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Listed below are 20 poker hands.

For each of the hands below, assume

it was dealt to you from a full shuffled

poker deck of 52 cards. You may stay

with the hand you are given or discard

any number of its cards. Any discarded

cards will be replaced by a random draw

from the remaining 47 cards. Your goal

is to improve the value of your hand as

much as possible. List which cards, if

any, you should discard from each of the

starting hands below:


1: 10♦ A♠ 10♥ 7♥ 8♣
2: K♠ 8♣ 9♣ 10♠ J♠
3: 9♣ 8♦ 8♥ 4♠ J♣
4: 6♥ K♦ A♦ K♥ 4♦
5: Q♠ 8♣ A♥ 6♦ 7♠
6: 3♣ K♠ 5♦ 7♦ 3♥
7: K♠ 7♥ 2♠ 5♠ J♣
8: J♦ 10♣ 10♥ 8♣ 9♠
9: 3♦ 6♠ 5♠ 4♠ 5♦
10: 5♠ 7♠ 5♦ A♠ Q♥
11: 4♠ 5♠ 6♠ 7♠ 7♣
12: 2♠ 3♣ 4♣ 5♣ 6♣
13: 3♠ 2♣ 3♣ 4♣ 5♣
14: 2♠ 2♣ 3♣ 4♣ 5♣
15: J♠ K♠ 2♥ 8♦ Q♠
16: 2♠ 3♠ 4♠ 5♠ 6♣
17: 3♦ 4♦ 6♦ K♦ 5♠
18: 2♦ 8♥ 5♥ 9♣ 6♦
19: 8♠ Q♦ A♠ K♠ 6♠
20: 3♣ 4♦ J♥ 2♣ 10♥
[/code]

Usual poker hand rankings apply.

So, for example, "round the corner"

straights are not allowed, although

an ace may be either high or low card

in a straight.

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I would discard


  1:  7♥  8♣ 

  2:  K♠

  3:  9♣  4♠  J♣ 

  4:  6♥  4♦ 

  5:  Q♠  8♣  6♦  7♠ 

  6:  K♠  5♦  7♦

  7:  7♥  J♣ 

  8:  J♦  8♣  9♠ 

  9:  5♦ 

 10:  7♠  A♠  Q♥ 

 11:  7♣ 

 12:  2♠ 

 13:  3♠ 

 14:  2♠

 15:  2♥  8♦

 16:  6♣ 

 17:  5♠ 

 18:  2♦  8♥  5♥  9♣  6♦ 

 19:  Q♦

 20:  3♣  4♦  J♥  2♣ 10♥

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I would discard


1: 7♥ 8♣
2: K♠
3: 9♣ 4♠ J♣
4: 6♥ 4♦
5: Q♠ 8♣ 6♦ 7♠
6: K♠ 5♦ 7♦
7: 7♥ J♣
8: J♦ 8♣ 9♠
9: 5♦
10: 7♠ A♠ Q♥
11: 7♣
12: 2♠
13: 3♠
14: 2♠
15: 2♥ 8♦
16: 6♣
17: 5♠
18: 2♦ 8♥ 5♥ 9♣ 6♦
19: Q♦
20: 3♣ 4♦ J♥ 2♣ 10♥

8 correct.

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1: 7♥ 8♣

2: K♠

3: 9♣ 4♠ J♣

4: 6♥ 4♦

5: Q♠ A♥

6: 5♦ 7♦

7: 7♥ J♣

8: 10♣

9: 3♦ 5♦

10: 7♠ Q♥

11: 7♣

12: None

13: 3♠

14: 2♠

15: 2♥ 8♦

16: 6♣

17: 5♠

18: 2♦

19: Q♦

20: J♥ 10♥

Edited by Balding Eagle
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1: 7♥ 8♣

2: K♠

3: 9♣ 4♠ J♣

4: 6♥ A♦ 4♦

5: 8♣ 6♦ 7♠

6: K♠ 5♦ 7♦

7: K♠ 7♥ 2♠ 5♠ J♣

8: 10♣

9: 5♦

10:7♠ A♠ Q♥

11:7♣

12:2♠

13:3♠

14:2♠

15:J♠ K♠ 2♥ 8♦ Q♠

16:None

17:5♠

18:2♦ 8♥ 5♥ 9♣ 6♦

19:Q♦

20:3♣ 4♦ J♥ 2♣ 10♥

Edited by Balding Eagle
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1: 7♥ 8♣

2: K♠

3: 9♣ 4♠ J♣

4: 6♥ A♦ 4♦

5: 8♣ 6♦ 7♠

6: K♠ 5♦ 7♦

7: K♠ 7♥ 2♠ 5♠ J♣

8: 10♣

9: 5♦

10:7♠ A♠ Q♥

11:7♣

12:2♠

13:3♠

14:2♠

15:J♠ K♠ 2♥ 8♦ Q♠

16:None

17:5♠

18:2♦ 8♥ 5♥ 9♣ 6♦

19:Q♦

20:3♣ 4♦ J♥ 2♣ 10♥

7 correct.

