Mathematical Logic

[GoldenSphynx]

Post any mathematical logic puzzle that you think is challenging, member with the right answer gives a new one. I'll start it off.

On the market of Covent Garden, mrs. Smith and mrs. Jones sell apples. Mrs. Jones sells her apples for two per shilling. The apples of Mrs. Smith are a bit smaller; she sells hers for three per shilling. At a certain moment, when both ladies both have the same amount of apples left, Mrs. Smith is being called away. She asks her neighbour to take care of her goods. To make everything not too complicated, Mrs. Jones simply puts all apples to one big pile, and starts selling them for two shilling per five apples. When Mrs. Smith returns the next day, all apples have been sold. But when they start dividing the money, there appears to be a shortage of seven shilling.

The Question: Supposing they divide the amount equally, how much does mrs. Jones lose with this deal?
Commence!!

[will]

that depends on the amount of apples they had in the first place

[GoldenSphynx]

Nope, there is a logical answer, just takes some work.

[eleet-2k2]

But you need a number of the original piles of apples.

[SSJ4 Gogitta]

The big pile of apples contains the same amount of large apples of half a shilling each (from mrs. Jones), as smaller apples of one third shilling each (from mrs. Smith). The average price is therefore (1/2 + 1/3)/2 = 5/12 shilling. But the apples were sold for 2/5 shilling each (5 apples for 2 shilling). Or 25/60 and 24/60 shilling. This means that per sold apple there is a shortage of 1/60 shilling. The total shortage is 7 shilling, so the ladies together started out with 420 apples. These are worth 2/5 × 420 = 168 shilling, or with equal division, 84 shilling for each. If Mrs. Jones would have sold her apples herself, she would have received 105 shilling.

she lost 21 shillings.

[GoldenSphynx]

Correct! Very nice job. Your turn to riddle us.

[GoldenSphynx]

Fine, then Ill just have to supply the riddles myself.

A banana plantation is located next to a desert. The plantation owner has 3000 bananas that he wants to transport to the market by camel, across a 1000 kilometre stretch of desert. The owner has only one camel, which carries a maximum of 1000 bananas at any moment in time, and eats one banana every kilometre it travels.

The Question: What is the largest number of bananas that can be delivered at the market?

[Curi0us_George]

Originally posted by GoldenSphinx86
Fine, then Ill just have to supply the riddles myself.

A banana plantation is located next to a desert. The plantation owner has 3000 bananas that he wants to transport to the market by camel, across a 1000 kilometre stretch of desert. The owner has only one camel, which carries a maximum of 1000 bananas at any moment in time, and eats one banana every kilometre it travels.

The Question: What is the largest number of bananas that can be delivered at the market?

1, plus any that the owner carries himself. The camel eats one before departng, adn one for each of the next 999 miles. So the camel arrives with one banana.

The owner can't get the camel to go back, unless he buys 999 bananas at the market to get the camel to make the return journey.

[SSJ4 Gogitta]

533.3333333

Im not going through the explination. it can be found on the net.

[Curi0us_George]

Originally posted by SSJ4_Gogitta
533.3333333

Im not going through the explination. it can be found on the net.

Well, I'm too mundane to break it down into multiple trips.

[Digipak]

That's in one trip buddy, 533 and 1/3.

[Hunter]

Posted By Digipak
That's in one trip buddy, 533 and 1/3.

That's what George said...READ!

Posted by Curious George
Well, I'm too mundane to break it down into multiple trips.

[rm']

Here's an interesting problem I had on a grade 13 Finite (statistics and probability) quiz. It's really just grade 9 math, but you only realize that once you've figured the problem out

A doctor was making a list of all the diseases that he came across in his practice, and how many people were afflicted by each disease. On average, there were 20 patients to each disease. If he doesn't count the flu, which has 92 patients, the average drops to 18. How many diseases are there (including the flu), and what is the maximum possible number of patients for a single disease?

[zootm]

ordinarily i could do that straight off, but:
[list=1][*]i hate statistics, give me some good proper maths[*]it's 3am[/list=1]

so i'm not gonna bother right now.

for those that want a logical question, try this one (but don't forget about rm''s one!)-

if it's raining, and i go outside, i'll get wet.
if i go outside, and i bring an umbrella, i won't get wet.
therefore, if i go outside and bring an umbrella it won't rain.

that can be proven using propositional logic. do it. mwah.

[zootm]

*thinks*
i can't do mental arithmetic at the best of times, but are there 46 diseases?

as for the other bit, bugger it, i'm going to bed.

[rm']

Goddamned Aristotle. I know to prove that, you need to use his rules for syllilogisms, unfortunately, I forget them at the moment.

Of course, a statement can be logical, but not true, as in this case.


And no, it's not 46.

[red1177]

112

[Curi0us_George]

Originally posted by zootm
for those that want a logical question, try this one (but don't forget about rm''s one!)-

if it's raining, and i go outside, i'll get wet.
if i go outside, and i bring an umbrella, i won't get wet.
therefore, if i go outside and bring an umbrella it won't rain.

that can be proven using propositional logic. do it. mwah.

If you accept the first two as true, then the third must be true.

Since the only constant is "outside", we throw it away and we are left with:

rain = wet
umbrella = !wet

umbrella = !(wet)
umbrella = !(rain)
umbrella = !rain

I don't really know how the logic should be written. That's good enough for me, though.

Digipak, the Camel makes multiple partial trips. That's what I was talking about.

[zootm]

if you can cause an assumption to create a contradiction, it must be false given the assumptions you use to cause the contradiction.

if you assume it's raining, and you bring an umbrella you get wet and you don't get wet, hence if you bring an umbrella, it doesn't rain, beacuse that would cause a contradiction.

the way you wrote it is wrong as = implies biimplication (if and only if), which is not present. it works here, however

here's a tricky one- is the following sentence true or false...

