Women Or Tigers? — Solution to Final Challenge!

Did you find out? Actually, it wasn’t that difficult after all …once you got on the right path. Here’s what I have come up with — tell me if you have the same, or another answer, by commenting on the post.

If the prisoner wants to know whether Room VIII is empty or not, the answer he gets must be crucial to solve the whole puzzle. So let’s take that room as our point of departure.

We know already that Room VIII cannot contain a woman, as its sign says it hosts a tiger, and the sign on the woman’s door is correct. Leaves us with two possibilities: Room VIII is either empty or hosts a tiger.

It will not help us much to assume it is empty. Then its sign can be either true or false. So let’s assume there is actually a tiger in Room VIII. This means the sign is not correct. The sign gives two statements, with an ‘and’ in between. If one or both statements are false, the sign is false, as it should be if there is indeed a tiger behind the door. The first statement ‘this room hosts a tiger’ is true (so we assume for now) — ergo, the second ‘Room IX is empty’ must be false for the sign to be false.

So Room IX is not empty. It contains either a woman or a tiger. Again, its sign gives two statements linked by an ‘and’. The first statement says the room contains a tiger. If there is a woman in it, the sign must tell the truth, which it then can’t as it talks about that tiger. So… it does contain a tiger. And, as with Room VIII, the second statement must be false in order for the sign to be false. The sign on Room VI therefore tells the truth (because the sign on Room IX, lying, says that it doesn’t). Let’s keep in mind (write down) that Room VI either hosts a woman or is empty.

The sign on Room VI says that the sign on Room III is false. This one gives two statements, this time linked by an ‘or’, meaning that both statements must be incorrect in order for the sign as a whole to be false. To take the second statement first (you’ll see in a second that in this case, that’s easier, in order to keep some sort of overview): ‘sign VII is lying’ …so sign VII must be true, and sign VII says the woman is not in Room I. Now that is good to know, let’s write it down too.

…and continue with the first of the two statements on Room III, which is also false (see the paragraph just above): ‘sign V tells the truth’. So it does not. Sign V is yet another two statements linked by an ‘or’. They are both false, according to the logic explained just before. For one, this means that Room II is indeed empty, as its sign states. OK. It also means that sign IV is lying.

Sign IV says that sign I is incorrect. So it is correct …that ‘the woman is in a room with an uneven number’. We know from two paragraphs further up that she is not in Room I. We also know that she is not in Room III as its sign is false. Same goes for Room V (go up just one paragraph from here), and for Room IX (go up three paragraphs from here).

Leaves us with one room carrying an uneven number: Room VII. A quick check: The sign on this room must tell the truth if the woman is indeed behind the door. It says that the woman is not in Room I. Indeed, as she cannot be in two rooms.

One doubt remains though! The premise for this string of logic was that Room VIII is not empty. But what if it is, what if that’s the answer the king gave the prisoner? Can he use that answer to follow a different logic and reach the solution in a different way, or even reach a different solution?

I would like to have your answers to this! Comment below if you have it!

Women Or Tigers? — The Final Challenge!

Now we’re finally talking! This seventh and last and ultimate logical challenge should be able to keep your brain cells busy for a little while. Read on.

‘It’s terrible!’, said the king. ‘I cannot seem to make the challenges difficult enough to have even one of those damn prisoners eaten! I will give it one more chance, one last challenge, which requires the prisoner to really exercise his thinking!’
The minister, as usual, agreed.

And the king was not exaggerating. The seventh prisoner was not given two, nor three doors to choose from — but no less than nine! The king told him that only one of the rooms hosted a woman. Each of the others might be empty …or host a hungry tiger.

He also said to the prisoner that the sign on the door hiding the woman would be true, those on the doors (if any) hiding a tiger would be lying, and those (if any) leading to an empty room could be either true or false.

