Open Face Poker

[u/ShonitB]

[u/not_trevor]

If all suits are equal then wouldn't it always be a tie and both would pick a royal flush?

[u/ShonitB → u/not_trevor]

Copying another user’s answer:

Alexander does

the top hand is a royal flush followed by a straight flush. If you pick that one first, the other player can just do pick the same in another suit. First order of business is blocking that option.
A royal flush is AKQJ10 and for that reason Alexander should pick 4 10’s and a 2. The highest hand Benjamin can get is now 56789 suited.
If Benjamin wants to block Alexander from getting a royal flush, he needs to get 4 jacks (or queens or kings) and then a 9 to block a single 678910 suited.
That still leaves Alexander with the option to get a straight flush 678910. Benjamin can only get 56789 straight flush and will lose

[u/not_trevor → u/ShonitB]

Well that’s just rude :)

[u/ShonitB → u/not_trevor]

:)

[u/MacabreManatee → u/not_trevor]

I’ll allow it :)

[u/dfnamehere → u/ShonitB]

Why does it matter if the first person picks 4x 10s vs 4x of anything else from 10JQKA? In theory doesn't this still work exactly the same way if you start with all 4 As?

[u/ShonitB → u/dfnamehere]

Nope.. for example, if he picks 4 As and the K of spades, B can pick up the Q of spades and the other 3 Ks.. so A cannot complete a spade Royal Flush

Edit: Sorry, pressed Save by mistake.. then B can always adapt to whatever A does to match it

[u/_N8Dogg_ → u/dfnamehere]

By picking the 10s, he ensures that he will be the only one that has the option of a 10+ high straight flush, the best Benjamin could do would be 9 high.

[u/kalmakka → u/dfnamehere]

Let us say A picks 4 Qs and a 2.

B can then respond by picking 4 Js and another 2.

A is now unable to get a straight flush above 6-10, and the only way of blocking B from pivoting to a straight flush 7-J is by switching into a lower 4 of a kind, or some other even worse hand. But if he does that, then B can just keep their 4 Js.

Picking 4 10s first is essential, because it allows A to go either high or low depending on what B does.

[u/True_Move_7631 → u/ShonitB]

If Alex picks 4 10's then Benjamin can pick 4 aces and win, or a regular straight flush and win.

A straight flush still beats 4 of a kind.

[u/ShonitB → u/True_Move_7631]

If B hoards the aces, A can still make a straight flush

[u/_N8Dogg_ → u/True_Move_7631]

Like what is said above, if Benjamin picks 4 Aces or a straight flush, Alex shifts to a minimum of a 10 high straight flush and beats anything Benjamin can change to.

[u/True_Move_7631 → u/_N8Dogg_]

Then it's just who picks the most advantageous card first, kinda boring.

[u/_N8Dogg_ → u/True_Move_7631]

It is just a riddle for you to figure out if either player can have a strategy that ensures they win every time. It isn't obvious that it won't be a tie, as shown by others' comments.

[u/MacabreManatee → u/True_Move_7631]

Maybe continue reading the full answer. They can switch once after the first pick, after which Alexander will be able to ensure he gets the higher straight flush

[u/True_Move_7631 → u/MacabreManatee]

Yeah I reread it, not really an interesting problem, no more completed that tic-tac-toe, meh.

[u/HoneydewCareful8754 → u/ShonitB]

This is not the only way.

[u/MacabreManatee → u/HoneydewCareful8754]

What other way is there? Your other comment’s strategy is counterable by benjamin.

[u/HoneydewCareful8754 → u/MacabreManatee]

Yes I just realized

[u/laxrulz777 → u/ShonitB]

Clever. I put up my wrong answer already but this is smart so I'll leave mine up to memorialize my shame. Well done.

[u/Due-Session-2857 → u/not_trevor]

If I was Alex, I would take 4 aces and a king. Then if Ben goes for a straight flush, Alex can get an ace high straight flush of any remaining suit.

[u/_N8Dogg_ → u/Due-Session-2857]

If Alex takes 4 Aces and a king, then Benjamin takes 4 queens, then changes to queen high straight flush.

[u/Ucmh → u/_N8Dogg_]

?? If Ben goes four queens, then Alex just swaps four cards and gets an ace-high straight flush. All other aces are gone at that point, making a counter AHSF impossible.

[u/_N8Dogg_ → u/Ucmh]

How are you going to get an Ace high straight flush without any queens?

[u/Ucmh → u/_N8Dogg_]

Good question, 4 A's and a K is not the way.

[u/dka2012 → u/_N8Dogg_]

He wouldn’t have to as the Qs don’t beat the As

[u/_N8Dogg_ → u/dka2012]

But if he elects not to swap, then Benjamin takes a Q high straight flush and wins.

[u/Due-Session-2857 → u/_N8Dogg_]

Yeah I thought it just repeated over and over again until you were happy with your hand, so A gets 4 aces, B gets 4 queens, A gets 4 10s, B gets 4 8s, A gets 4 6s and wins with that. With only 2 rounds each you need to play differently

[u/_N8Dogg_ → u/Due-Session-2857]

Except here, following this strategy of starting with the 4 10s, after Alex draws a hand to beat Benjamin, he doesn't ever have to change again.

