[Six points]

How about seven points?

[Six points]

Can you put six points on the plane, so that the distance between any two of them is an integer, and no three are collinear?

[more difficult dissection: the square]

It use to be one of my favorites. I found it in Mathematical Snapshots,

a great book (devoted to recreational math) by Hugo Steinhaus.

[Probability of a rope cut exactly in half, in a single slice]

there are only three possibilities of this experiment after the cut.

1. Left peice is bigger

2. right peice is bigger

3. left peice = right peice

so the probability he calculated is 1/3

It would be true, if all three possibilities were equally probable.

Which is not the case.

[Playing with bases]

The statement is false for every base that is divisible by a square of some prime and true otherwise.

So it it is false for 8, 9 and true for 5, 6, 7 (and 10).

[Death Penalty]

I would quickly swallow picked paper. My executioners will have no choice but to look at the other paper (with "DEATH" written on it) and they will deduce that I've picked up one with "LIFE".

[Slicing a pizza a whole different way]

any explanation please

A problem is similar to (see my solution).

You are adding new point on the circle and drawing new chords. For each chord you count intersections with old chords and add 1, this gives you the number of new regions. This is how you arrive at the formula. Last step is to simplify it.

But to be honest I just guessed the solution

I took the formula from above mentioned problem and made an adjustment for that extra 1s. Few tries and it worked.

Edit: Guessing helped me only in the last step (simplification). I needed non-simplified formula to verify guessed one.

[Slicing a pizza a whole different way]

I was to hasty with my previous answer.

Here's another try:

n(n-1)(n-2)(n-3)/24 + n(n-1)/2 + 1

[Slicing a pizza a whole different way]

n(n-1)(n-2)/6 + n

[Replacing Tiles]

a) How many tiles are ruined on a 4 by 6 floor?

b) How about a 63 by 81 floor?

a) 4 + 6 - gcd(4,6) = 4+ 6 - 2 = 8

b) 63 + 81 - gcd(63,81) = 63 + 81 - 9 = 135

[Replacing Tiles]

m + n - gcd(m,n)

[An oldie a professor once told me on Area and Perimeter]

Anyone cares to show why the other two solutions I've found (in post #5) are wrong?

[Two Ten Ton Tough Elephants]

Do the reckoned years start with winter or with summer?

Commutativity of multiplication makes it irrelevant, I believe.

[An oldie a professor once told me on Area and Perimeter]

I got 3 solutions: (4,13), (4,61), (16,73).

[Reciprocals and Cubes]

I would go with binomial formula:

(x+y)³ = x³ + 3x²y + 3xy² + y³ = (x³ + y³) + 3xy(x+y)

We know that

x³ + y³ = 4

and

1/x + 1/y = -1

or

x + y = -xy.

Let's introduce new variable:

z = x + y = -xy.

We can now substitute binomial formula with:

z³ = 4 - 3z²,

which is equivalent to

(z-1)(z+2)² = 0.

Last thing to do is to solve system of equations:

x + y = z

xy = -z

for z = 1 and for z = -2.

For z=1 we get "golden ratio" solutions: (1 ± sqrt(5))/2,

and z=-2 case has no real solutions.

[Calling all positive integers ...]

4x10

²⁴⁹

[Discursive Judges]

It obviously depends on jurisdictional procedure. If each judge votes guilty/not guilty, then the accused will be released, since only 1 out of 3 judges votes "guilty". But if judges vote on P, then vote on Q - he will be convicted (both P and Q will be voted "true" with 2 votes out of 3).

[Squaring a Bishop]

It couldn't find 10x10 square.

[Squaring a Bishop]

More squared bishops in attached file.

bishop.txt

[The seeking mountain climber]

The audience took 12 hours, which leaves 16 hours for climbing/descending. Since the seeker descended three times faster than he climbed, descending took three times less time then climbing, so he was climbing for 12 hours and descending for 4 hours. Climbing for 12 hours at the rate 1.5 mph means that the trail was 18 miles long. (Descending for 4h at the rate 4.5 mph gives the same 18 miles.)

[Matching values of Die throws pt. 3]

if they are equivalent then why don't they produce the same expected values as Bushindo pointed out.

They do of course. Bushindo made a silly error evaluating your formula.

[Matching values of Die throws pt. 3]

There are kⁿ ways in which n dice can each show k or less.

For the highest score to equal k, we must subtract those cases for which each die shows less than k; these number (k − 1)ⁿ.

So, k is the highest score in kⁿ − (k − 1)ⁿ cases out of 6ⁿ.

In other words, p_n(k), the probability that the highest individual score is k, is (kⁿ − (k − 1)ⁿ)/6ⁿ.

The expected value, E(n), of the highest score is the sum, from k = 1 to 6, of k · p_n(k).

Hence E(n) = [6(6ⁿ−5ⁿ) + 5(5ⁿ−4ⁿ) + 4(4ⁿ−3ⁿ) + 3(3ⁿ−2ⁿ) + 2(2ⁿ−1ⁿ) + 1(1ⁿ−0ⁿ)]/6ⁿ. = 6 − (1ⁿ + 2ⁿ + 3ⁿ + 4ⁿ + 5ⁿ)/6ⁿ.

7 - (1^n + 2^n + 3^n + 4^n + 5^n + 6^n)/6^n =

= 6 + 6^n/6^n - (1^n + 2^n + 3^n + 4^n + 5^n + 6^n)/6^n =

= 6 - (1^n + 2^n + 3^n + 4^n + 5^n)/6^n

[Matching values of Die throws pt. 3]

7 - (1^n + 2^n + 3^n + 4^n + 5^n + 6^n)/6^n

For the highest score to equal k, we must subtract those cases for which each die shows less than k; these number (k − 1)ⁿ.

So, k is the highest score in kⁿ − (k − 1)ⁿ cases out of 6ⁿ.

In other words, p_n(k), the probability that the highest individual score is k, is (kⁿ − (k − 1)ⁿ)/6ⁿ.

The expected value, E(n), of the highest score is the sum, from k = 1 to 6, of k · p_n(k).

That's the obvious and natural approach. And that's exactly how I got the formula.

When you say "close but", you mean that the formula is wrong?

[Squaring a Bishop]

BISHOP
IMPOSE
SPORTS
HORNET
OSTEAL
PESTLE

[Matching values of Die throws pt. 3]

7 - (1^n + 2^n + 3^n + 4^n + 5^n + 6^n)/6^n