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2: K♠ 8♣ 9♣ 10♠ J♠
3: 9♣ 4♠ J♣
4: 6♥ A♦ 4♦
5: Q♠ 8♣ 6♦ 7♠
6: K♠ 5♦ 7♦
7: K♠ 7♥ 2♠ 5♠ J♣
8: J♦ 8♣ 9♠
9: 3♦ 6♠ 4♠
10: 7♠ A♠ Q♥
11: 4♠ 5♠ 6♠
12: none
13: 3♠ 2♣ 3♣ 4♣ 5♣
14: 3♣ 4♣ 5♣
15: J♠ K♠ 2♥ 8♦ Q♠
16: none
17: 3♦ 4♦ 6♦ K♦ 5♠
18: 2♦ 8♥ 5♥ 9♣ 6♦
19: 8♠ Q♦ K♠ 6♠
20: 3♣ 4♦ J♥ 2♣ 10♥
1:  A♠ 7♥ 8♣ 

Edited by plainglazed
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2: K♠ 8♣ 9♣ 10♠ J♠
3: 9♣ 4♠ J♣
4: 6♥ A♦ 4♦
5: Q♠ 8♣ 6♦ 7♠
6: K♠ 5♦ 7♦
7: K♠ 7♥ 2♠ 5♠ J♣
8: J♦ 8♣ 9♠
9: 3♦ 6♠ 4♠
10: 7♠ A♠ Q♥
11: 4♠ 5♠ 6♠
12: none
13: 3♠ 2♣ 3♣ 4♣ 5♣
14: 3♣ 4♣ 5♣
15: J♠ K♠ 2♥ 8♦ Q♠
16: none
17: 3♦ 4♦ 6♦ K♦ 5♠
18: 2♦ 8♥ 5♥ 9♣ 6♦
19: 8♠ Q♦ K♠ 6♠
20: 3♣ 4♦ J♥ 2♣ 10♥
1:  A♠ 7♥ 8♣ 

8 correct.

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Listed below are 20 poker hands.

For each of the hands below, assume

it was dealt to you from a full shuffled

poker deck of 52 cards. You may stay

with the hand you are given or discard

any number of its cards. Any discarded

cards will be replaced by a random draw

from the remaining 47 cards. Your goal

is to improve the value of your hand as

much as possible. List which cards, if

any, you should discard from each of the

starting hands below:


  1: 10♦  A♠ 10♥  7♥  8♣

  2:  K♠  8♣  9♣ 10♠  J♠

  3:  9♣  8♦  8♥  4♠  J♣

  4:  6♥  K♦  A♦  K♥  4♦

  5:  Q♠  8♣  A♥  6♦  7♠

  6:  3♣  K♠  5♦  7♦  3♥

  7:  K♠  7♥  2♠  5♠  J♣

  8:  J♦ 10♣ 10♥  8♣  9♠

  9:  3♦  6♠  5♠  4♠  5♦

 10:  5♠  7♠  5♦  A♠  Q♥

 11:  4♠  5♠  6♠  7♠  7♣

 12:  2♠  3♣  4♣  5♣  6♣

 13:  3♠  2♣  3♣  4♣  5♣

 14:  2♠  2♣  3♣  4♣  5♣

 15:  J♠  K♠  2♥  8♦  Q♠

 16:  2♠  3♠  4♠  5♠  6♣

 17:  3♦  4♦  6♦  K♦  5♠

 18:  2♦  8♥  5♥  9♣  6♦

 19:  8♠  Q♦  A♠  K♠  6♠

 20:  3♣  4♦  J♥  2♣ 10♥

Usual poker hand rankings apply.

So, for example, "round the corner"

straights are not allowed, although

an ace may be either high or low card

in a straight.

Some clarification please. In the bolded passage, what do you mean by "improve the value of the hand as much as possible"?

Does it mean discard the hand so as to maximize the chance of drawing a better hand, and that all better hands count equally? E.g. if we have a pair, then an improvement to 2 pairs counts the same as an improvement to a royal flush?

Does it mean discard some cards so that we maximize the expected 'value' of the new hand for some payoff scale? If so, what is that payoff scale?

