Xmass Present Math Puzzle

[CraigD]

For XMass, I got a Quarto board and pieces. :)

 

What does the sequence 1, 2, 3, 4, 5, 8, 13, 15, 6, 11, 9, 16, 7, 12, 14, 10 have to do with it?

[Turtle]

For XMass, I got a Quarto board and pieces. :)

 

What does the sequence 1, 2, 3, 4, 5, 8, 13, 15, 6, 11, 9, 16, 7, 12, 14, 10 have to do with it?

 

:xmas_sheep: 0001, 0010, 0011, 0100, 0101, 1000, 1101, 1111, 0110, 1011, 1001, 0000, 0111, 1100, 1110, 1010 a draw?? :xmas_rudolph:

[Turtle]

For XMass, I got a Quarto board and pieces. :)

 

What does the sequence 1, 2, 3, 4, 5, 8, 13, 15, 6, 11, 9, 16, 7, 12, 14, 10 have to do with it?

 

perfect players & one of the only two draws using all 16 pieces? (the other being 14, 13, 12, 11, 10, 7, 2, 16, 9, 4, 6, 15, 8, 3, 1, 5)

 

i have some variants for you to try craig.

 

...Players take turns choosing a piece which the other player must then place on the board. A player wins by placing a piece on the board which forms a horizontal, vertical, or diagonal row of four pieces, all of which have a common attribute (all short, all circular, etc.). A variant rule included in many editions gives a second way to win by placing 4 matching pieces in a 2x2 square....

 

try using the other 3 tetrominos as ways to win.

[CraigD]

:xmas_sheep: 0001, 0010, 0011, 0100, 0101, 1000, 1101, 1111, 0110, 1011, 1001, 0000, 0111, 1100, 1110, 1010 a draw?? :xmas_rudolph:

perfect players & one of the only two draws using all 16 pieces? (the other being 14, 13, 12, 11, 10, 7, 2, 16, 9, 4, 6, 15, 8, 3, 1, 5)

You’ve guessed along the right lines – more, I suspect, by knowing how math puzzle fans like us behave when given a simple board game as by analyzing the sequence :)

 

The sequence {1, 2, 3, 4, 5, 8, 13, 15, 6, 11, 9, 16, 7, 12, 14, 10} is the canonic order in which the places, counting left to right and top to bottom, are placed to check for wins as quickly as possible. I was doing this to answer the question “is a draw possible”.

 

The answer is “yes”, for both the standard (I tetromino only) and published variant (I or square tetrominos) rules. The first I-only non-win, counting systematically, is

1357 1358 1367 2457

1368 1458 2458 2467

1457 1468 2358 2368

2357 2367 2468 1467

1/2060. 0,0,0,9,0,0,3,0,2,0,0,0,1,2,1,0

It occurs on the 2060 try. For the I or square rules, is

1357 1358 1367 2457

1368 2458 2367 2468

1457 1468 1458 2368

2357 2467 2358 1467

1/44. 0,0,0,9,0,0,3,6,0,0,0,3,1,0,1,0

It occurs on the 44 try. Both are pretty common – from counting a million or so try, about 1 every 22 for the I only rules, 1 every 32 for the I or box rules.

 

My notation, by the way (nothing brilliant, just the first one to come to mind), is 1=black, 2=white, 3=square, 4=round, 5=short, 6=tall, 7=hollow, 8=solid, though it doesn’t matter if you switch the attributes around, as long as you preserve their exclusivity rules (1 and 2 can’t be in the same piece, etc).

 

i have some variants for you to try craig.

 

try using the other 3 tetrominos as ways to win.

Roger that. :thumbs_up

 

I’m out of play time ‘til this evening, but will check all the tetromino variants then.

 

Tonight, assuming my professional and Xmass social scene is fairly calm and bright, I expect to actually play the game as its maker intended, with another human. ;)

[Turtle]

You’ve guessed along the right lines – more, I suspect, by knowing how math puzzle fans like us behave when given a simple board game as by analyzing the sequence :)

 

The sequence {1, 2, 3, 4, 5, 8, 13, 15, 6, 11, 9, 16, 7, 12, 14, 10} is the canonic order in which the places, counting left to right and top to bottom, are placed to check for wins as quickly as possible. I was doing this to answer the question “is a draw possible”.

 

mmmm... i don't quite grok yet. did you mean "pieces", not "places"?

 

...My notation, by the way (nothing brilliant, just the first one to come to mind), is 1=black, 2=white, 3=square, 4=round, 5=short, 6=tall, 7=hollow, 8=solid, though it doesn’t matter if you switch the attributes around, as long as you preserve their exclusivity rules (1 and 2 can’t be in the same piece, etc).

 

 

Roger that. :thumbs_up

 

I’m out of play time ‘til this evening, but will check all the tetromino variants then.

