)

which re-arranged as (r-1)r(r+1) is more obviously the product of three consecutive numbers

one of which has got to be divisible by 3

Yes I'd forgotten that simplified proof.

There is a less elegant proof that r^2 -1 is divisible by 3 when r is not itself divisible by 3.

Let r = 3n - 1

then r^2 =  (3n-1)*(3n-1)     =     9n^2 - 6n + 1  

So r^2 -1 =     9n^2 - 6n       =    3n(3n -2)

Let r = 3n - 2

then r^2  = (3n - 2)*(3n - 2)  =  9n^2 - 12n + 4

So r^2 -1  = 9n^2 - 12n + 3   = 3(3n^2 - 4n + 1)

Lord of the Wings JPEG

The Lord of the Wings Bookworm

I was re-arranging my bookshelf the other day. My three volumes of “Lord of the Wings” needed dusting but I noticed they were in the “correct sequence” as you might expect and as shown in the diagram (although in practice they were tightly packed together - you could almost say "stuck together)

However, much to my dismay, I discovered that a bookworm had tunnelled its way from Page 1 of the trilogy, right through to the last page of the trilogy.

Now the three volumes are identical in size and construction. They are hard-backed, with the front and back each 1/8 of an inch thick. The pages are together three inches thick in each volume.

How long was the tunnel that had been bored by the bookworm ?

notnaim man posted:

I make that 1 1/2 inches

I'm going to give you 9/10

I can see that you recognise the solution, ie where the first and last pages are.

You probably got slightly confused with the thickness of each volume being three inches, not each volume being one inch.

"The pages are together, three inches thick in each volume"

Man on a Rope JPEG

A rope is passed over a pulley which is attached to the rigid roof.

At one end of the rope is a weight.

The weight exactly balances a man hanging onto the other end of the rope.

ie it’s in static equilibrium.

The man starts to climb the rope. Don't rely on the picture, most people climb a rope using their feet together.

What happens ?

 

The rope is fully flexible and the pulley is frictionless. I therefore don’t think it matters whether the rope/pulley contact is frictionless or not. But if you consider this aspect to be important, say so and allow for it in your explanation.

You might well be right sjb......

....the opinion of most  mathematicians and scientists today is that the man and the weight will always remain opposite each other. So, as the man climbs the rope, so the weight rises. (when I say "most", that seems to be the conclusion in quite a few Googled links !!!!)

Lewis Carroll, he of Alice in Wonderland fame, suggested that the weight neither rises nor falls. But it isn't entirely clear whether he meant in relation to the man (as above), or in relation to the surroundings !

O f course, neither of the above seems IMHO to take acceleration into account.

But I suppose, that if the man climbs slowly, then it might be possible to ignore acceleration.

But when he stops climbing, no matter how slowly he was climbing, I have this inner feeling that the system as a whole continues to move, unless a little bit of friction exists between the rope/pulley or in the pulley bearing !

I have £1000 in £1 coins.

I also have 10 bags in which to put those coins.

Each bag will be sealed and labelled showing the number of coins therein.

I wish to distribute the coins to those bags, such that going forward I am able to simply pick up one or more bags and without opening them or redistributing the coins further, be able to take any sum of £££ from £1 to £1000.

How should I distribute the coins.

I only need to be able to make one such transaction, not multiple transactions.

Hi IB,

I can see where you are coming from with the "weights" analogy, but.....

.....which bag, or combination of bags, from your list would you hand to me if I asked for (say) £11 ?

or perhaps (say) £67 ?

The idea is that I know how much is in each bag (I put the coins into the bags and labelled them accordingly) and in future need to be able to simply pick up one or more bags eg in a bit of a hurry !......so as to take ANY sum of whole pounds from £1 to £1000

Cheers, Don

Having not used physical weights with balances I misremembered the sequence, and should have stopped to check but didn’t as that was 10 bags. Correct sequence tmake any number is of course 1,2,2,5,10,10,20,50,100,100,200,500

However that would require 2 bags too many, so I’m looking for the hidden twist, but so far to no avail, other than wondering about your need to state that you only need to make one transaction, which I had taken as read. I suppose you could have 3 single coins in your pocket, so not needing the first 2 bags, but I think that is outwith the meaning.

No, there is no hidden meaning. It’s just straightforward. 

10 bags contain all 1000 coins. We just need to put the right number of coins into each bag so that by selecting one or more bags you can pick up any amount of £££ from £1 to £1000.

If the bags were then replaced, you could select some other amount by selecting a different combination of bags ........etc etc

so, no hidden meanings, no twist of words, just a straightforward puzzle !

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