Jay Reevo
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I've noticed that Limes gets a lot of attention over at the 1 Player Guild. Seeing photos of it left me unenthusiastic for a long while until I watched a Rahdo doing a run-through. It struck me that there must be.... well, I didn't know how many card placement possibilities there couple be. I've been at it for a while but all but given up trying to figure out how to work it out. I figure that there is a factorial element (the order and choice of 16 out of 24 cards) multiplied by the placement and rotational placements. Anyone fancy working it out?


(Image by Bill Kunes)

A quick look at the mechanics (well, my understanding, anyway) without any thought to the actual game-play, meeple placement or scoring... just the laying mechanic and the possible placement options. I think that this covers the individual variables -

- There's a shuffled, numbered deck of 24 cards. Only the top sixteen will be used.
- Cards are played from the deck one at a time.
- The cards need to be permanently placed orthogonally (so no diagonal placement), thus having four placement options (apart from the first card) with four rotational options.
- The card also has to fit inside a 4x4 grid, which is fluid in that the grid doesn't really become concrete until there are four vertical and horizontal cards in play. (The first card could be anywhere in the grid to begin with but would move around as others are placed next to it.)
- This continues until the sixteenth card is played and placed in that 4x4 grid.
- The remaining eight cards are not used.

I thought that this would be a good one to flex my brain and an exercise but I'm so out of my depth. However, I'm still so interested in the answer... well, not so much the answer but how this equation would be solved seeing as there are quite a few variables (including diminishing placement options). I was also wondering if it may not even be the ability to give a concrete answer due to the floating first card.

Hopefully someone can provide an answer and/or explanation.

Thanks for looking in.

My head aches!

p.s. I just ordered the game.


 
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Josh Jennings
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This seems pretty straight-forward to me, but please correct me if I'm wrong (it happens often!).

First we need to select just 16 of the 24 cards for use. That is a simple combination C(24,16) = 735,471 combinations.

Now we need to orient the 16 cards into 1 4x4 grid. Note that although the order in which they are played can differ, the end result is always the same. The cards will be arranged in a 4x4 grid using the 16 cards we've selected. If you want to cover the different possibilities for playing the cards then it becomes a much harder problem because there are many different ways to arrive at the same 4x4 grid.

The number of different 4x4 grids from 16 cards is simple permutation. P(16,16) = 20,922,789,888,000. So there are 20 trillion permutations with just 16 cards assuming no rotation.

Next we take into account the rotation. Each card can be rotated in one of four orientations. That means that there are 4^16 = 4,294,967,296 combinations of rotations for the 16 cards.

Our final answer is all of those numbers multiplied together. The result is: 66,091,408,588,783,063,199,121,408,000 different board layouts.
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Byron S
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There are 24P16=1.5388x10^19 ways to choose the order of the 16 tiles from the pile of 24. Beyond that, there are 4 ways to independently orient each tile, so you multiply that by 4^16 = 4,294,967,296. However, you will get some duplicate layouts due to symmetry, so divide by... 4. The final count is somewhere around...
16,522,852,147,195,765,799,780,352,000 possible layouts.
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Josh Jennings
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runtsta wrote:
There are 24P16=1.5388x10^19 ways to choose the order of the 16 tiles from the pile of 24. Beyond that, there are 4 ways to independently orient each tile, so you multiply that by 4^16 = 4,294,967,296. However, you will get some duplicate layouts due to symmetry, so divide by... 4. The final count is somewhere around...
1.65 x 10^28 possible layouts.


Ooh, I forgot to divide by 4. Good catch!

That would mean 16,522,852,147,195,765,799,780,352,000, to be precise. ;)
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col_w
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About the same amount of layouts as there are molecules in half a tonne of water.
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Koldie wrote:
Shannon number. The theoretical number of legal positions on a chess board.
The number is often compared to the number of atoms in the universe.

Color me naive, but I am constantly amazed that human brains can contemplate (let alone compare) highly abstract, disparately-sourced numbers like these.

It's quite an achievement for organs that evolved from crude survival engines. We should all be very proud of our brains, however we use them.
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Christopher Dearlove
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davek wrote:
Koldie wrote:
Shannon number. The theoretical number of legal positions on a chess board.
The number is often compared to the number of atoms in the universe.

Color me naive, but I am constantly amazed that human brains can contemplate (let alone compare) highly abstract, disparately-sourced numbers like these.


These are all small numbers compared to others that are considered in other contexts. Read about Knuth's arrow notation for really big numbers.

But still finite. Now look at transfinite numbers. Not just small ones like the number of integers or the number of real numbers (not the same) but bigger than that. I recall a supervisor who having shown how to construct a transfinite number of staggering size then described it a short small.