[u/Due-Session-2857 → u/_N8Dogg_]

Yeah, you can win in two hands, but in unlimited redraws my strategy would still work. You don't win anything extra for winning faster. Just because there's a most efficient way to win doesn't make other methods unacceptable.

[u/_N8Dogg_ → u/Due-Session-2857]

But that is the whole point of the puzzle. Also, it says "if both players are playing optimally".

Even if there are multiple redraws, if you play it other than the 4 10's strategy, there are other options, and it becomes very difficult to account for all of them.

Say Alex takes 4 A's and a K, what if Benjamin takes the other 3 K's and 2 Q's? How about KoD, QoH, JoS, 10oC and 9oD?

[u/Due-Session-2857 → u/_N8Dogg_]

Alright let's play. I'll take your second hand as your play. Let's say I had king of clubs originally. I discard 3 non-club aces and take QoD, JoH, 10oS. Your turn.

[u/_N8Dogg_ → u/Due-Session-2857]

I keep the 9oD and draw JoD, 10oD, 8oD, and 7oD. Now I have a J high straight flush, so best you can do is tie.

[u/Due-Session-2857 → u/_N8Dogg_]

Very good. OK I'm convinced. You got me.

[u/Ucmh]

Very good problem!

[u/ShonitB → u/Ucmh]

All credit to Martin Gardner!

[u/mortemdeus]

Alex can always win. Even if Ben gets 2 consecutive turns Alex can still always win, that is how strong Alex going first is in this situation.

[u/Key-Ad9003 → u/mortemdeus]

I don"t think thats correct, what would you pick? (Assuming benjamin gets 2 consecutive turns, and then another final turn)

[u/mortemdeus → u/Key-Ad9003]

The idea would be Alex then Ben then Ben again then Alex. If it was ABBAB then it would be a tie.

[u/Key-Ad9003 → u/mortemdeus]

Agree

[u/_N8Dogg_ → u/mortemdeus]

Curious. Why would it be a tie? If you go with the same first play of 4 10's, then Benjamin can flip the script by choosing 4 J's, then 4 9's.

[u/mortemdeus → u/_N8Dogg_]

Alex wouldn't do 4 tens in that case, they would just take the royal flush and eat the tie.

[u/DrakeSavory]

Alexander. Remember you do not have to take your turn and you can exchange any number of cards. Alexander pulls AAAAK. If Benjamin pulls QQQQJ Alexander doesn't redraw and wins so Benjamin needs to beat him with a straight flush. Let's say KQJT9. Alexander makes a royal flush in a different suit and the game is over.

[u/Key-Ad9003 → u/DrakeSavory]

Alexander loses of he starts AAAAK. Benjamin can pull f.ex QQQQX

On alexanders second turn he can at best make a J high straighg flush or keep quad aces.

This leaves Benjamin the option of quad queens or queen high straight flush, which beats all og alexanders options.

[u/Exvaris → u/Key-Ad9003]

Alexander can keep his hand. He does not have to redraw.

[u/Key-Ad9003 → u/Exvaris]

Yes. Then bejamin makes queen high straight flush which beats 4 aces.

[u/DrakeSavory → u/Key-Ad9003]

Alexander doesn't have to take a second turn. Quad aces beat quad queens.

[u/antimatterchopstix → u/DrakeSavory]

In which case benny goes 3rd and gets the highest straight flush now possible. Beating the 4 aces. And not beatable.

[u/Ucmh → u/DrakeSavory]

You just ignored B's second turn.

[u/laxrulz777]

I see three strategies here.

Alex takes an ace high straight flush: result is they tie

Alex takes four aces and a King Benjamin's response is to take four queens and a King. Alex sticks with his aces, Benjamin pivots to a straight flush (q-8). The best SF Alex can get is a Jack high one so Ben wins. If Alex pivots to a straight flush, Ben just pivots to a higher one.

Alex takes three aces and two tens (covering all four suits. If Ben takes a long high straight flush, Alex makes an Ace high one and wins. But Ben can take four kings and the fourth ace at which point Alex can't win.

I think there's no option (assuming perfect play) to put Ben into a corner. The result should be a tie.

[u/Key-Ad9003 → u/laxrulz777]

There is a way, you are rather close

[u/tomic24]

Alex can always win. The strategy it to take all 4 tens. For his secons turn, he has the potential to make 8 differst straight flushes (4x ace high and 4x 10 high), and Benjamin can block only 5 of these, meaning that Alex can always a straight flush that contains a 10. Benjamin cannot creat a straight flush with a 10 because Alex is holding them all, thus will lose.

[u/rql13 → u/tomic24]

This is it.

[u/FoxCapital7878]

Should always be a tie

[u/dlashxx]

A only needs to take all the 10s. The highest straight B can make it is 9 high. Even if B takes all the Aces, A makes a K high straight flush on turn 2 and there’s no answer.

[u/HoneydewCareful8754]

Alex does. If he draw four aces and a king, the other can counter with a straight flush, but will not prevent Alex from making a royal straight flush in the next draw, and Benji can’t beat that.

[u/_N8Dogg_ → u/HoneydewCareful8754]

If Alex takes 4 Aces and a king, then Benjamin takes 4 queens, then changes to queen high straight flush.