Here is what the signs said:

ROOM I: THE WOMAN IS IN A ROOM WITH AN UNEVEN NUMBER

ROOM II: THIS ROOM IS EMPTY

ROOM III: EITHER SIGN V TELLS THE TRUTH, OR SIGN VII IS LYING

ROOM IV: SIGN I IS LYING

ROOM V: SIGN II IS LYING OR SIGN IV TELLS THE TRUTH

ROOM VI: SIGN III IS LYING

ROOM VII: THE WOMAN IS NOT IN ROOM I

ROOM VIII: THIS ROOM HOSTS A TIGER, AND ROOM IX IS EMPTY

ROOM IX: THIS ROOM HOSTS A TIGER, AND SIGN VI IS LYING

The prisoner looked at the doors, read the signs, and reflected for a long time. He then said, angrily: ‘This challenge cannot be logically solved! That is not fair!’
‘Yes, I know!’, replied the king, laughing.
‘Easy for you to laugh’, said the prisoner and added: ‘At least give me one hint: Is Room VIII empty or not?’
The king was a fair person, and, impressed by the prisoner’s clear-mindedness, he did tell him whether Room VIII was empty or not.
When the prisoner had this answer, he also knew, after quite a bit more thinking, where the woman was hiding.
Which room did he choose?
Take a deep breath, and probably pen and paper, and a bit of time and peace. And do post your answer and how you reached your conclusion. ‘Cause I don’t have the solution anymore…
I hope you have enjoyed these two weeks of logical challenges. And may you find your woman too, or your tiger.

Women Or Tigers? — Solution to Sixth Challenge

A bit more tricky, or at least time-consuming, yesterday’s challenge, wasn’t it?

The solution is that the woman is in Room I and the tiger in Room II. This is the only combination that fits with the signs on the doors and the conditions given as to whether they tell the truth or not.

But let’s look at the different possibilities in detail anyway.

If the woman is in Room I, that sign tells the truth, meaning that Room III is empty. The tiger must thus be in Room II, which fits with the sign on that door telling a lie.

This already rules out the second of second possibility, which would be a woman in Room I, an empty Room II, and a tiger in Room III — ‘cause then the sign on Room I would lie and cannot contain the woman.

If the tiger is in Room I, this sign is lying, meaning that the woman would be in Room III. She cannot be, as the sign on her door must tell the truth. (And, just for the thought experiment, if instead Room III is empty, the sign on the tiger’s door would not be lying as it is supposed to.)

Leaves us two theoretical possibilities, both implying that Room I is empty. As the sign on the empty room can both tell the truth or a lie, we have to move on to assumptions about who or what is in Room II. If it’s the woman, the sign on her room tells the truth …but it doesn’t, as Room I cannot both be empty and contain the tiger.

If instead the tiger is in Room II, it fits that that sign is lying. It leaves Room III for the woman …but that sign says the room is empty, which is a lie.

Therefore, the only valid possibility is the first one: Woman in I, tiger in II, and Room III is empty. The sixth prisoner can walk away a happy man with his new bride!

Women Or Tigers? — The Sixth Challenge

Pretty cool solution yesterday, wasn’t it? Though still not that difficult. Hang on for the last two challenges as this is where the real fun begins.

The king told the sixth prisoner that one of the three rooms would host a woman, another one a tiger, and the last one would be empty. He also explained that the sign to the door hiding a woman would be correct, the one hiding a tiger would lie, and the one to the empty room could be either lying or telling the truth.

Here is what the signs read:

ROOM I: ROOM III IS EMPTY

ROOM II: THE TIGER IS IN ROOM I

ROOM III: THIS ROOM IS EMPTY

The prisoner happened to know the woman who was behind one of the three doors, and he was eager to marry her. So even though it would be better to open the door to the empty room than the one hiding the hungry tiger, he clearly preferred to find the woman behind the door.

Can you figure out where the woman is, and where the tiger is?

This is the before-last challenge. Its solution will be published here tomorrow, and the day after tomorrow follows the seventh, last and ultimate challenge …to which I do not have the solution! Can you help?

Women Or Tigers? — Solution to Fifth Challenge

One or none of the three signs tell the truth. Let’s assume that the one on Room I does tell the truth. That would mean there is a tiger in that room. It would also mean that the two other signs are both lying. For Room II that could fit, as it would hide a tiger. However, that would make the sign on Room III correct — so we can rule out that possibility.

