The worlds first riddle!

Cube
2x^2+2y=1
y=x/3
cube side (√19-1)/6

Rap

number of spheres

=∑X(X+1)/2

Rap

The perpendicular distance from any vertex of a triangle to the side opposite that vertex.

To calculate the volume of a triangular pyramid, we first compute its height. We do this by applying the Pythogorian theorem to a right-triangle constructed by dropping a perpendicular from the apex of this pyramid to its base -it passes through the centroid of the B1 equilateral triangle. For a unit equilateral triangle, the centroid is at a distance [1/( v3)] from its vertices. The right-triangle therefore has base [1/( v3)] and and hypotenuse Y1 = [(v3)/ 2] .




The top row is counted as the 0th row and the 1 at the beginning of each row is the 0th entry for that row. Each entry is the sum of the two above. The kth entry on the nth row is the binomial coefficient.

(n)
(k)


A magic hexagon of order is an arrangement of close-packed hexagons containing the numbers 1, 2, ..., , where is the th hex number such that the numbers along each straight line add up to the same sum.

The magic constant for an order ^n hexagon would be


the first few of which are 1, 28/3, 38, 703/7, 1891/9, 4186/11, ... which requires 5/[2n - 1] to be an integer for a solution to exist. But this is an integer for only n=1 (the trivial case of a single hexagon) and n=3


I like Rap's answer for cubes.


"truth 25% of the time"
"What is the probability that the first statement is true?"

How about ¼.

I'm not sure what problems Try was solving, but...

PYRAMID
1. n(n+1)(n+2)/6
2. 10 + (10/3)(n-1)sqrt(6)

HEXAGON
2 / (sqrt(6)+sqrt(2))
Rotate the hexagon so that a 45/45/90 triangle is formed in one corner, and the hexagon touches the square with 4 of its 6 vertices. ie, 2 diameters of the hexagon are touching the square. This must be the optimal positioning, since rotating the (fixed sized) hexagon either way will increase the size of square required for it.

CUBE IN CONE
3 / (1 + 3/sqrt(2))
For a cone of height h and radius r, a cube of side length a sitting centered on the base will touch the lateral surface at four points. If we cut through two diagonal corners of the cube, we'll find two triangles similar to h:r : h-a:a/sqrt(2) or a:r-a/sqrt(2).
Solving either gives the same answer: h:r = h-a:a/sqrt(2) , giving:
==> a * h / sqrt(2) = rh-ra ==> a * (h/sqrt(2) + r) = rh ==> a = rh / (r + h/sqrt(2)

INTEGER
368/27, 242/27
Step 1: let A=3+sqrt(c) and B=3-sqrt(c)
Step 2: we are interested in all values for c, for which A^(1/3) + B^(1/3) = n, where n is an integer.
Step 3: cubing step 2, A +3(A^2B)^1/3 +3(AB^2)^1/3 + B = n^3
Step 4: recognizing that A+ B =6, regrouping the middle terms of step 3 yields: 6 + 3(AB)^1/3(A^1/3 +B^1/3) = n^3
Step 5: recognizing from 2 that A^1/3 + B^(1/3) = n, rewrite as: 6 + (3n)(AB)^1/3 = n^3
Step 6: bringing all but AB to the right hand side, (AB)^1/3 = (n^2)/3 - 6/n
Step 7: cubing each side, AB = n^6/27 - 8/(n)^3 - 6n^3/9 + 4
Step 8: AB = 9 - c, substituting above and solving for c: c = 5 + 8/(n)^3 + [18(n)^3 - (n)^6] / 27
Step 9: the only values for n that produce positive rational number for c are
n=1, c = 368/27 and n=2, c = 242/27

TRUTH
1/10
The probability we are looking for is the conditional probability P(A|B) of the first fellow's statement being true (event A) provided the second fellow claims that it is (event B) indeed so.

Let's examine the other two probabilities in the standard definition: P(A|B)·P(B) = P(AB).

AB is the concurrent event of the statement being true and the second fellow saying so, which only happens when both of them tell truth. The probability of this event is 1/4·1/4 = 1/16: P(AB) = 1/16.

The second fellow might have made his claim provided both of them either told truth or both lied, which means that P(B) = 1/4·1/4 + 3/4·3/4 = 10/16. From here, P(A|B) = (1/16)/(10/16) = 1/10.

Still working through the stacking spheres and one of the rational numbers but I have some questions on the cube in the cone

Let x be the length of one side of the cube, then the diameter of the circle on the cone that contains 4 corners of the cube is √2x m. When I look at the cross section of the cube and the cone through the apex of the cone and the diagonal of the cube I see a triangle with a 1m base and a 3m height inscribing a rectangle √2x m wide and x m tall.

