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 [quote="L'lanmal"][b]WRAP UP[/b] This has been pretty well discussed. The original game is isomorphic to tic-tac-toe as Thok described. For most people, this is easier to visualize than it is to think about the list of words. You can force a draw by taking the center square, or "lamb": [code]bet each gem still lamb clog mutt fraud bug (cup)[/code] The game described in the referenced article was also Tic-Tac-Toe, but he used different words including the titular "hot". Those words did not support the addition of "cup" or an equivalent, so I changed them even for this first half of the puzzle. The revised game is an easy win for the 2nd player by utilizing the each/clog/cup diagonal. "Lamb" and "Clog" are both winning words, amongst others. Trojan Horse and lostdummy both point out that the game is still winnable under slightly stricter constraints. Trojan Horse suggests you eliminate the extra diagonal by replacing "cup" with "pun" or "each" with "axe". The game become winnable (but with some difficulty) by starting with "clog", or symmetrically with "still". I had not considered or noticed this. lostdummy additionally suggests keeping the extra diagonal but detach the word from the bottom row, through either adding "cry", or placing additional conditions on a win on the bottom row. Then the game is still winnable with "clog" (or perhaps "each"). If you do both constraints at the same time, then you just have a game with at least the constraints of tic-tac-toe, and it is a draw with perfect play. So, we just wasted a bunch of time playing Tic-Tac-Toe![/quote]
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L'lanmal
Posted: Thu Oct 25, 2012 10:17 pm    Post subject: 1

WRAP UP

This has been pretty well discussed.

The original game is isomorphic to tic-tac-toe as Thok described. For most people, this is easier to visualize than it is to think about the list of words. You can force a draw by taking the center square, or "lamb":
Code:
bet    each   gem
still  lamb   clog
mutt   fraud  bug  (cup)

The game described in the referenced article was also Tic-Tac-Toe, but he used different words including the titular "hot". Those words did not support the addition of "cup" or an equivalent, so I changed them even for this first half of the puzzle.

The revised game is an easy win for the 2nd player by utilizing the each/clog/cup diagonal. "Lamb" and "Clog" are both winning words, amongst others.

Trojan Horse and lostdummy both point out that the game is still winnable under slightly stricter constraints. Trojan Horse suggests you eliminate the extra diagonal by replacing "cup" with "pun" or "each" with "axe". The game become winnable (but with some difficulty) by starting with "clog", or symmetrically with "still". I had not considered or noticed this. lostdummy additionally suggests keeping the extra diagonal but detach the word from the bottom row, through either adding "cry", or placing additional conditions on a win on the bottom row. Then the game is still winnable with "clog" (or perhaps "each"). If you do both constraints at the same time, then you just have a game with at least the constraints of tic-tac-toe, and it is a draw with perfect play.

So, we just wasted a bunch of time playing Tic-Tac-Toe!
lostdummy
Posted: Mon Oct 15, 2012 11:35 am    Post subject: 0

That makes more sense ;p

BTW, if added word is 'cup' (U&C), then limiting to "need 4 U" will not help, and 1st player will still win.

Same thing if added word is 'cry' (just C), first player still win.

But if added word is 'pun' (just U), then limiting first player to "need 4 U" will result in draw, which means second player win if he takes bet.

This definitely makes "first player need 4U to win" better limiting condition than previous "first player can't start with U", since even without starting with U first player was able to win in all above scenarios.
L'lanmal
Posted: Fri Oct 12, 2012 9:12 pm    Post subject: -1

As in:

Before you can object to Skinny so cavilierly wagering with your winnings, the gambler counters. "You're on! But I'm no fool, there are now four words with u's in them on the list. Three u's shouldn't be a win, you should need all four."

(To be clear, three of any of the rest of the letters is still a win.)

And yes, clog -> gem -> bet is a pretty clear win for the first player with two fork threats. I missed that.

Not typed from my phone.
referee
Posted: Fri Oct 12, 2012 9:00 pm    Post subject: -2

he needs four words with u... or three words with any other letter.
lostdummy
Posted: Fri Oct 12, 2012 8:18 pm    Post subject: -3

L'lanmal wrote:
For the pun variant, clog can be countered by gem, i believe. I tried to construct such that the extra diagonal was necessary.

I don't think 'gem' can help.

Since this is hard to actually prove, I made program that either find best solution/move, or can play against me.

When finding best solution, he confirms that 1st player can win, and suggest one way to do that is with 'clog'. When playing against him, if I start 1st with 'clog', he can not beat me (confirming 1st player can win).

When it start 1st and select 'clog', if I select 'gem', it select 'bet'. And after that its more or less visible that I lose.

Example responses for my 2nd move, after: clog -> gem -> bet:
- lamb -> mutt -> still -> bug -> he wins
- mut -> lamb-> he wins
- still -> bug-> lamb-> mut -> he wins
- etc... ;p

L'lanmal wrote:

Here's a version I considered... skinny can pick any word first, but needs all 4 words with u's to win.

