The worlds first riddle!

ANTS
I assume that if any ant can't make a legal move, then none of the ants move.

Max edges for two ants = 6
A's route: 1-2-3-4
B's route: 3-1-4-2

Max edges for three ants = 3
A's route: 1-2
B's route: 2-3
C's route: 3-1

Max edges for four ants = 4
A's route: 1-2
B's route: 3-1
C's route: 2-4
D's route: 4-3

Just for kicks, one ant can traverse 5 edges.
Route: 1-2-3-1-4-2

THREE PYRAMIDS
You would see eight or ten surfaces depending on how tall the ship is and how close to the island it is. There are 15 total surfaces. The base of the first pyramid is face down, and four other faces are used for attachment. That leaves ten. The top two faces make angles of almost 16 degrees with the horizontal plane. If you can get to a point above the tilted planes (climb a mast and/or move farther away from the island), you will see those surfaces.

Mark:

ANTS
I assume that if any ant can't make a legal move, then none of the ants move.

You assume correctly.

Max edges for two ants = 6

Max edges for three ants = 3 I thought the 3 ants could still cover all 6 edges without violating the rules.


Max edges for four ants = 4



Just for kicks, one ant can traverse 5 edges.
Route: 1-2-3-1-4-2

I never thought of that, very good.


THREE PYRAMIDS:

I tended to agree with Whim, and then there were edits…now no answer matches mine!

I have two surfaces of three (sides). Two surfaces of two (ends). As there was only one base on the ground, two bases. Total = 6.

Comments, objections and legal argument welcome.



Talking about more pyramids:

Marilyn vos Savant is a column in Parade Magazine. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."

In her Parade Magazine column of January 5, 1997, Marilyn claimed that 4900 balls would be required to build a four-sided equilateral pyramid (with a square base and equilateral triangles on the four sides).

Was she right


Now, how about:

If one side of the bottom layer of a triangular pyramid of balls has 12 balls, how many are there in the whole pyramid

How can you have fewer than six visible triangles (two per pyramid)? Those plus the two squares should be visible at sea level from any distance. Visibility of the top two triangles is dependent on the location of the viewer.

MARYLIN
4900 are required for a pyramid with a 24x24 base. However, only 5 are required for a pyramid with a 2x2 base. Perhaps she was looking for the first non-trivial solution where the total is a square number.

If N is the number of balls on a side, the total is
(N^3)/3 + (N^2)/2 + N/6

TRIANGULAR PYRAMID
364

If N is the number of balls on a side, the total is
(N^3)/6 + (N^2)/2 + N/3

Try, I edited my answer because I didn't use three identical pyramids.
I used one pyramid with a square base and stuck two tetraedras (spelling?) to its sides. That way you get 4 surfaces you can see. Two triangle shaped ones and two trapezium (three triangles in same plane) shaped ones.

Later I saw it needed to be three identical square base pyramids. So I changed my answer.

Whim

First, the no quibble answers.

Mark came up with the perfect solutions.

MARYLIN
4900 are required for a pyramid with a 24x24 base. However, only 5 are required for a pyramid with a 2x2 base. Perhaps she was looking for the first non-trivial solution where the total is a square number.

If N is the number of balls on a side, the total is
(N^3)/3 + (N^2)/2 + N/6

TRIANGULAR PYRAMID
364

If N is the number of balls on a side, the total is
(N^3)/6 + (N^2)/2 + N/3



Now, it is the ancient Pharaoh's curse on the riddle setter.
I am sure it was my poor description that caused the confusion.

Mark
"How can you have fewer than six visible triangles (two per pyramid)? Those plus the two squares should be visible at sea level from any distance. Visibility of the top two triangles is dependent on the location of the viewer."

Bearing in mind, we are not counting triangles, only surfaces. Three identical pyramids are joined together, but have only one base. The two sides of three form one surface, albeit made of three triangles. Two ends of one triangle each, and the underside of two square bases.

Whim
"I used one pyramid with a square base and stuck two tetraedras (spelling?) to its sides. That way you get 4 surfaces you can see. Two triangle shaped ones and two trapezium (three triangles in same plane) shaped ones.

Later I saw it needed to be three identical square base pyramids. So I changed my answer."

As the bases do not touch the ground, as I read it, you are still correct.

Perhaps it is easier to imagine, three identical pyramids built in a row with their bases touching. Then tilt the two ends, so the three points meet. Although you would see eight triangles, they now only make four surfaces (two sides and two ends). Plus two bases.


Back 2 basics.

The sum of the second and fifth terms of an arithmetic sequence
is equal to the sixth term of the sequence. If this common value is 25, what is the tenth term of the sequence


Politician A lies on Mondays, Tuesdays, and Wednesdays but tells the truth on the other days of the week.
Politician B lies on Thursdays, Fridays, and Saturdays but tells the truth the other days of the week.
One day, both of them said, "Yesterday was one of my lying days."
On what day did they say this


A square has a side length of x units.
The square's length is then increased by 2 units and its width is increased by 9 units. By how many square units does the area of the new rectangle exceed the area of the square

Answer in terms of x.

BACK 2 BASICS
Deja vu. See Nov. 5.

No, no, no. The three triangular faces on each side are not coplanar. If the two attached pyramids were tetrahedrons, then they would be. The dihedral angle between adjacent triangular faces of a square pyramid with equilateral triangular faces is approximately 109.47 degrees. Therefore, when a pyramid is tilted against another to attach triangular faces, the angle between adjacent triangular sides is about 141 degrees (360 - 109.47*2). They are not in the same plane.

The dihedral angle between adjacent faces of a tetrahedron is approximately 70.53 degrees. If you tilt the tetrahedron against the square pyramid, the angle between the two visible sides is 180 degrees. They are in the same plane.

Build two square pyramids with equilateral triangular faces and put them together. It will be very obvious.

If you consider the union of two non-coplanar, intersecting planes to be one surface, then the answer you seek is one.

1 - 4 - 7
The most obvious answer is probably 10.

HAT - FISH - HEAVEN

The only link I see is "7th heaven"

Eggs:
Assuming you boil them together in the same pot ... 10 minutes.


Vice President:
The President will still be the head of the Executive branch.

Pyramids:
I know you've conceded... but I have to say that I agree with Mark on this one.

WHAT TIME IS IT?
1:45 (or 1:35)

EGGS
I agree with MerlinsGodson, 10 minutes.

Merlin, "…I have to say that I agree with Mark on this one."

Mark, "I agree with MerlinsGodson"

Where I ask is Whim when you need him, he sometimes agrees with me. I will have to change my shrink.

Eggs:
The simple answer is 10 minutes. In practice, with more eggs it takes longer for the water to reach boiling point, so 15 to 20 minutes would be more accurate. However, you avoided 120 minutes.


Mark:
WHAT TIME IS IT?
1:45 (or 1:35)

1:45 will do nicely. Why I hear you ask? Well, the man gave a quarter to two, etc.

BTW what was thinking behind 1:35?


Two convicts are locked in a cell. There is an unbarred window high up in the cell. No matter if they stand on the bed or one on top of the other they can't reach the window to escape. They then decide to tunnel out. However, they give up with the tunnelling because it will take too long. Finally, one of the convicts figures out how to escape from the cell.

What is his plan


Use the digits 1, 2, 3, 4, 5 and 6 once only, in this multiplication sum to make it correct.

? ?
x ?
-------
? ? ?

54
*3
-----
162


Whim

Two convicts :


They start to pile up the sand from tunneling. And when there is enough sand in the cell they escape with the help of the bed ontop of the pile.

Whim