So how about the sign on Room II… if it tells the truth, the two other signs do not… but the one on Room I should then contain a woman. As there is a woman in only one of the rooms, this possibility can be discarded as well.

Maybe the sign on Room III is correct, making the two others lie. That would mean there is a woman in Room I, a tiger in Room II, and as the signs do not give any indications of what is behind the third door, there could be a tiger. In other words, this could be a possibility.

Still, let’s check if all three signs are lying: Woman in Room I, tiger in Room II… but then the sign on Room III is not lying. So that one’s out as well.

The prisoner should open the door to Room I this time!

Women Or Tigers? — The Fifth Challenge

‘Dammit!’, the king was in a rage, ‘Both prisoners chose the right door again. I have to make it more difficult, for my own and for my people’s entertainment. I will add one more room, place a tiger in two of them, and a woman in the last one. Let’s see how the prisoners will handle that challenge!’ he said.
‘What a superb idea!’ said the minister.

The three signs read:

ROOM I: THERE IS A TIGER IN THIS ROOM

ROOM II: THERE IS A WOMAN IN THIS ROOM

ROOM III: THERE IS A TIGER IN ROOM II

The king told that day’s prisoner that no more than one of the signs was correct.

Where was the woman?

Women Or Tigers? — Solution to Fourth Challenge

So how was yesterday’s logical exercise? Still too easy? Let’s crank it up a notch then. That’s what our friend the king did. But that’s for tomorrow. Let’s first look at the (best) answer to yesterday’s challenge.

If the sign on the door to Room I tells the truth, there would be a woman in both rooms. That, however, is impossible, as Room II would have to contain a tiger to tell the truth.

Therefore, the sign on Room I must be lying. There is thus a tiger behind it. And it does matter which door you choose — ergo there must be a woman in the other room, Room II. Also, it then fits that the sign on this door is lying too.

Women Or Tigers? — The Fourth Challenge

This challenge was a favourite of the king’s.

The signs on the doors for the second prisoner to choose between read:

ROOM I: IT WON’T MAKE A DIFFERENCE WHICH DOOR YOU CHOOSE

ROOM II: THERE IS A WOMAN IN THE OTHER ROOM

Remember that if there was a woman behind door #1, the sign on the door told the truth — but if there was a tiger behind it, the sign told a lie. As for the sign on door #2, it was the opposite: If there was a woman behind it, the sign would be false, and if there was a tiger behind it, the sign would be true.

Also remember that both rooms might contain a woman, or both a tiger, or of course there might be a woman in one room and a tiger in the other.

What would you choose, if you were in this prisoner’s shoes?

Women Or Tigers? — Solution to Third Challenge

The prisoner got it right this time as well, even though the challenge had become a little trickier. This was the logic that made him make the right choice:

As the two signs say exactly the same thing, they are either both true or both false. This means there must be a woman in one of the rooms and a tiger in the other — according to the instruction given by the king for this day’s two challenges.

If there is a woman in Room I, its sign tells the truth. If the sign tells the truth, there is also a woman in Room II. And if there is a woman in Room II, the sign on its door tells a lie. So that is not possible.

Ergo, Room I must contain a tiger. This fits with the sign on its door telling a lie. Also, as there must be a woman in Room II, it fits that that sign tells the truth.

So Room II it is! But what about the next prisoner’s challenge? It’s one of the king’s favourite challenges, so check it out. Tomorrow.

Women Or Tigers? — The Third Challenge

‘The first two days of my new scheme have been a fiasco!’, the king said to his minister. ‘Both prisoners solved the logical problem I had set up for them. I will make the challenges more difficult!’
‘Splendid idea!’, said the minister.

On this day, as it was a public holiday, two prisoners had to face the challenge. They were both given the instruction that if there was a woman behind door #1, the sign on the door told the truth — but if there was a tiger behind it, the sign told a lie. As for the sign on door #2, it was the opposite: If there was a woman behind it, the sign would be false, and if there was a tiger behind it, the sign would be true.

There was still the possibility that both rooms could contain a woman, or both a tiger, or of course there might be a woman in one of the rooms and a tiger in the other.

