A: I don’t know my number.

B: I don’t know my number.

C: I don’t know my number.

A: My number is 25.

What are the numbers on the other two hats?

This is a general solution for 3 hats. The 3 players are A,B,C. Hat numbers are expressed as triples in that order eg (1,2,3). Player order is A,B,C, and A2 means player A second turn.

We start out by finding solutions for the smallest possible hat numbers. The logic is unchanged if all hat numbers are then multiplied by the same

factor: (1,2,3) is logically equivalent to (10,20,30).

Solutions

A1: (2,1,1)

Interpretation: If A1 knows his hat number, then (2,1,1) is the only solution; if he is silent, then (2,1,1) is excluded as a solution.

Rationale: If A1 sees two identical hats he knows his must be the sum, not the difference. Otherwise he must pass, since his hat is ambiguous.

Example: On the first round, A announces his hat is number 50. The other two hats must both be 25.

B1: (1,2,1) (2,3,1)

Interpretation: If B1 knows his hat number, then these are the only solutions; if he is silent, then these are excluded as solutions.

Rationale: (1,2,1) as for A1; (2,3,1) because (2,1,1) was excluded so B1 knows his must be the sum, not the difference. Otherwise he must pass, since his hat is ambiguous.

Example: On the first round, B announces his hat is number 33. The other two hats must 22 and 11 (in that order).

C1: (1,1,2) (2,1,3) (1,2,3) (2,3,5)

Interpretation: If C1 knows his hat number, then these are the only solutions; if he is silent, then these are excluded as solutions.

Rationale: (1,1,2) as for A1; (2,1,3) because (2,1,1) was excluded so B1 knows his must be the sum, not the difference; (1,2,3) because (1,2,1) was excluded; (2,3,5) because (2,3,1) was excluded. Otherwise he must pass, since his hat is ambiguous.

Example: On the first round, C announces his hat is number 25. The other two hats must 10 and 15 (in that order).

A2: (3,2,1) (4,3,1) (3,1,2) (4,1,3) (5,2,3) (8,3,5)

Interpretation: If A2 knows his hat number, then these are the only solutions; if he is silent, then these are excluded as solutions.

Rationale: For the previous 2 sets of excluded solutions, replace the A value by the sum of the other two values. The general principle is that a previously ambiguous possible solution (sum or difference) has become unambiguous (sum only), and is either the solution or is excluded.

Example: On the second round, A announces his hat number is 25. The other two hats are 10 and 15 (in order).

Example: On the second round, A announces his hat number is 60. The other two hats could be 40,20; 45,15; 20,40; 15, 45; 24, 36. He knows, but we don’t.

This system of numbering can be extended to arbitrarily large puzzles as a purely mechanical exercise. Hopefully I made no mistakes in the calculations, but in any case I am confident the method is correct. This puzzle has been presented elsewhere, but so far I have not seen this particular method for solving it described elsewhere.

]]>First: what do you mean by “god”?

If you mean “god” as a concept, obviously it exists. Virtually everyone on the planet has some kind of concept of a god, and it is possible to have a conversation with references to a god and be confident of some meaningful exchanges of information.

If you mean “god” as a physical entity that currently exists then the relevant facts are all the scientific information we have gathered about the physical world. None of those facts indicate the existence of a physical god, so we can be exactly as certain that there is no god as we can be that there is no unicorn, no Easter Bunny, no Santa Claus etc. In my view that is a high level of certainty, but whether it leaves room for doubt could be a matter of opinion.

If you mean “god” as something else eg a non-physical entity, or an entity that existed in the past but no longer, then the burden of definition is yours to bear. If you can’t define what you mean, then you can’t expect me to argue whether it exists or not. That is “agnostic by definition”.

In my view of the world, “god” exists as a concept of human invention and does not exist as a real, physical entity. I think the facts available are sufficient to reach that conclusion with a high degree of certainty, and that the process of getting to that point is highly rational. Regarding other kinds of god, I deal with those on a case by case basis as anyone offers a definition of what “god” means.

Aren’t you glad you asked?

]]>I am directly involved in the awarding process in two organisations, and I have to say it’s not easy to do. No-one wants to give an award for “just doing your job”, but if you have a candidate for recognition within their chosen profession or speciality, what exactly is it that they should have done to justify that award? Not an easy question.

