A Wager with Words

[L'lanmal]

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.

[Thok]

I presume the original game is essentially Tic-Tac-Toe

[Thok]

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.

[Trojan Horse]

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.

[Trojan Horse]

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.]

[DejMar]

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.

[lostdummy]

[L'lanmal]

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]

[referee]

he needs four words with u... or three words with any other letter.
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[L'lanmal]

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.

[lostdummy]

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]

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":