[[0,1] to (0,1)]

If x=0, f(x)=1/2;

if x≠0 and 1/x is an integer, f(x)=x/(1+2x);

otherwise, f(x)=x.

Very impressive! Did you just devise this yourself?

[It's a trap!]

.The password, clues 2 and 5 are correct the rest are lies.

Hmm, I misread the teaser thinking it was 12 letters,

'The Password ' or 'Thepasswordis'?

2 and 5 are true, then 1 must be true as well (3 and 4 are false).

TTFFT: "Thepasswordis" (the main answer)

TFFTT: "notinthishint"

FTTTT: (any 13 letter sequence in the 2nd sentence other than "Thepasswordis")

TFTFT: ([almost] any 13 letter sequence excluding previous answers)

Maybe you can refine the OP to allow for only one possible answer.

[It's a trap!]

notinthishint

[Four Numbers]

The six products listed can obviously be split into 3 pairs of products, each which equal a*b*c*d.

So by listing out all possible products that can be found by multiplying any two of the 5 given products, we're guaranteed to find exactly 2 of these products that equal a*b*c*d.

If one tries doing this for 2,3,4,5,6 only one end product is repeated twice (12) ensuring that 12 = a*b*c*d.

Once we have this, finding the last product is easy.

the two pairs of procucts that multiple to 12 are (2,6) and (3,4).

Thus the last pair must be (5,2.4), giving the sixth product as 2.4

The product of the 6 products (as mentioned in the first sentence) is a³b³c³d³ = (a*b*c*d)³.

(a*b*c*d)³ = 2*3*4*5*6*x where x is the sixth product.

12³ = 720x

x = 1728/270 = 2.4

[mental diagnosis]

The doctor's anaylsis is flawed, in that it assumes absolute probabiltiy, when, in truth, people believe propositions with subjective probability.

[Averages Part II]

...was to pick a number S (ideal for differences), calculate and average all the differences D, then calculate the average to be S+D.

Because 35 is the mode, it makes a good choice for S (6 differences will be 0). The respective differences are:

0,13,0,5,15,0,0,5,115,0,5,0,10,10

These sum to 158, so the average difference D is 178/14 = 12+10/14 = 12+5/7

So the average is 35+12+5/7 = 47+5/7 = 47.714285(repeating)

[magic triangles]

9
0 4
7 6 3
1 5 2 8
(sum = 13)

[Are you a shoplifter?]

Is the OP correct?

With the coin setup, it should either change to be (a) answer the truth for heads or change to be (b) respond 'no' for tails. Otherwise, it doesn't make sense from a psychology standpoint that the person would feel at ease participating, there will be no doubt when they answer about shoplifting.

For example, if you go with (b), then if they shoplifted, they will be comfortable responding "no" (a lie) as there is a 50% chance they are only responding because the coin dictactes it.

With the given example, if they respond "no", you will no for sure that they shoplifted. Who would be amenable to that?

[Another spin on a classic: Transporting Apples]

Assume the business is 3000 miles away. The driver will consume all the apples.

Assume the business is one inch away. Almost any scheme delivers all the apples.

So, don't we need to know how far away the business is?

Or should we take the distance to be x<3000 and optimize for each x?

oops. I meant to also state that the business is 1000 miles away.

How does that qualify as local?

[50% increase]

Let's the value be m and its shifted version n.

2/3*n = m

m and n are both integers, so n-m=1/3*n is also an integer.

Therefore 3 | n .

As their digits have the same sum, 3 | m.

Therefore 3 | n-m --> 3 | 1/3*n --> 9 | n.

As their digits have the same sum, 9 | m.

Following similar logic from above, above 9 | n-m --> 9 | 1/3*n --> 27 | n.

So you need only account for 1/27 of the integers for a brute strength solution (which speeds it up significantly)

I got 285714 and 571428.

[Shuffle to Double]

0 is not a positive integer.

24.

[An Arbitrary sum?]