"this sentence is false"

[rm']

Neither. It's a paradox, and paradoxes don't have yes or no values.

[J. Burnaway]

Let D=number of diseases

therefore, number of patients=20 D (avg. of 20 people per disease)

Number of diseases, not including the flu = D-1

Flu patients = 92

Therefore,

18(D-1) + 92 = 20D (18 average patients for all the diseases, not including flu, plus flu patients, equals 20 patients average for all diseases)

18D-18+92=20D
92-18=2D
D=37

There are 37 total diseases.

[rm']

Exactly. And part B?

[J. Burnaway]

And the raining thing:

just looking at it quickly, the first sentence is contradicted by the second - if it's raining, and I go outside, I get wet (this sentence doesn't say if you do or do not have an umbrella, therefore we must assume that even if you have an umbrella, you get wet).
The next sentence states that if you bring an umbrella, you won't get wet - which contradicts the first sentence.

If we change the first sentence to, if it's raining, and I go outside without an umbrella, I get wet...we have:

1. If A(raining) and B(go outside) but not C(have umbrella) -->D(get wet)
2. If B(go outside) and C(have umbrella) -->not D(get wet)
3. Therefore if B(go outside) and C(have umbrella) --> not A(raining) is unproven, because we only have A(raining) as a given, not as a result. The C(have umbrella) variable leads to not D(get wet) regardless of condition A(raining).

I confused myself.

[zootm]

rm'- nobody knows... it is a paradox- but nobody has a good true or false value for it- there are 3 camps in the logic community-
[list=1][*]it's true and false[*]it's neither true nor false[*]it's not a sentence. at all.[/list=1]

as for J. Burnaway's solution of the diseases thing- that's the calculation i did! bloody mental arithmetic! i told you it was late... lol...
also, J- in point 1 you post for the raining- !C is not a condition of that one that's the flaw in the assumptions that lets you prove something crazy...

attempt @ part b
the maximum amount of people in one disease can be worked out as it would be the number if all the other diseases have one afflicted person (as the doctor had to come across them). so if the maximum patients-per-disease is M, and we use the average for the 36 non-flu diseases, this average is 18, hence

18 = (M+35)/36
648 = M+35
613 = M

so the most that one disease can have is 613...

[J. Burnaway]

Sorry, forgot about part B.

Maximum number of patients for a single disease is: 613

1 disease @ 613 patients
Flu @ 92 patients
35 diseases @ 1 patient (each must have a minimum of 1 patient, or he wouldn't be counting the disease)

= 740 patients

This of course assumes that each person only has one disease.

[zootm]

Originally posted by J. Burnaway
This of course assumes that each person only has one disease.

now that's a good point...

[eleet-2k2]

Good job zootm. That seems right, but I havent really tried.

[J. Burnaway]

By day, a snail climbs up from the bottom of a well. The well is almost 27 feet deep. He climbs up 3 feet a day. At night he sleeps, and because the walls are slippery, he slides back down 2 feet before he wakes up and starts climbing again.

How many days does it take him to get out of the well?

[zootm]

25 days... each of the first 24 days he climbs 3 and slides down 2 (poor guy ) and the last day he climbs 3 and escapes...

[J. Burnaway]

Good job - not too difficult a question, but most people just divide 27 by 3 and say "9 days"

[zootm]

i thought the obvious wrong answer would be 27 days... meh.

[J. Burnaway]

You're right, I was lost in space for a second there!

[Curi0us_George]

Originally posted by J. Burnaway
And the raining thing:

just looking at it quickly, the first sentence is contradicted by the second - if it's raining, and I go outside, I get wet (this sentence doesn't say if you do or do not have an umbrella, therefore we must assume that even if you have an umbrella, you get wet).
The next sentence states that if you bring an umbrella, you won't get wet - which contradicts the first sentence.

If we change the first sentence to, if it's raining, and I go outside without an umbrella, I get wet...we have:

1. If A(raining) and B(go outside) but not C(have umbrella) -->D(get wet)
2. If B(go outside) and C(have umbrella) -->not D(get wet)
3. Therefore if B(go outside) and C(have umbrella) --> not A(raining) is unproven, because we only have A(raining) as a given, not as a result. The C(have umbrella) variable leads to not D(get wet) regardless of condition A(raining).

I confused myself.

The first two statements do not contradict. That's the entire basis of the third statement. If it is raining, and you go outside, you will get wet. If you go outside with an umbrella, you will not get wet. Therefore, if you go outside with an umbrella, you must not get wet, or one of the first two would be false.

You have to assume that the first two are both correct. Neither of them truly are, but for the sake of the logic puzzle, we remove the logic from the real world, and treat them as facts.

[J. Burnaway]

Ok, I can see now that they don't contradict. And if the first two are true, then the third must be also. So the first sentence is incorrect!

Next question?

[zootm]

as i think i mentioned earlier, the problem is based on false assumptions... it's an abstract argument. it is, however, a valid one. if the assumptions were correct, the conclusion would be also

[Bilbo Baggins]

As long as 1+1=2, we should all be happy.

[eleet-2k2]

But I need more advanced math than kindergarten.

[rm']

No you don't. Until you are in a position of responsibility in which you have to deal with mortgages and other kinds of loan repayments, pretty much all of the math you require to manage your life is intuitive.

[SSJ4 Gogitta]

1
11
21
1211
111221
312211
13112221


What comes next in that sequence, and why?

[bballer182]

Not so much a question but a paradox.

When you stand 100 feet from a wall and once every minute you can move closer to the wall by cutting the distance between you and the wall in half. how long will it take you to get to the wall?