A strange request
- Can you put some tiles in my garden, please?
- Can do. Which colours are they?
- Some are black and some are white.
- What pattern do you have in mind?
- The tiles will make a straight walkway only one tile wide.
- And the pattern?
- I don’t care. But I do have one condition.
- One moment. Let me write it down.
- Put the tiles down in such a way that there are at least two tiles of the same colour and the tile midway between them has the same colour too.
- Is this a prank call?
- Do you mean it can’t be done?
- No, it is just too easy. Three white tiles in a row will do.
- I see. I am sorry, I guess I explained it badly. I want you to put the tiles down in a random fashion.
- You mean we should throw a coin to decide if the next tile should be black or white?
- Yes.
- We can do that, but what is your problem?
- Will I be guaranteed to get two tiles of the same colour with the tile midway between them of the same colour too?
- That is a different question. I will switch you over to Mr. Knowles. He knows more about this. Just one minute.
February 26th, 2009 at 4:09 am
I don’t think I fully understand but I’ll take a stab at it.
If you have room for just three tiles, then you’d flip a coin once – heads means three blacks while tails means three whites.
These requirements can only be met with an odd number of tiles ? 3.
Flip a coin to determine the color of the first, middle, and last tiles. Flip it again for each remaining tile.
February 26th, 2009 at 6:13 am
I think the problem is this: For any combination of colours, will one always have two tiles and the tile midway between them with the same colour?
For 5 tiles one may have BWWBW, so the answer is no.
For 7 tiles one may have BWWBWWB, so the answer may be yes since the first and last tile meet the condition. But are there combinations of 7 tiles where the condition is not met?