The first prisoner was shown two doors with the following signs:

ROOM I: THERE IS A WOMAN IN BOTH ROOMS

ROOM II: THERE IS A WOMAN IN BOTH ROOMS

So which door should the prisoner choose?

Women Or Tigers? — Solution to Second Challenge

Got yesterday’s challenge? Got it solved?
Well, it wasn’t very difficult, was it?

Here’s the solution:

Since either both signs tell the truth or both tell a lie, it means that if the sign on Room I is true, the one on Room II is true as well. Therefore, there must be a tiger in Room I and a woman in Room II.

It is not possible that both signs tell a lie. That would imply, according to sign #1, that there is no woman, only tigers — and as sign #2 says there is a tiger, this one cannot be lying as well.

Hence, both signs tell the truth, and the prisoner will find his bride behind the door to Room II.

Tomorrow we’ll be back with a bit more of a challenge. Do check it out, it’s fun!

Women Or Tigers? — The Second Challenge

So the first prisoner who was subject to the eccentric king’s new scheme not only saved his life but could even leave the arena with a lovely wife.

The king now changed the signs on the two doors and chose new tennants for the rooms behind them. The signs now read:

ROOM I: THERE IS A WOMAN IN AT LEAST ONE OF THE ROOMS

ROOM II: THERE IS A TIGER IN THE OTHER ROOM

‘Are the signs telling the truth?’, asked this second day’s prisoner.
‘Either they are both true or they both tell a lie’, was the king’s reply.

Which room should the prisoner pick?

Women Or Tigers? — Solution to First Challenge

Let us assume that the sign on the door to Room I tells the truth. This would mean that there is a woman in that room, and a tiger in Room II.

It would also mean that the sign on the door to Room II tells a lie. This cannot be the case, as it says that there is a woman in one room and a tiger in the other.

Therefore, the sign on Room II must be true: There is indeed a woman behind one door, and a tiger behind the other.

As there is a true and a false sign, the sign on Room I must be the false one. In other words, Room I contains the tiger, and Room II the woman. The prisoner, unless he is tired of life, should choose door #2.

Easy no? Tomorrow’s challenge will be slightly less straight-forward. So stay tuned.

Women Or Tigers? — The First Challenge

The day came when the king started his new experiment for punishing — or rewarding — his prisoners. The king explained to the chosen prisoner that each of the two rooms could contain either a woman or a tiger …but that there might be a tiger in both rooms, or a woman in both rooms.

‘What if there is a tiger in both rooms?’, the prisoner asked.
‘Well, then you are just unlucky’, was the king’s reply.
‘And if there is a woman in both rooms?’, asked the prisoner.
‘Then… you’re just lucky!’, replied the king, ‘Stupid question, isn't it?’
‘OK, but let's assume then that there is a woman in one of the rooms and a tiger in the other. What will happen then?’, the prisoner continued.
‘In that case it is for you to pick the right one, simple as that!’
‘But… how can I know which door to pick?’
The king then pointed at the signs he had put on the doors to the two rooms:

ROOM I: THIS ROOM CONTAINS A WOMAN, AND THE OTHER ONE CONTAINS A TIGER

ROOM II: ONE OF THESE ROOMS CONTAINS A WOMAN, AND ONE OF THEM A TIGER

‘Is it true what the signs say’, the prisoner now asked.
‘One of them tells the truth. But the other one tells a lie’, was the king’s answer.

If you were the prisoner, which door would you choose (provided that you prefer to meet the woman rather than the tiger…)?

Women Or Tigers? — Introduction

Over the next two weeks, I will post seven fun and increasingly complicated mathematical exercises. But hey, read on! I said they were fun! The seven exercises are part of the same story. Each other day I will give you one challenge to think about and solve, and each other day you will get the solution …except on the last day, as I actually no longer have the answer to that last and most complex challenge. So stay tuned! :)

You may know the short story ‘The Lady, Or The Tiger?’ by Frank R. Stockton. It tells how a prisoner is forced to choose between two doors — one room contains a lady, and the other one a tiger. If he chooses the former, he will marry the lady, but if he chooses the latter, he will probably end up as lunch.