 

Tonight, assuming my professional and Xmass social scene is fairly calm and bright, I expect to actually play the game as its maker intended, with another human. ;)

 

i got my binary notation from the link on strategy at the bottom of your wiki page link.

 

http://web.archive.org/web/20041012023358/http://ssel.vub.ac.be/Members/LucGoossens/quarto/quartotext.htm

...Symmetries

Naively there are 16 board configurations with zero pieces on the board, corresponding with the sixteen possible pieces your opponent could have chosen for you to put. In reality all of them are identical. The easiest way of seeing this is, is by mapping the pieces to bit strings of length four. The pieces are then

 

0000, 0001, 0010, 0011, …, 1111

 

The first bit represents the color, the second the size, the third roundness, the fourth solidness. Black is zero, white is one, tall is zero, small is one, and so on.

 

Given an arbitrary piece xxxx we can always reduce it to 0000 by toggling the appropriate bits. Toggling bits preserves quarto’s. ...

 

from their "so on" i got this key then for the bit strings:

 

first bit, color: black=0 white=1

second bit, size: tall=0 short=1

third bit, roundness: round=0 square=1

fourth bit, solidness: solid=0 hollow=1

 

 

i did some counting of possible arrangements of tetrominoes on a 4x4 board and came up with this preliminary count. (attached drawing not the complete list.) seems like there are many names for each tetromino; i picked one each. a "horizontal" placement has the long axis horizontal & a "vertical" placement, the long axis vertical.

 

line: 4 vert + 4 horiz + 2 diagonal = 10

square: 9

L & J: 12 vert + 12 horiz = 24 if counting diagonal placements, add another 16 i think. ?

S & Z: 12 vert + 12 horiz = 24 diagonal 16?

T :12 vert + 12 horiz = 24 diagonal +16?

 

that's all i got.

 

ps i have not actually played the game.

[Turtle]

You’ve guessed along the right lines – more, I suspect, by knowing how math puzzle fans like us behave when given a simple board game as by analyzing the sequence :)

 

The sequence {1, 2, 3, 4, 5, 8, 13, 15, 6, 11, 9, 16, 7, 12, 14, 10} ...

 

I’m out of play time ‘til this evening, but will check all the tetromino variants then.

 

Tonight, assuming my professional and Xmass social scene is fairly calm and bright, I expect to actually play the game as its maker intended, with another human. ;)

 

:D seems all i have today is play time. can't get your game outa my mind. :lol: anyway, i thought it might not be obvious now what i did. first, i presumed - :naughty: - that you knew the binary coding, and then i put 2 & 10 together to further presume that you had taken the binary code for each piece & converted it to decimal. on that presumption i presumed that your sequence is the order of pieces played in a game that plays all pieces. of course that says nothing about where on the board each is played or if such a game is possible. i would need an expert for that. :hihi:

 

so then, having read something at the strategy page on how under some circumstances you can swap all the bits, i immediately (:hihi:)took it out of context & then i took your sequence, converted the elements to binary, swapped all the bits in the resulting binary sequence & then converted that new sequence back to decimal and posted it. :eek: again, i don't know if that is a valid operation.

 

 

so here's my notes on all that.

first bit color: black=0 white=1

second bit size: tall=0 short=1

third bit roundness: round=0 square=1

fourth bit solidness: solid=0 hollow=1

 

1, 2, 3, 4, 5, 8, 13, 15, 6, 11, 9, 16, 7, 12, 14, 10

 

1 0001 black/tall/round/hollow

2 0010 black/tall/square/solid

3 0011 black/tall/square/hollow

4 0100 black/short/round/solid

5 0101 black/short/round/hollow

8 1000 white/tall/round/solid

13 1101 white/short/round/hollow

15 1111 white/short/square/hollow

6 0110 black/short/square/solid

11 1011 white/tall/square/hollow

9 1001 white/tall/round/hollow

16 0000 black/tall/round/solid

7 0111 black/short/square/hollow

12 1100 white/short/round/solid

14 1110 white/short/square/solid

10 1010 white/tall/square/solid

 

0001, 0010, 0011, 0100, 0101, 1000, 1101, 1111, 0110, 1011, 1001, 0000, 0111, 1100, 1110, 1010

flip:

1110, 1101, 1100, 1011, 1010, 0111, 0010, 0000, 1001, 0100, 0110, 1111, 1000, 0011, 0001, 0101

 

1110 14

1101 13

1100 12

1011 11

1010 10

0111 7

0010 2

0000 16

1001 9

0100 4

0110 6

1111 15

1000 8

0011 3

0001 1

0101 5

 

14, 13, 12, 11, 10, 7, 2, 16, 9, 4, 6, 15, 8, 3, 1, 5

 

please check me for errors!?

