Logic Puzzles and reasoning

[Valerion]

This thread is for logic puzzles. Unlike the other Puzzles thread, I am not just looking for the first person to post the correct answer. Instead, post whatever you think is the right answer, with no explanation at all. After some time the puzzle master will post the correct answer and the reasoning behind it.

Do not look it up on the net, and if you have seen the puzzle before, you can mention it, but please don't answer it.

Puzzle 1 - Answer posted on 27 July:
Given: A set of cards where each card have a number on one side, and a letter on the opposite side

Rule: If a card has a vowel on one side, it must have an even number on the other side.

Look at this deck of four cards: [E] [P] [8] [5]

Question: Which card or cards must be turned over in order to determine whether this particular deck follows the rule?

[Galahad]

The cards labelled "E" and "5" will have to be turned over. The remaining two are ambiguous.
(This is assuming the deck of cards follows a standardized pattern with what is on their sides.)

[Leeward]

Just [E].

[Rakuen Growlithe]

I got asked this on IRC. Gave the right answer then panicked and changed it. Was too much pressure. x.x

[Galahad]

Will the answer be revealed today?

[Valerion]

You need to turn over 2 cards, [E] and [5]. Turning over the other 2 are wrong, because they cannot give you information. Full explanation here (I changed the card values, but the rest is the same), but in short this was some first-order logic. Interesting that 4% get it right in the abstract case, but in a practical case it rises very high. Clearly humans have issues reasoning in abstract logic.

[Galahad]

Magnificent!
When will the next puzzle be posted, and by whom?

[Valerion]

Magnificent!
When will the next puzzle be posted, and by whom?

Whomever wants to go next.

[Galahad]

Given that there has not yet been a response, I shall take the liberty of doing so, please.

What is the missing number, indicated by '?', in this sequence?:

22344/221324, 5/15, 110002/213012, 32/1312, 667/?

I created the puzzle, so there is no solution available online. The answer shall be revealed on the evening of 30 July (tomorrow).

[Leeward]

I tried, I failed. I tried hex and binary, I tried flipping, I tried factorisation. I can see there's some sort of symmetry transformation but I just can't put my finger on it.

[Galahad]

The rules of this thread do not mention hints, and I know you are intelligent enough as not to need them.
However, I shall say this:
Those are not fractions. Focus on simple patterns in the digits (for example - in the numbers to the right of the slashes, do you see something interesting about every second digit?), not complex mathematical ones as a whole.

A bright child may actually solve this quicker than an educated adult, simply due to the difference in approach.

[Leeward]

I didn't treat them as fractions from the start, but now that you made it obvious it was no fun solving it. :'(
2617

[Galahad]

(... My sincerest apologies. I forgot I had to reveal the answer and bring an end to my turn. o_o)

Leeward is correct. The answer is 2617.

Explanation: Each term has a "numerator" - a number on the left of the forward slash - and a "denominator" - a number on the right of the forward slash. There is no relationship between each term; instead, the only relationship that exists is between a numerator and a denominator. Each numerator is randomly generated and independent; however, the denominator "counts" the frequency of each digit in the numerator in the following way:

Let us say the numerator is 1152.
From left to right, there are two 1s, one 5, and one 2.
Hence, the numerator will 21 15 12.
211512.

Similarly, the denominator for 7772 will be three 7s, one 2: 37 12. 3712.

The pattern thus emerges that each odd digit in the denominator is the frequency (number of occurrences) the next digit, in the numerator.

[Galahad]

It would seem no-one else is keen to begin a new puzzle. Hence, with the hope that I have been forgiven for my lack of punctuality with the previous puzzle, I shall present another one of my own making (with inspiration from a puzzle game I played):

Observe the following grid of numbers:

Magic Square.png (4.47 KiB) Viewed 4408 times

Thou shalt not criticize Galahad's use of MS paint!

This is what is known as a magic square. It works as follows:

• All the digits in a column (vertical), row (horizontal), or diagonal must add up to the same amount.

• Only a single digit can occupy each of the blocks. (i.e no number greater than 9)

• Each of the digits from 1 to 9 may only be used once. You may not use a digit more than once. 0 cannot be used.

The 1 and 2 have already been inserted, leaving digits 3 to 9. Indicate where they fit in, in order to satisfy the above conditions.

The solution shall be revealed on Monday (30 August) evening.

[Trace]

It would seem no-one else is keen to begin a new puzzle. Hence, with the hope that I have been forgiven for my lack of punctuality with the previous puzzle, I shall present another one of my own making (with inspiration from a puzzle game I played):

Observe the following grid of numbers:

Magic Square.png

Thou shalt not criticize Galahad's use of MS paint!

This is what is known as a magic square. It works as follows:

• All the digits in a column (vertical), row (horizontal), or diagonal must add up to the same amount.