Maybe I didnt understand what you meant there, but how can he ever get all 4 words with U, if whoever plays second can select one word with U, and immediately prevent Skinny from winning ?
L'lanmal
Posted: Fri Oct 12, 2012 3:07 pm    Post subject: -4

For the pun variant, clog can be countered by gem, i believe. I tried to construct such that the extra diagonal was necessary.

Here's a version I considered... skinny can pick any word first, but needs all 4 words with u's to win. (This preserves the tic-tac-toe view more perfectly, but does a worse job of drawing attention away from the new diagonal.)

sent from my non-i phone
lostdummy
Posted: Fri Oct 12, 2012 12:20 pm    Post subject: -5

Trojan Horse wrote:
Wow. This game is pretty interesting with "pun" instead of "cup". If I'm not mistaken Skinny wins after all

Yes, first player wins with 'pun' too, and actually same start can be used for both cup/pun: 'clog'

Thok's tic-tac-toe analogy is quite good, although it may mislead with 10th word since regular tic-tac-toe would require continuous 3 cells, so mutt-fraud-X-cup would not look like win in tic-tac-toe, even if they are win here ;p

Related to different options for 10th word, in original 9 words there are 8 lines/groups of 3 words : 3x (ELUTAGBM) . In addition , there is 2xC and 1x (SOF).

Therefore possible additions as 10th word are:
1) add letter not present in 3x or 2x ('xyz')
2) add new 3 word/letter (C only had 2, so for example 'cry')
3) add 4th word/letter ('pun' is 4th U)
4) any combination of above ('cup' = 2+3)

Basically, #1 does not change from original, and as any tic-tac-toe, (with original 9 words and 8 groups of x3 ) that is 'draw', or in other words second player win.

But even #2 (making 9th x3 group, with letter C) will ensure that 1st player win.

Also, #3 on its own (expanding one 3x group to 4x, in this case U) will ensure that 1st player win.

Therefore, any combination of those #2 and #3 (as 'cup') will even more ensure that 1st player win ;p
DejMar
Posted: Thu Oct 11, 2012 2:46 pm    Post subject: -6

Thok's comparative take on the words shows us that the old member might be a WOPR. And the best thing would be not to play the game with the newer restrictions.
Trojan Horse
Posted: Thu Oct 11, 2012 5:00 am    Post subject: -7

Wow. This game is pretty interesting with "pun" instead of "cup". If I'm not mistaken... [Skinny wins after all. It is a first-player win, but the full proof is pretty complicated.]
Trojan Horse
Posted: Thu Oct 11, 2012 4:03 am    Post subject: -8

Here I was thinking that the second version was a pretty complicated game. I was running through possibility after possibility, and I was thinking that Skinny probably had put himself into a bad bet.

Then I read Thok's comment, and I realized that I missed the clog-each-cup combination.

I must be blind.

Okay, so let's say there was no such combination. Let's say the 10th word was "pun" instead of "cup". Everything else is still the same; the first player can't start with a word with a u in it, and the second player wins if all the words are used up. Then who wins? I think it's a second player win, but I haven't gone through all the possibilities yet.
Thok
Posted: Wed Oct 10, 2012 10:40 pm    Post subject: -9

Here's the board as tic tac toe

bet each gem
still lamb clog
mutt fraud bug

cup can essentially be placed next to bug (so each/clog/cup form a diagonal) and lamb would seem to be a winning word in both cases.
Thok
Posted: Wed Oct 10, 2012 9:30 pm    Post subject: -10

I presume the original game is essentially Tic-Tac-Toe
L'lanmal
Posted: Wed Oct 10, 2012 9:20 pm    Post subject: -11

You arrive at The Grey Labyrinth sports bar to find it packed. As the sport of the day is chess, there is plenty of time for discussion and wagering between moves.

An old member trolls the crowd. "A wager," he says. "We alternate taking words from this list. The first person to collect three which share a common letter wins. I'll give you the win if we run out of words, but in exchange I get to pick first. Any takers?"

Here is his list:
bet
bug
clog
each
fraud
gem
lamb
mutt
still

"Sure," you immediately reply. After he grabs "bet", you make your selection.

A short time later, you console the gambler just before you collect your winnings. "There's nothing you could have done better. There simply is no way to force a win."

Skinny Dakota speaks up. "Care to go double-or-nothing? Add "cup" to the list, and I'll go first."

Before you can object to Skinny so cavilierly wagering with your winnings, the gambler counters. "You're on! But I'm no fool, there are now four words with u's in them on the list. You can go first, but on your first move you can only pick from the remaining six words."

"Done!" replies Skinny.

What word did you choose in your game to win? And are your winnings safe, or has Skinny finally bitten off more than he could chew?

Inspired by "The Game is Hot." Leo Moser, Recreational Mathematics Magazine, Vol. 1, June, 1961, pages 23-24 as described in Mathematical Carnival. Martin Gardner, Mathematical Association of America, 1989, pages 208-211.