 

thinking over & using the bit codes, i got onto comparing qualities. as in, every piece has at least 1 feature in common with 14 other pieces and every piece shares nothing in common with just one other piece. i haven't worked it out yet for 2 & 3 features in common. :lol: anyways, not even sure this comparison is any aid to playing the game . :shrug:

 

that's it...for now. . . . .

 

Edit: every piece shares 2 qualities with 3 other pieces & every piece shares 3 qualities with only 1 other piece. :smart:

[CraigD]

The sequence {1, 2, 3, 4, 5, 8, 13, 15, 6, 11, 9, 16, 7, 12, 14, 10} is the canonic order in which the places, counting left to right and top to bottom, are placed to check for wins as quickly as possible. I was doing this to answer the question “is a draw possible”.

mmmm... i don't quite grok yet. did you mean "pieces", not "places"?

I mean, in my tortured prose, “… the order of the places in which pieces are placed to check for wins as quickly as possible”. That is, place the first 10 pieces like so:

1 2 3 4

5 8 _ _

6 _ 9 _

7 _ _ 10

 

PS:

I added all the tetrominos, and checking the first million boards, found no non-wins, leading me to suspect that there are none. Will play around some more to see if where the boundry between non-wins possible and impossible lies. :)

 

If you consider the standard rules to be for the I tetromino only, then when adding other tetrominos to the victory rule, diagonals should be accounted for, too, eg not just

1 1 1 0

0 0 1 0

0 0 0 0

0 0 0 0

but

1 0 0 0

0 1 0 0

0 0 1 0

0 1 0 0

etc.

[Turtle]

I mean, in my tortured prose, “… the order of the places in which pieces are placed to check for wins as quickly as possible”. That is, place the first 10 pieces like so:

1 2 3 4

5 8 _ _

6 _ 9 _

7 _ _ 10

 

:confused: still not clicking for me, but no worries. this all seems rather similar to my box printing problem and you remember how many weeks it took me to get your explanation right in my head on that one.

 

PS:

I added all the tetrominos, and checking the first million boards, found no non-wins, leading me to suspect that there are none. Will play around some more to see if where the boundry between non-wins possible and impossible lies. :)

 

If you consider the standard rules to be for the I tetromino only, then when adding other tetrominos to the victory rule, diagonals should be accounted for, too, eg not just

1 1 1 0

0 0 1 0

0 0 0 0

0 0 0 0

but

1 0 0 0

0 1 0 0

0 0 1 0

0 1 0 0

etc.

 

i suggest then that when using tetrominos other than line & box, that you not use line (or box).

 

roger diagonals. :thumbs_up i didn't include them in my board drawings, but mentioned them in the text.

...L & J: 12 vert + 12 horiz = 24 if counting diagonal placements, add another 16 i think. ? ...

 

this would give 40 possible winning placements for L&J, T, and S&Z. as you hint, this many positions may force wins, so perhaps some finer divisions such as declaring wins only for horizontal & vertical L's - but not their mirror J's - which would leave only 12 winning placements. another option then might be playing wins for only diagonal L's which would give 8 winning placements possible.

 

i played a few online games; ich bin nicht so gut. :doh: i lost them all, though i did force one game to a full board & got a little satisfaction from that. here's the link to the game; i found it at that original wiki page you gave. thnx for the mind stretching holiday fun!

 

Quarto Online

[Turtle]

hey craig! (and other interested readers. ;) ) i think you're gonna like this! :hihi: not quite quarto, but polyominos up the game board. credit where credit is due; pam brought it to my attention. check it:

 

Ancient puzzle gets new lease of 'geomagical' life

An ancient mathematical puzzle that has fascinated mathematicians for centuries has found a new lease of life.

 

The magic square is the basis for Sudoku, pops up in Chinese legend and provides a playful way to introduce children to arithmetic. But all this time it has been concealing a more complex geometrical form, says recreational mathematician Lee Sallows.

 

He has dubbed the new kind of structure the "geomagic square", and recently released dozens of examples online.

 

...

Tetris bricksIn his geomagic square, each digit of the grid is replaced by a Tetris-like shape called a polyomino, which is made up of different numbers of identical squares. Crucially, there must be a way to combine the polyominos in each row, column and diagonal to build a single master shape (see picture).

 

The bricks can be in two, three, or, in theory, even more dimensions – though visually representing a 4D geomagic square would be challenging. ...

 

[phillip1882]

let's say you have three properties per attribute and 6 attributes: (blue, yellow, red; small, medium, large; triangle, circle, square; border, double border, filled; 1 shape, 2 shapes, 3 shapes; stripes, checkerboard, solid)

how big should the board be? what would be the optimal shape?

(fewest draws, but hard for either player to win consistently)