• Only a single digit can occupy each of the blocks. (i.e no number greater than 9)

• Each of the digits from 1 to 9 may only be used once. You may not use a digit more than once. 0 cannot be used.

The 1 and 2 have already been inserted, leaving digits 3 to 9. Indicate where they fit in, in order to satisfy the above conditions.

The solution shall be revealed on Monday (30 August) evening.

I think I figured out the answer. I'll PM you the solution, and if you confirm that it's right I'll post it in a couple days, with a brief explanation. I want to give everyone else a chance first though.

[Rakuen Growlithe]

The solution shall be revealed on Monday (30 August) evening.

I am so eager to see how this happens.

[Galahad]

29 August. As you can see from the time signature, I posted this at approximately midnight, and simply confused the two dates.

[Randall]

The solution shall be revealed on Monday (30 August) evening.

I am so eager to see how this happens.

Likely cross-site scripting and SQL injection exploits.

[Leeward]

The solution shall be revealed on Monday (30 August) evening.

I am so eager to see how this happens.

Likely cross-site scripting and SQL injection exploits.

To corrupt the calendar into showing impossible dates? Seems a bit circuitous.

[Galahad]

Some supervillains wish to take over the world. My agenda is to change calendars, it seems.

[Faanvolla]

#GregorianMustFall

[Galahad]

Trace submitted a correct solution to me via PM.

The solution is...

Magic Square Solution.png (12.11 KiB) Viewed 4316 times

This digit configuration satisfies all three conditions. Each row, diagonal and column adds up to 15, and no digit is used more than once.

There is a pattern in solving it. Firstly, one should fill the centre block because 4 of the 8 columns, rows and diagonals use it. Why use 5? Because that is the middle or median digit between 1 and 9.

Once you have 5, solving the remainder becomes easier. If you write out 1 to 9 and ignore 5, you can pair the leftmost digit, 1, with the rightmost digit, 9. So, 9 would go opposite to 1. You then move on the second leftmost and second rightmost: 2 and 8.

Continue this pattern, and you will have solved the square.

[Trace]

Well, you beat me to posting the solution. My thought process on solving it was a little different. Based on the fact that 1 and 2 were given, I made the assumption that being the lowest numbers, they had to share columns with the highest number, 9, placing it in the top center. From there, our methods were similar. You probably gave a better explanation for the answer than I could anyway...

[Galahad]

On the contrary, Trace: Your explanation is easier to follow than mine and just as correct! Our two methods are very similar, but I appreciate your contribution. Thank you for the solution and the explanation.

I have given two puzzles in a row, now: it may be time for someone else to give us something to solve.

[Rakuen Growlithe]

it may be time for someone else to give us something to solve.

You are the mayor of a town. Half the town isn't speaking to the other half of the town. A few houses on one end of the town want to declare independence. Scattered through the town are some people that want anyone who disagrees with them to be kicked out the town. A second group wants them to be kicked out of the town and starts to riot if you disagree with them. If you kick them out then a third group promises to riot. Many people all throughout the town are upset that the other groups are fighting with each other. Others throughout the town get equally upset if you "suppress" the fighting groups. What do you do?

[Raven Song]

Go to the pub and have a drink

[Tetsudra]

What do you do?

Feed everyone Marmite.

This was a serious solution proposed to stabilize the Middle East. The theory was that a lack of zinc makes people argumentative, and the average Middle Eastern diet does not include high levels of zinc. So by carpet-bombing them with Marmite, you could remove the base chemical deficiency that's causing the conflict.

[Galahad]

Very subtle, Rakuen.

I am afraid I am going to have to ask for some clarification before attempting to answer, as there are some points that, at least from my view, are somewhat ambiguous:
1) Will the group wishing to declare independence accept as residents any of those who initially might not?
2) When you say that one group wants any who disagree with it to be kicked out, do you mean disagree with them as a group (i.e they hold a shared belief), or individually?
3) You mentioned that the "third group" will riot if one of the two other disagreeing groups is kicked out. Which of the other two groups is being referred to?
4) Can I assume the two halves of the town simply cannot talk to each other?

[Rakuen Growlithe]

Are countries where they eat more Marmite less argumentative than other places? I like Raven's answer though. I have to say I don't have an answer though. No matter how much gets clarified there are always new groups and new fights and new interests that are in conflict. Don't think there is a solution. I guess that means it's a bad puzzle...

[Tetsudra]

Are countries where they eat more Marmite less argumentative than other places?

Marmite specifically? Maybe not, but let me show you two maps. The first, a map showing countries hostile to the Christian faith, graded by risk of physical persecution. So take this one as a Religious Diversity Intolerance Index:

The second map, percentage of population that is Muslim - otherwise known as a Inverse Likelihood of Bacon Consumption map:

One possible conclusion is systemic religious intolerance. The other conclusion is that bacon makes you peaceful, and a survey of 1/1 bacon eaters (myself) validates this hypothesis.