Calling the segments of the triangle base that are not within the rectangle y, then
2y+√2x=1m
and
from the height to width ratio
y/x=(1/2)/3
or
y=(1/6)x
2(1/6)x+√2x=1
1/3x+√2x=1
(1/3+√2)x=1
(1+3√2)/3x=1
so
x=3/(1+3√2)m

And the two liar problem has the second liar answering on the condition that heard the first response?

Rap

base radius is 1m; so triangle base is 2m

yes, second tribesperson heard the first statement

Then
2y+√2x=2
and
y/x=1/3
so
2x/3+√2x=2
2x+3√2x=6
x=6/(2+3√2) m

Rap

Which is equivalent to the answer I posted.

Mathematical truth is not the same thing as empirical truth. The latter involves predictions, such that an empirical statement is true if and only if the experiences that it predicts come to pass. However, mathematical statements, being a subset of 'analytical statements,' are true if and only if they agree with other predesignated statements.

In essence, math is merely the analysis of descriptions. It is a procedure where nothing more is said than A = A. This means that mathematical calculations are simply an attempt to find out if two entities, functions, quantities or relations are the same. If they turn out to be the same, we call that equation "true," meaning the two things are the same thing. If they turn out to be different, then we call that equation "false," meaning the two things are not the same thing.

All of mathematics comes down to this art of answering the one simple question "are these two things the same?" in all those instances where the two things concerned can be described with zero ambiguity. If they cannot be so described, the problem cannot be addressed mathematically, unless the ambiguity itself is unambiguous, in which case we can only use statistics to do the figuring. Failing that, all that remains is ordinary language, and the inevitable errors of ambiguity inherent in it.

Therefore, I would be obliged if you could enlighten me on the Truth/lies question.

Whilst your answer is without a doubt correct, I am having great difficulty in understanding what relevance the second man has.

"You encounter two members of a tribe who tell the truth 25% of the time. One makes a statement, and the second says it's true. What is the probability that the first statement is true?"


For example, suppose there had been only one man who told the truth half of the time.

Would it be fair to assume that there was a 50:50 chance of the truth?

Now, what difference would it make if 10 other men gave an opinion?

Take the same point with a fair coin heads/tails. Would it make any difference if 1000 people said the next flip was going to be tails?
No, the odds would stubbornly remain at 50:50.


Perhaps you can enlighten me, as it would appear I have misplaced my brain. Thanking you in advance.

check on the rational numbers-there was a transcription error in step six, but it was corrected in step 7.

Neat piece of algebraic manipulation.

rap

Try:

Let's say Rap lies 99% of the time. Whim flips a fair coin, and Rap (who observes the flip) says it was heads. I then offer you the following wager:

If the coin is heads, you pay me $2.
If the coin is tails, I pay you $1.

Do you accept the wager? Absolutely (99 times out of 100 you'll win a dollar from me), but not if you believe the probability of heads is 1/2.

Rap's statement provides additional information that, in this case, outweighs the information that the coin is fair.

In a way, it's similar to the problem with the two coins where one is two-headed. If you choose a coin and see that one side is heads, the probability that the other side is heads is affected by the distribution of heads on the coins. The fact that you are looking at a head rules out one of the four possible initial outcomes.

Likewise, the fact that Rap says the coin is heads rules out (probabilistically) certain initial outcomes.

I hope that is enlightening.

[size=8]CATS
5

TREES
51

NAME
Frances - France - franc[/size]

[

Rap

Potato

Why, yes it is.

Rap

2:37AM Just a silly guess, didnt really work it out. Am I close?

Alibi
[size=7]The prof's body temperature is given by
from
dT/ΔT=-Cp*t and the initial condition (t=0) that ΔT=28ºT then
ΔT=28exp(-Cp*t)
Where ΔT is the temperature difference between the body and the room
Cp is the body heat loss constant
T is the time since death
@ 9:00AM ΔT=10ºF
@ 10:00AM ΔT=8ºF
so
10=28exp(-Cp*t) and 8=28exp(-Cp(t+1)) so Cp=ln(10/8)
going back to the body temp @ 9:00AM
t = ln(10/28)/ln(8/10)=4.614 hrs
so the prof bought the chalk at about 4:30AM and your studying til 12:00 midnight alibi needs some serious improvement.
Check
ΔT@9:00AM is about 28exp[-ln(10/8)*4.5]=10.25ºF
ΔT@ 10:00AM is about 28exp[-ln(10/8)*5.5]=8.20ºF[/size]
Rap