But no one really understands these. They just know how to define them, reason about then, and understand their properties.
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Jay Reevo
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Wow! You lot are incredible! Thanks so much for all your replies.

thermogimp wrote:
This seems pretty straight-forward to me


My brain is still wrestling with all the info but I think I can see how the math works. However, I do have some questions about the mathematical vocabulary if you're still with me ...

thermogimp wrote:
The number of different 4x4 grids from 16 cards is simple permutation. P(16,16) = 20,922,789,888,000.


I understand how the factorial works in this instance. P(16,16) can be written as 16! correct? 16x15x14x13x13 etc.

runtsta wrote:
There are 24P16=1.5388x10^19 ways to choose the order of the 16 tiles from the pile of 24.


This one above threw me though and was the reason that I gave up in the first place. I was thinking that I would have had to started with 24! but then stopped when the equation finished after 16. I had no idea how something like that would be calculated. Is that what this means?

24P16

Byron, would you mind elaborating on the vocabulary of this please? How would it be calculated on a... calculator!?


Dearlove wrote:
These are all small numbers compared to others that are considered in other contexts.


Mind blowing stuff.

davek wrote:
We should all be very proud of our brains, however we use them.


Well, it should be "We You should all be very proud of our brains [...]"

Well, I am amazed at how people can work this kind of thing out. I just don't have the vocab so struggle with this kind of thing. I'm fascinated by it although I was very average at it in school but learn to adore it as an adult (I'm now 44). I spend quiet a bit trying to work out this kind of thing and enjoy the challenge. Last week I worked out the area of circle's segment that was half of its radius after seeing a some statistic that was inaccurately visualised. It took me 48 minutes! But... I was very pleased with myself! Actually, I'm grateful for being curious enough to think about the problem, even if I am unable to work it out myself.

davek wrote:
We should all be very proud of our brains, however we use them.


col_w wrote:
About the same amount of layouts as there are molecules in half a tonne of water.


OK, that got me... I'm going to have a lay down! Thanks for the input everyone.
 
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The simple bit is the permutations and combinations.

Suppose we have n objects, all different. (Extending this to when they aren't all the same can be done, but let's not worry about that now.)

The easier case to explain is permutations. I want r of these objects, in order. There are obviously n ways to pick the first one, then (n-1) ways to pick the second, down to (n-r+1) ways to pick the r-th. To get the total number, multiply these all together.

You can express this as multiplying all the numbers up to n together, then dividing by the product of all the numbers up to (n-r). Or using factorials, the answer we want, permutations of r objects from n, or nPr, is given by:

nPr = n! / (n-r)!

But if you actually want to calculate it, don't use that, use the product form, because it is both quicker, and sometimes n! is too big, although nPr is not.

Now let's consider combinations of r objects from n. One way to do that is to say we can group together all the permutations that produce the same combination. Which is how many ways to order r objects. Which is actually rPr, but more usefully is r!

So now we have that the number of combinations nCr is given by:

nCr = nPr / r! = n! / (n-r)! / r!

Note that nCr = nC(n-r). That should be obvious if you think about it.

Note that to calculate nCr it's actually possible for nPr to be too big (overflow) but nCr not to. Calculating nCr so that it only overflows if the final result overflows is possible, and if you really want to know how, ask. But mostly you don't need that. Except if doing it by hand, cancel factors before multiplying, it's easer - and always possible.

On a calculator, look for the nPr and nCr buttons. If your calculator hasn't got them, one that does will cost you about a pound here. How you use them is typically type number, press nPr or nCr, type number, press = or Ans.
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thermogimp wrote:
runtsta wrote:
There are 24P16=1.5388x10^19 ways to choose the order of the 16 tiles from the pile of 24. Beyond that, there are 4 ways to independently orient each tile, so you multiply that by 4^16 = 4,294,967,296. However, you will get some duplicate layouts due to symmetry, so divide by... 4. The final count is somewhere around...
1.65 x 10^28 possible layouts.

Ooh, I forgot to divide by 4. Good catch!

That would mean 16,522,852,147,195,765,799,780,352,000, to be precise.

Just looking at the board shown, tiles 20, 24, and 11(? to the right of 24) appear to have rotational symmetry. Tiles 12 and 23 appear to be rotations of each other. The rest of the tiles...? I don't see an inventory of tiles in the game page.
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Jay Reevo
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... and it just got more complicated... well, I guess less complicated, really! Less cards placement options due to the rotational symmetry on some cards.

Thanks all. My brain hurts with all the thinking. I'm doing my best to suppress the though of adding seven maples into the mix!

You lot are incredible.
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Christopher Dearlove
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On the subject of frightening large numbers, when Rubik's cube came out, it was advertised as "more than three billion" permutations. (Actually I remembered it as three million, but apparently not.) In fact there are more like 43 quintillion. Which is more than three billion, but in the same way that the sun is more than ten metres away. (Actually even that shows that an astronomical unit is a hard to comprehend size, so the shortfall may not be as obvious.)
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