The worlds first riddle!

Mark:

2007 TRIANGLES

I get:
84169 triangles
502 isosceles (including equilateral)
1 equilateral
83667 scalene

0 right
26880 acute
57289 obtuse




Those answers rock!

However, for the sceptics…
Let the lengths of the sides be a, b and c. We avoid duplicates by ordering them a<=b<=c. The triangle inequality is now equivalent to c<a+b. To get the right perimeter, we need b=2007-a-c. Then a<=b means 2a+c<=2007, b<c>=2007, and c<a+b means 2c<2007.

We could simply count the triangles systematically. Note that 2007/3<=c<2007/2. For each value of c, the inequalities in the previous paragraph imply 2007-2c<=a<=(2007-c)/2.

With a and c being integers, there are 1+2+4+5+7+8+...496+497+499+500+502 solutions. Grouping terms (500+1=499+2=497+4...), this works out to 502+334*501/2=84,169. When a hits either of its bounds the triangle is isosceles.

This happens for every c for the lower bound and every odd c for the upper bound, a total of 502 times. One of these is when both bounds are hit, giving the equilateral triangle.



An amusing alternative method of counting:

Each triangle corresponds to a lattice point (a point with integer coordinates) in the region of the (a,c)-plane satisfying our inequalities. This region is the triangle with vertices v1=(669,669), v2=(1,1003) and v3=(502,1003). The points on (v1,v2] -- the line from v1 to v2, excluding v1 and including v2 -- and (v1-v3] give isosceles triangles; the point v1 gives an equilateral triangle; the remaining points (on (v2,v3) and inside the region) give scalene triangles.

It is easy to see that there are 334 lattice points on (v1,v2], 167 lattice points on (v1-v3] and 500 lattice points on (v2-v3). To find the number of lattice points inside the region, we use Pick's Theorem, which states that the area of a simple lattice polygon is one less than the sum of the number of lattice points in its interior and half the number of lattice points on its boundary. The number of interior lattice points is thus the area (501*334/2) plus 1 minus half the number of boundary lattice points (1+334+167+500), which works out to 83,667.





Every year at this time we host the Riddles Mastermind competition and 2007 is no exception.

This year's mega prize is the biggest and best -


a) What are ID ten T mistakes


b) When is four half of five


c) You have a dime and a dollar, you buy a dog and a collar, the dog is a dollar more than the collar, how much is the collar


d) On my way to the fair, I met 7 jugglers and a bear, every juggler had 6 cats, every cat had 5 rats, every rat had 4 houses, every house had 3 mouse's, every mouse had 2 louses, every louse had a spouse. How many in all are going to the fair


e) There was an airplane crash, every single person died, but two people survived. How is this possible


f) That attorney is my brother, testified the accountant. But the attorney testified he didn't have a brother. Who is lying


g) The Mississippi River is the dividing line between Tennessee and Arkansas. If an airplane crashed exactly in the middle of the Mississippi River, where would the survivors be buried


h) What came first, the chicken or the egg



Good luck!

MASTERMIND
[size=7]a) ?
b) ?
c) a nickel
d) 1 (you)
e) they were married
f) neither - the accountant is a woman
g) survivors aren't buried
h) chicken[/size]

[edit] found answers to a and b with Google.

I have been directed by the Politburo to point out that question:

h) What came first, the chicken or the egg?

Requires the correct scientific answer!


Thank you.

Mark:

MASTERMIND

a) (ID10T)
b) (F(IV)E)
c) a nickel
d) 1 (you)
e) they were married
f) neither - the accountant is a woman
g) survivors aren't buried
h) chicken

H) Dinosaurs were laying eggs before chickens were invented.


Congratulations, for the fourth year in succession you are the official winner - collect your prize from the foyer of A2K Towers! :wink:



Using pennies, nickels, dimes, quarters, half-dollars and dollar coins, in how many ways can you make change for a dollar

But were they laying chicken eggs (not that that is what was asked for)?

Using pennies, nickels, dimes, quarters, half-dollars and dollar coins, in how many ways can you make change for a dollar

If you don't count swapping a dollar coin for a dollar as making change then there are 292 different ways. You want them listed?
taking a cue from Mark I did it in white...I could be wrong - :wink:

It took like half the morning to get from the bunk house to the barn; what is this white stuff?


Miss Mi:

CHANGE for a DOLLAR

If you don't count swapping a dollar coin for a dollar as making change then there are 292 different ways.

Hey, lucky guess!...No, I'm only joshing with ya, It was an inspired guess.

"You want them listed?"

Regretfully I must decline your kind offer; life is too short.

"taking a cue from Mark…"

I didn't know you played pool, that's great!



CPHAIER


DALENCE

I was wondering…

How many; all positive real solutions are there to the simultaneous equations:

(1) x + y2 + z3 = 3
(2) y + z2 + x3 = 3
(3) z + x2 + y3 = 3

Grunt.

About the chicken and the egg, I fancy a dinosaur lay that mutant egg which was the first chicken.

CPHAIER = pine chair


DALENCE = line dance

SIMULTANEOUS EQUATIONS
[size=7]at least one: (1,1,1) :wink: [/size]

Hey, lucky guess!...No, I'm only joshing with ya, It was an inspired guess.


..actually - math whiz hubby helped me..you know there is no way I could do it on my own...

Solipsister contributes to the rich mix of English literature with:

" Grunt."