A king had read the story, and it gave him an idea. ‘This is exactly how I should like to treat my own prisoners,’ he said to his minister, ‘but I will not leave things to blind chance. I will put a sign on each door, and in each case I will tell the prisoner something about these signs. If the prisoner is smart and knows how to think logically, he will save his life, and will even win himself a beautiful bride.’

‘What a brilliant idea, your highness,’ said the minister.

Does this intrigue you? If yes, check out my blog tomorrow for the first challenge. We will start very lightly and build up from there…

Why is the protection of fundamental rights needed in the EUand how has the initial problem been resolved (until now)?

1. Introduction: Why are fundamental rights needed
If members of a society do not have rights, there is no or only a weak base for making rules for that society. Rights reflect values, and it is upon values — intended to prop up the common good — that a society is built. Rules cannot exist without rights, as they are based upon them, rather than vice versa: It would not make sense for a society to construct rules and then base the rights on them afterwards.

Nevertheless, this is more or less what happened during the first half century of the European Communities’, and later the European Union’s, existence. In the beginning, the formalised cooperation between six independent countries was mainly a political project calling for a common approach within a few rather specific areas. With time, this cooperation grew larger in terms of participating countries, wider in terms of policy areas falling under its wings, and deeper in terms of involving not just the upper spheres of political decision-makers but also going further and further down into people’s lives.

This is where a common approach to the basic aspect of rights starts to become a necessity. Each society, each member state of the European Union is built on rights for its citizens or inhabitants — be it explicitly, formally or informally. These rights may differ from country to country, and in an ever more tightly-knit community between nations, agreeing on a set of common rights for each and every of its meanwhile half a billion inhabitants would seem to be essential in order to continue the progression towards common goals that serve the European Union as a whole, and each person as an individual.

2. The past
Certain specific rights were laid down already in earlier treaties, such as equality between the sexes in 1957 by means of Article 119 of the Treaty establishing the European Economic Community ¹ (now Article 157 in the Lisbon Treaty), and consumer protection in 1992 by means of Article 129a of the Treaty on European Union ² (now Article 12 in the Lisbon Treaty).

Perhaps more importantly, the Treaty on European Union also established that:

‘The Union shall respect fundamental rights, as guaranteed by the European Convention for the Protection of Human Rights and Fundamental Freedoms signed in Rome on 4 November 1950 and as they result from the constitutional traditions common to the Member States, as general principles of Community law.’ ³

This was the first time that a European Community or Union treaty directly mentioned fundamental rights. However, without establishing its own definitions of the rights, it instead referred to those set out by the Council of Europe 42 years before. As every member state of the European Union was already a member of the Council of Europe at its time of accession ⁴, it was nothing new to commit to this Convention. What was new was that the European Union as such committed to them, formally.

Until then, the protection of fundamental rights had been taken care of not through treaties but through case law. During the years, the European Court of Justice has ruled in a number of interesting cases involving questions of fundamental rights. One such case was the Schmidberger case, in which the Court of Justice ended up ruling that freedom of speech prevails over the free movement of goods principle. ⁵ Other examples include the Nold case, emphasising that the European Communities’ citizens are protected by international and national human rights principles, ⁶ and the Stauder case, which can be considered an early example of protection of personal data. ⁷

3. The present
Since late 2000, the fundamental rights of the citizens of and inhabitants in any European Union member state are protected by the Charter of Fundamental Rights of the European Union. ⁸ That is the theory. For, proclaimed by the three legislative institutions of the European — the Council, the Parliament and the Commission — there were doubts whether the charter was actually legally binding. Moreover, the United Kingdom had reserves about the Charter, as had both the meanwhile new member states Czech Republic and Poland when the Charter was referred to directly in the Lisbon Treaty, ⁹ amending the Treaty on European Union, and thus making it legally binding from 1 December 2009 when the Lisbon Treaty entered into force. All of these three member states were granted an opt-out in a Protocol to the Treaty.

The Lisbon Treaty also includes a Protocol to the Treaty on European Union, opening up for the possibility of applying for accession to the European Court of Human Rights ¹⁰ — again, as an entity, as the ‘European Union’ in itself, in addition to its member states already being members of the Court.