The guidelines used for some the awards I am involved in include:

1. “Took a risk, made a difference; encouragement in mid-career ”

2. “Distinguished contribution to the field of …”

3. “Outstanding and distinguished lifetime achievement in and contribution to the field of …”

4. “Very significant lifetime achievement in and contribution to the field of …”

5. “Extraordinary and long-term contribution to the … organisation”

They all imply something more than just “doing your job” but how much more?

Another question: in many cases we look to a recipient as a role model, for the young to emulate. What if you have someone who makes a truly outstanding contribution, recognised by everyone, but cheats on the tax? Or perhaps molests little girls? Where do you draw that line?

Let me know if you find some easy answers. I don’t know any.

]]>Type (a) laws only limit freedom to the extent that rights are in conflict: the protection of the rights of the victim restricts the rights of the perpetrator. Most laws in this category are a net positive for personal freedoms, and laws such as those in a Bill of Rights are entirely about freedoms.

Type (c) laws are relatively neutral on freedoms. The laws that set up the parliament, the police force, courts, ATO, banks, industry, contracts, public transport, infrastructure, corporations, etc have little impact on personal freedom. The agencies they create may have an impact, but not the laws themselves.

Type (b) laws are the ones we should worry about. They include laws on topics like public drunkenness, affray, most traffic laws, public nuisance, censorship, etc, etc. These are the “do-good” laws that sound great in theory but add up to the nanny state.

So what about banning the burqa?

I don’t favour a ban, because I see it as type (b), regulating conduct with no great contribution to the protection of rights. However, there are some undesirable aspects and it would be relatively easy to introduce 2 specific laws, to make it:

- A criminal offence for a person to engage or attempt to engage in any commercial, contractual or regulated transaction or activity, or the creation, signing, production of any document related to personal identity, without exposing one’s full face, except with the prior express and written permission of the other part(ies) or the relevant regulating organisation as the case may be;
- A criminal offence for a person to impose or attempt to impose any obligation or demand or exemption on any other person or organisation on the grounds of any religious belief, principle or claim.

The rationale is that you are personally free to do what you like but transactions with other people give them the right to know who they are dealing with, and you can hold what beliefs you like but not impose them on others.

So, you can wear your burqa, but you cannot buy or use a ticket for a train, tram or bus; cannot drive a car; cannot go shopping; cannot buy food etc unless you are willing to remove it whenever you interact with anyone; and you cannot use “freedom of religion” as an excuse to force your requirements onto others.

These are type (a) laws that protect the freedoms and rights of the people and organisations you interact with against unwelcome religion-based demands and obligations. I would like them to extend to a number of other religious groups, but we won’t go into that now.

]]>The argument is over the relative speed of one back tire spinning, compared to the road below.

The following are given:

1. Vehicle speed 120 km/h.

2. Engine at redline, consulting gearbox ratios extrapolates this to 100 km/h in reverse.

3. Open diff, so one tire does not slip at all, but remains in contact with the road while the other spins backwards.

So, does anyone have an idea what the relative speed difference was under those conditions?

His view, by assuming the same ratios as when driving at slow speed in reverse, results in a rear wheel speed equivalent to something over 400 km/h.

My view goes something like this. First, it seems inescapable that if the gearbox engages immediately and if there is no slippage in the torque converter, then the instantaneous situation would be that the engine must be rotating backwards. That simply isn’t possible, so we have to assume major slippage in the transmission.

Assuming there is slippage even for a brief period, then where is it? The torque converter is by far the best candidate. We know that a heavy load can switch a torque converter out of coupling mode (that happens if you tow something too heavy in too high a gear). So we get the following sequence of events.

1. Reverse gear engaged in gearbox.

2. Front shaft of gearbox reverses, breaks torque converter out of coupling mode (stator locks).

3. Engine goes to full power, peak revs.

4. Torque converter passes magnified engine torque back to gearbox.

5. Gearbox passes torque (reverse direction) back to diff and to wheels.

6. Torque exceeds limiting friction of one tyre, wheel slows down and may reverse.

7. Steady state is reached when torque transmitted by the engine is equal to torque transmitted from two wheels, one slipping.

If this is the situation, then it’s not possible to predict exactly what the slipping wheel is doing, but I would guess rotating slowly forwards (same direction as road).

Google was not my friend.

]]>Utepils (Norwegian) means “outside beer”. It refers to the highly pleasurable activity of sitting outside enjoying a beer, especially the first warm day of spring.

Drachenfutter (German) means “dragon fodder”. This is the gift a husband gives his wife when he’s been a naughty boy, in the hope of not having to sleep on the sofa.

Attaccabottoni (Italian) means “button attacker”. Someone who starts conversations and won’t let you get away.