Pairing the two lists are in reverse order ensures that each number from the lower half is matched with a number from the upper half. That is, each pair will consist of a number (1 to n) and a number (n+1 to 2n).
As the differences are being added the end value will be

SUM(n+1 : 2n) - SUM (1 : n)

((2n/2)(2n+1) - (n/2)(n+1)) - (n/2)(n+1))

(2n/2)(2n+1) - (2)(n)(n+1)/2

(n)(2n+1) - (n)(n+1)

2n²+n - (n²+n)

2n²+n - n²-n

n²

[Every Natural Number can be Unambiguously Described in Fourteen Words or Less]

But now n is "the smallest natural number that cannot be unambiguously described in fourteen words or less".

That sentence is not a true description of n because it veracity would lead to a contradiction (Similar to "This statement is false"). Therefore it is an untrue statement, and thus a false description of n.

[My first sequence]

16

14

No, sorry.

[Blindman's Chess]

Is he unable to see or does he put up blinds?

[Sequence (repost)]

This is a re-post of to "bump" it up.

11, 12, 13, 22, 15, ?, 17, 32, 23, ?, 111, ...

Fill in the ?'s

[My friend and I play in a chess tournament]

# of eigth graders

------------------------
5 (1 pt each)
6 (2 pts each)
10 (5 pts each)
15 (8 pts each)
30 (16 pts each)

[9, 99 or 999?]

Will your response contain the same number of letters as the number's digits?

yes - 999

no - 9

(no answer) - 99

But as phrased, if the number were 999 a response of either 'yes' or 'no' is equally valid. However, clearly, we need a question that can give 3 different types of responses, a yes, a no, or a lack of response...

I would ask: what answer has as many letters as digits in the number you are thinking of? If "Yes" the number is 999, if "No" it is 99; silence implies it is 9.

Could you provide a truthful response which contains the same number of letters as the number's digits?

yes - 999

no - 9

(no answer) - 99

If he says "Yes", the number might be either 999 or 99, because the truthful response he

could have provided might have been either "Yes" or "No".

(for 99) He couldn't provide any response as they (1) wouldn't be truthful or (2) wouldn't contain an equitable number of letters to digits.

[9, 99 or 999?]

Will your response contain the same number of letters as the number's digits?

yes - 999

no - 9

(no answer) - 99

But as phrased, if the number were 999 a response of either 'yes' or 'no' is equally valid. However, clearly, we need a question that can give 3 different types of responses, a yes, a no, or a lack of response...

I would ask: what answer has as many letters as digits in the number you are thinking of? If "Yes" the number is 999, if "No" it is 99; silence implies it is 9.

Could you provide a truthful response which contains the same number of letters as the number's digits?

yes - 999

no - 9

(no answer) - 99

[9, 99 or 999?]

Will your response contain the same number of letters as the number's digits?

yes - 999
no - 9
(no answer) - 99

[My first sequence]

ok then can you tell me this?

what will come after 111?

211? or something else?

2213

[a^2 -b^2 = (a+b)(a-b)]

a² - b²

(a + b - b)² - b²

(a + b)² + b² - 2b(a + b) - b²

(a + b)² - 2b(a + b)

(a + b)(a + b -2b)

(a + b)(a - b)

[My first sequence]

11, 12, 13, 22, 15, ?, 17, 32, 23, 111, ?

Fill in the ?'s

are you sure 111 will follow 23 and nothing is between then?

as per me,

16 in 1st blank

211 in 2nd blank

My apologies. The last question mark should be before the 111:

11, 12, 13, 22, 15, ?, 17, 32, 23, ?, 111

16

101

Nope, try again.

[Number game 1]

(sum=27)

12 6 9 | 3 4 5 7 8

12 4 5 6 | 3 7 8 9

3 4 5 6 9 | 12 7 8

...

(sum=36)

31 5 | 2 4 6 7 8 9

...

(sum=45)

31 5 9 | 24 6 7 8

...

A better question would be "How many different pairs of groups (with the same sum) can be formed?"

[My first sequence]

11, 12, 13, 22, 15, ?, 17, 32, 23, 111, ?

Fill in the ?'s

are you sure 111 will follow 23 and nothing is between then?

as per me,

16 in 1st blank

211 in 2nd blank

My apologies. The last question mark should be before the 111:

11, 12, 13, 22, 15, ?, 17, 32, 23, ?, 111