h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

That's very likely to be the answer Czar Dene is seeking. I came up with the same solution a couple of days ago, but dismissed it as being too prosaic.

Too prosaic? You should have answered in verse

Icy is correct to say that it is binary. But I had the broken lines as nil. But I accept your answer. Your turn.

How many ways are there of tiling a floor, if your tiles are regular polygons whose sides have to fit edge-to-edge (i.e. no gaps and no skewing them, as with a brick wall).

5,505,005?

I'll accept either of two answers, but sadly that's not one of them. It is a multiple of one of them, though

Surely the number of permutations depends on the number of tiles needed to cover the floor?

If I have 16 square plain white tiles to fit a 4x4 floor, then any of 16 can be placed top left, any of 15 can fit next to it, any of 14 next to that, and so on - I may be wrong (I'm happy to be corrected) but doesn't that amount to 16! (2.09227899 x 10^13)?

On the same basis (and I'm beginning to have serious doubts here ), 100 square plain white tiles would give some 9.33262154 x 10^157 permutations.

The problem is complicated further when you recognise that each tile has 4 edges, and so may be placed 4 ways...

Where *is* Julzes when you need him?

I'm looking for the number of distinct tiling patterns, so 'squares' would be one pattern.

If the outermost edges (those touching the walls) have to be straight, then equilateral triangles work; if the edges can be ragged, then hexagons will do the job.

On the other hand, if mixing shapes and/or sizes in the same design is permitted, I'd have to think again (or maybe not ).

On the third hand, neither 2 nor 3 go into 5,505,005...

The square is just one type of rectangle, so I suppose I should add rectangles as another shape. BUT 4 doesn't go into 5,505,005 either. Perhaps the answer is five?

Rectangles aren't generally regular (only in the case of a square).

Squares, triangles and hexagons are three of the tilings I'm looking for, but I'm also looking for tilings which use more than one type

And the answer isn't 5.

Lost patience and went to Google for the answer.

Still, at least we've managed to clarify the question a bit.

Like to post the answer, so we can move on?

I have nothing to post, so let's let someone else have the pleasure.

I'll post it then, and we'll have a footrace for the next puzzle.

As you probably found in your Google search, the classic answer is the set of 11 Archimedean tilings.

Three of these are the regular ones you found, using the same polygon throughout: http://mathworld.wolfram.com/RegularTessellation.html

The other eight use more than one type of polygon: http://mathworld.wolfram.com/SemiregularTessellation.html

But, depending on your definition of tiling, you can define lots of similar ones (a hexagon can be replaced by six triangles, etc). In fact, there is an infinite number.

***

OK, next puzzle, anyone?

Spoilsport!

"In two years," said Dad to Roy and Joy, "Joy's age will be three-quarters of Roy's and in 10 years your combined ages will be three-quarters of mine. At the moment the total of your ages is a third of mine."
How old are they?

Looks like it's just you and me, Czar Dene . Perhaps you can attract a few more people from the other games threads?

As usual, I don't have a new teaser so I'll just add another clue (hope you don't mind):

The three ages (currently) add up to 40 years.

It's a shame we don't have more subscribers

I'll post one, but I'd better solve this one first:

I'll leave the working out to others, but there are three equations in three variables, ages J, R and D, and it all works out to Roy being 6, Joy being 4 and Dad being 30.

OK, here's a new one:

There are an odd number of meerkats in a field. Each meerkat is suspiciously eyeing the nearest meerkat. How do you know that at least one meerkat isn't being watched?

... I should also add that the meerkats are a distinct distance apart.