This may be a Solipsism, however a Grunt may refer to; but not restricted to:

a deep guttural sound
slang for an infantryman
A character in the G.I. Joe universe who is, in fact, an infantryman.
"The Grunt", a 1970 instrumental recording by The J.B.'s
the fantasy novel Grunts
fishes in the family Haemulidae called grunts
the death metal singing style
the class Grunt in Conker Live & Reloaded
a Covenant enemy in Halo: Combat Evolved
a dessert comprised of stewed or baked fruit covered with a rolled biscuit or cookie dough and baked. Certain grunts called "slumps" are similarly made but inverted before serving, so that the crust is on the bottom.
a Warcraft orcish warrior.
Gruntz a puzzle/strategy game.
a Finnish noise/power electronics artist called Grunt
a type of fish the Grunt-fish
Often associated with VN Commodores, grunt refers to the lack of power, handling and general drivability. Grunt is said to justify why faster cars are not as good, because "they lack grunt mate".
Grunt RX-10 - a sci-fi project and book, a Matrix parody


"About the chicken and the egg, I fancy a dinosaur lay that mutant egg which was the first chickenStormy:

CPHAIER = pine chair


DALENCE = line dance

Good to see that you still have power in all that snow. Start the holidays early, any problems; tell them I said it would be ok! :wink:


Miss mi Mark:

SIMULTANEOUS EQUATIONS
at least one: (1,1,1)


Clearly one solution is x = y = z = 1. And that this is the only solution.

From (1) − (2),
x(1 − x2) + y(y − 1) + z2(z − 1) = 0 (4)
Similarly,
y(1 − y2) + z(z − 1) + x2(x − 1) = 0 (5)

Next, from (4) − z • (5),
x(x − 1)(1 + x + xz) = y(y − 1)(1 + z + yz) (6)
Similarly,
y(y − 1)(1 + y + yx) = z(z − 1)(1 + x + zx) (7)

Since x, y, and z are positive, the factors (1 + x + xz), (1 + z + yz), and (1 + y + yx) are all positive.
Hence (x − 1), (y − 1), are (z − 1) are all negative, all zero, or all positive.

That is, x, y, and z are all less than 1, equal to 1, or greater than 1.
We have already accounted for the second case. The other two cases are inconsistent with equations (1), (2) and (3).

Therefore, the only positive real solution of the simultaneous equations is x = y = z = 1.

Source: Polynomials, by Edward J. Barbeau.


Mark continues, " :wink: "


Woah! Slow down, you are so talkative.




CWTEHRILL


SDRPIEVGED


REBCYCLE

Please note: − = - (Minus)


Stupid system!

[size=7]winter chill
driving speed
recycle bin[/size]

Mark:

CWTEHRILL = winter chill
SDRPIEVGED = driving speed
REBCYCLE = recycle bin

Damn, that was good!


There are two empty urns in a room. You have 50 white balls and 50 black balls.

After you place the balls in the urns, two random balls will be picked, each from a random urn.

The first ball will not be returned to the urns after it is removed.

Distribute the balls (all of them) into the urns to maximize the chance of picking:

a) two white balls
b) one white and one black ball (in any order.)

I'm thinking:

a:
Distribute them as such: 50 of white in one, 50 of black in the other.
That should create a 25% chance of getting two white balls.
(I can't think of any other scenarios off the top of my head )

b.
Same situation as the first, except now there is a 50% chance of success.

P.S.: Thought of these quickly while procrastinating on a History assignment

BALLS
[size=7]I haven't calculated the probabilities, but I'd say:

a) two white balls in one urn, all other balls in the other urn
b) one of each in one urn, all other balls in the other urn

Or, to generalize, put what you want to draw in one urn, and put the rest in the other urn.[/size]

TehMeh:

a:
Distribute them as such: 50 of white in one, 50 of black in the other.
That should create a 25% chance of getting two white balls.
(I can't think of any other scenarios off the top of my head )

I think you will find Mark and I like better odds than that. :wink:

b.
Same situation as the first, except now there is a 50% chance of success.

Getting better.

P.S.: Thought of these quickly while procrastinating on a History assignment


History is like trying to nail jelly to the wall. Those who cannot learn from it are doomed to repeat it.



Mark:

BALLS
I haven't calculated the probabilities, but I'd say:

a) two white balls in one urn, all other balls in the other urn
b) one of each in one urn, all other balls in the other urn

Let's not generalize!


a - To maximize the chance of picking 2 white (W) balls, the best way is indeed to put 2 W balls in the urn U1 and 48 W balls + 50 B (black) balls in the urn U2.

There are 4 four possibilities to choose the two urns: U1 & U1, U1 & U2, U2 & U1, U2 & U2, each of them having the same probability 1/4.

So the probability P(1a) to pick the two W balls is equal to:

P(1a) = 1/4 * [2/2 * 1/1 + 2/2 * 48/98 + 48/98*2/2 + 48/98 * 47/97 ] = 10537 / 19012 = 0,554228...
It is easy to check that if the urn U1 contained a higher number of W balls or a certain number of B balls, the probability P(1a) should decrease.

b - Same reasoning: we put one W ball and one B ball in the urn U1. So the urn U2 contains 49 W balls and 49 B balls.

So the probability to pick one W ball and one B ball (in this order) is equal to:

1/4 * [1/2 * 1/1 + 1/2 * 49/98 + 49/98 * 1/2 + 49/98 * 49/97]
The probability to pick one B ball and one W ball (in this order) is the same.

So the requested probability to pick one W ball and one B ball (in any order) is equal to:

P(1b) = 1/4 * [ 1 + 2 * 49/98 + 2 * 49/98 * 49/97] = 243 / 388 = 0,626288....



EBSUSS


GMAUCHEN

* The sum of the digit is 6.
* Each digit is different.
* The number is odd.

What is the greatest 4-digit number that has all of the characteristics listed above