[Trumpace]

What kind do you like? Can we have a go at some?

Here is one from Martin Gardner:

You have an 8x8 mutilated chessboard: a1 and h8 have been removed (the two end squares of the diagonal). You are given 31 dominoes (2x1 rectangles), can you cover the mutilated chessboard with those 31 dominoes completely?

[whereagles]

That's a classic one. Here's the solution (hidden):

Spoiler

A1 and H8 have the same color (black). If you remove 2 black squares, there will be 2 unmatched white squares which you'd have to cover with 1 domino piece. But that is impossible because every domino piece always has to cover a black and a white square.

[Trumpace]

Right whereeagles!

Are we the only 2 interested in puzzles?

[jikl]

This is a sort of puzzle...

Name a US state that is two antonyms separated by one letter. (Example: BlackQWhite)

Sean

[Blofeld]

Answer hidden:

Spoiler

Connecticut

Another domino one: how many different ways are there of tiling a 2 by n board with n dominoes? (ignoring the pips on the dominoes)

[Sigi_BC84]

I liked this one:

You are given 9 black marbles, 11 red marbles and 13 white marbles.
You are also given a magic device with a hole at the top and a dispenser at the bottom that works as follows:

• if you insert two marbles of different color, the box eats the marbles and dispenses two marbles of the remaining color (eg. insert a red and a white, get two blacks in return etc.)
  
• if you insert two marbles of the same color, the box eats the marbles and dispenses two marbles, one of each remaining color (eg. insert two blacks and get one red and one white in return etc.).
  
• the box has an endless power supply, ie. it does the above as often as you want

Your task is to recolor your supply of marbles so that you end up with 33 marbles of the same color (either all black, all red or all white).

How can that be done, if at all? If not, why not?

--Sigi

Hint (hidden):

Spoiler

It helps to know about basic modular arithmetic.

[Blofeld]

Marbles:

Spoiler

Modulo 3, any operation does the same thing (adds or subtracts one) to all three totals. So in particular the difference between any two totals remains unchanged (mod 3). But none of the differences are zero (mod 3) to begin with, and to get all of them the same colour needs two of the totals to be zero, and hence the difference between them zero (also zero mode 3). So we can't do it. At best we can get 32 of one colour and one of another.

[luke warm]

here are a couple of old ones... not hard, just old

a farmer has a hound, a fox, and a hen... he has to get himself and all 3 animals across a river, but his boat is so small that it will only hold him and one animal at a time... he can't leave the fox and hen alone, the fox will eat the hen... if he leaves the hound and fox alone, the hound will kill the fox... how does he cross the river with his 3 animals? like i said, not hard just old

the other relatively easy one goes like this (hope i remember it correctly)... there is a room that contains two doors... nothing distinguishes the doors, the only difference is that one door leads to heaven, the other to hell... the room also has in it two men, exact twins... they are the same in every way except that one always tells the truth and one always lies... they also happen to know which door leads to heaven and which to hell

you can ask either twin one and only one question... what question do you ask to determine which door leads to heaven?

[Trumpace]

Blofeld, on Jul 12 2006, 07:20 AM, said:

Another domino one: how many different ways are there of tiling a 2 by n board with n dominoes? (ignoring the pips on the dominoes)

Hidden:

Spoiler

Fibonacci

[barmar]

Twins:

Spoiler

Ask either one "If I asked your brother which door goes to Hell, what would he say?" The answer will be the door to Heaven.

[Sigi_BC84]

How about this one (ridiculously hard):

What is the maximum number of completely bounded volumes that can be formed by three interpenetrating cubes, considering only the surfaces of the cubes as boundaries and counting only volumes that are not further subdivided?

(BTW I don't know the answer but it's supposed to have been posted somewhere on the net.)

--Sigi