Saudade (Portuguese) means “sadness”, but really refers to an intense nostalgia for the past or missing friends or anything really.

I’m sure these are all words we could use from time to time.

]]>A. Alf meets Bert, and asks him how many children he has, and of what sex.

The puzzle: if Bert has two children and at least one of them is a boy, what are the odds that he actually has two boys?

B. Alf now asks the same questions of several more people, until he gets to Charlie.

The puzzle: if Charlie has two children and at least one of them is a boy, what are the odds that he actually has two boys?

C. Alf goes back to Bert, and asks him what days any boys were born on.

The puzzle: if Bert has two children and at least one of them is a boy born on a Tuesday, what are the odds that he actually has two boys?

D. Alf now asks the same questions of several more people, until he gets to Dave.

The puzzle: if Dave has two children and at least one of them is a boy born on a Tuesday, what are the odds that he actually has two boys?

This strange puzzle and/or variants of it was originally set by Martin Gardner. The surprising thing is that the exact same question can have two different answers depending on the assumed context from which the odds should be calculated. There are enough clues here to make the answers fairly obvious.

This puzzle is unsatisfactory, and in some ways it was intended to be so. You don’t have to agree with my answers, but if you don’t at least know why.

Odds may be defined as the number of favourable outcomes as a fraction of the total number of outcomes, all outcomes being equally likely. The intention was as follows.

A. Total outcomes: 2 (other child could be boy or girl). Favourable: 1. Answer 1/2.

B. Total outcomes: 3 (boy/boy, boy/girl or girl/boy). Favourable: 1. Answer 1/3.

C. Same as A. Answer: 1/2.

D. Total outcomes: 27 (boy Tue/boy any=7, boy not Tue/boy Tue=6, boy Tue/girl any=7, girl any/boy Tue=7). Favourable: 13. Answer 13/27.

The problem is not the maths, but the ambiguous language used to frame the questions. In particular, it is very difficult to be sure how many total outcomes were possible at the time the question was asked. I’m happy for anyone to disagree with my interpretation, because the point was really to show the gap between the verbal question and the underlying maths.

This puzzle was prompted by this article.

http://sciencenews.org/view/generic/id/60598/title/When_intuition_and_math_probably_look_wrong

http://www.tomjubert.com/irrational

It’s a download, text-based propositional logic with a cute back story. Only 10 questions, but it certainly made me think.

Enjoy!

]]>If you choose an answer to this question at random, what is the chance you will be correct?

A: 25%

B: 50%

C: 0%

D: 25%

The answer is none. You will not get a correct answer by choosing at random.

The problem is set up to look like a typical liar’s paradox and it gets close, but not this time. The explanation goes something like this.

There are only two ways I know to get accurate answers for questions of this kind that depend on probabilities.

1. Enumerate the cases and count the results.

2. Turn it into maths, and apply standard theorems and transformations.

Assuming we’re not mathematicians, enumerating the cases is the way to go.

There are 4 equally probably choices that could be made: A,B,C,D. We treat each choice as a separate trial which either succeeds (the answer is correct) or fails (the answer is incorrect); there are no fractional results for a trial. The required overall probability of a correct answer is the number of successful trials divided by 4.

Trial 1: Answer A is correct if and only if it is correct and all other answers are incorrect. However, answers A and D are identical so if answer A is correct then answer D must also be correct. Therefore answer A is wrong. Fail.

Trial 2. Answer B is correct if it is correct and exactly one other answer is correct. However, if answer B is correct then answers A, C and D must be incorrect. Therefore answer B is incorrect. Fail.

Trial 3. Answer C is correct if no answers are correct. Answer C cannot be both correct and incorrect, therefore answer C must be incorrect (in which case there is no contradiction). Fail.

Trial 4. Answer D: same logic as answer A. Fail.

Result: 4 fails, there is no chance of picking the correct result by chance.

You might think by inspection that this outcome would make Answer C correct, but it does not. When you try to choose it in a trial by this careful procedure you find out that Answer C is actually incorrect, like all the others. Appearances can be deceptive. The paradox is that there appears to be a correct answer, and yet you can never choose it.

The point here is that this is a failed paradox. There is no “correct” answer to this question ante facto, because marking takes place after an answer has been selected. The reason why that is so is because we don’t know the answer until after the selection is made. The consequence is that the chance of making a correct selection is nil, even if you make selection C (0%).

I did a bit of Googling, but couldn’t find any authoritative analysis. Just lots of chatter.

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