Meanwhile, in 2007, the European Union Agency for Fundamental Rights was established. ¹¹ This advisory body was set up to observe and verify whether the people of European Union have the rights granted to them by the agreed charters.

4. The future
In a time where the European Union, the entire European continent, and the World as a whole seems to be in a phase of transition, discussions on fundamental rights mushroom perhaps more than ever. Ever more apocalyptic predictions on our Planet’s ecological balance together with a financial end economical crisis raise questions on how our current way of life can, could, should or even must be reshaped into a future society based on sustainability.

This is why fundamental rights in a not very distant future might have a less individualistic focus, along the line of the ideas of Jeremy Rifkin, adviser to the European Union and to heads of state around the World:

‘We are so used to thinking of property as the right to exclude others from the use or benefit of something that we’ve lost sight of the fact that in previous times property was also defined as the right not to be excluded from the use or enjoyment of something. […] In a collaborative economy, the right of inclusion becomes more important in establishing economic and social relationships than the right of exclusion.’ ¹²

by Bjørn Clasen

Sources
¹ European Economic Community: Title III — Social Policy, Chapter 1 — Social Provisions, Article 119 of Treaty establishing the European Economic Community (Publishing Services of the European Communities, Luxembourg 1957). Quoted here from Centre Virtuel de la Connaissance sur l’Europe (http://tiny.cc/rometreaty)
¹ European Communities: Title XI — Consumer Protection, Article 129a of Treaty on European Union, signed at Maastricht on 7 February 1992. Official Journal of the European Communities C 191 of 29 July 1992 (Office for Official Publications of the European Communities, Luxembourg 1992), p 25
³ Ibidem, Title I — Common Provisions, Article F(2). p 5
⁴ Member states of the Council of Europe, 2011, Wikipedia 11 December 2011 (http://en.wikipedia.org/wiki/Member_states_of_the_Council_of_Europe)
⁵ Court of Justice of the European Union: Judgment of the Court of 12 June 2003, Case C-112/00 — Eugen Schmidberger, Internationale Transporte und Planzüge v Republik Österreich. European Court reports 2003 (Office for Official Publications of the European Communities, Luxembourg 2003), pp I-5694 – I-5725
⁶ European Court of Justice: Judgment of the Court of 14 May 1974, Case 4-73 — J. Nold, Kohlen- und Baustoffgroßhandlung v Commission of the European Communities. European Court reports 1974 (Office for Official Publications of the European Communities, Luxembourg 1974), p 491
⁷ European Court of Justice: Judgment of the Court of 12 November 1969, Case 29-69 — Erich Stauder v City of Ulm - Sozialamt. European Court reports 1969 (Office for Official Publications of the European Communities, Luxembourg 1969), p 419
⁸ European Parliament, Council, Commission: Charter of Fundamental Rights of the European Union. Official Journal of the European Union C 303 of 14 December 2007 (Office for Official Publications of the European Communities, Luxembourg 2007), pp 1-35
⁹ European Union: Title I — Common Provisions, Article 6 of Consolidated version of the Treaty on European Union (Publications Office of the European Union, Luxembourg 2010), p 19
¹⁰ European Union: Protocol relating to Article 6(2) of the Treaty on European Union on the accession of the Union to the European Convention on the Protection of Human Rights and Fundamental Freedoms, to the Treaty of Lisbon amending the Treaty on European Union and the Treaty establishing the European Community, signed at Lisbon, 13 December 2007. Official Journal of the European Union C 306 of 17 December 2007 (Office for Official Publications of the European Communities, Luxembourg 2007), p 155
¹¹ Council of the European Union: Council Regulation (EC) No 168/2007 of 15 February 2007 establishing a European Union Agency for Fundamental Rights. Official Journal of the European Union L 53 of 22 February 2007 (Office for Official Publications of the European Communities, Luxembourg 2007), pp 1-14
¹² Jeremy Rifkin: From Belongings To Belonging, in The Empathic Civilization — The Race To Global Consciousness In A World In Crisis (Jeremy P. Tarcher/Penguin, New York 2009), pp 541-542