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Discard these:

1: 7♥ 8♣

2: K♠

3: 9♣ 4♠ J♣

4: 6♥ 4♦

5: Q♠ A♥

6: 5♦ 7♦

7: 7♥ J♣

8: 10♣

9: 5♦

10: 7♠ A♠ Q♥

11: 7♣

12: -

13: 3♠

14: 2♠

15: 2♥ 8♦

16: -

17: K♦

18: 2♦

19: Q♦

20: 3♣ 4♦ 2♣

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I agree with bushindo. Are we weighing risk vs reward? Otherwise you'd just...

keep one card in the royal flush range (or more than one if they're suited) and discard the rest hoping for a royal flush every time regardless of the changes of actually getting it.

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Some clarification please. In the bolded passage, what do you mean by "improve the value of the hand as much as possible"?

Does it mean discard the hand so as to maximize the chance of drawing a better hand, and that all better hands count equally? E.g. if we have a pair, then an improvement to 2 pairs counts the same as an improvement to a royal flush?

Does it mean discard some cards so that we maximize the expected 'value' of the new hand for some payoff scale? If so, what is that payoff scale?

What I mean is to make the expected value of the resulting hand as large as possible.

Since there are only 7462 different values for poker hands, we can give the worst

possible hand a value of 1,the best possible a value of 7462, and all the others

appropriate values in between. I hope that's clear.

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What I mean is to make the expected value of the resulting hand as large as possible.

Since there are only 7462 different values for poker hands, we can give the worst

possible hand a value of 1,the best possible a value of 7462, and all the others

appropriate values in between. I hope that's clear.

sorry superprismatic, but am still a little uncertain as to the goal. am thinking we're looking for the discards that result in most likely improving your hand in any way? otherwise, discarding none if the odds are less than 50% that you would impove?

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sorry superprismatic, but am still a little uncertain as to the goal. am thinking we're looking for the discards that result in most likely improving your hand in any way? otherwise, discarding none if the odds are less than 50% that you would impove?

With each possible way of discarding, you have some

expected value for the hand that result when you replace

these cards from the 47 cards remaining in the deck.

I wish to find the discard which gives the highest

expected value for the resulting hand. Perhaps this

will help:

How I determine the best draw

There are 2.6 million different poker hands but

there are only 7,462 different values for those

hands. I wrote a program that computes the value

of a hand -- using the values 1 to 7,462. That

isn't very hard to do. I could have just made

a big (2.6 million-long) lookup table, but I wanted

to be able to use it on machines with small

memories. So, I optimized the thing to do on the

order of a million hands a second on one core of

a 6-core AMD processor. I tested it a lot, but

I could have some small bugs.

For this problem, I exhaused on all possible ways

to discard (32) and repopulate from 47 cards

(about 1.7 million) for each of the hands. I

computed the average value of a hand resulting

from each of the discard repopulation possibilities.

The discard which gives the best average value wins.

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  1: 10♦  A♠ 10♥  7♥  8♣   -   discard 7♥  8♣

  2:  K♠  8♣  9♣ 10♠  J♠   -   K♠

  3:  9♣  8♦  8♥  4♠  J♣   -   9♣   4♠  J♣  

  4:  6♥  K♦  A♦  K♥  4♦  -   6♥  4♦ 

  5:  Q♠  8♣  A♥  6♦  7♠  -  8♣ 6♦  7♠ 

  6:  3♣  K♠  5♦  7♦  3♥  -  3♣  5♦  7♦  3♥   [my thinking is that it's much more likely to get a high pair by discarding 4 than to get 2 pair by keeping the 3s]

  7:  K♠  7♥  2♠  5♠  J♣   -   7♥  2♠  5♠  J♣   

  8:  J♦ 10♣ 10♥  8♣  9♠  -  10♥  [not sure if this is smart--expected value might be higher for keeping 10s]

  9:  3♦  6♠  5♠  4♠  5♦  -  5♦

 10:  5♠  7♠  5♦  A♠  Q♥ - 5♠  7♠  5♦   Q♥

 11:  4♠  5♠  6♠  7♠  7♣  -  7♣  [9/52 chance of getting another spade; 8/52 chance of getting 3 or 8]

 12:  2♠  3♣  4♣  5♣  6♣  -  2♠  [risky, but 2/52 chance of straight flush, 9/52 chance of flush makes it worth it, I think]

 13:  3♠  2♣  3♣  4♣  5♣  -  3♠ 

 14:  2♠  2♣  3♣  4♣  5♣  -  2♠  

 15:  J♠  K♠  2♥  8♦  Q♠  -  J♠  2♥  8♦  Q♠ 

 16:  2♠  3♠  4♠  5♠  6♣  -   6♣

 17:  3♦  4♦  6♦  K♦  5♠  -  5♠

 18:  2♦  8♥  5♥  9♣  6♦  -  All? 

 19:  8♠  Q♦  A♠  K♠  6♠  -  A♠  K♠

 20:  3♣  4♦  J♥  2♣ 10♥  -  J♥ 

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This seems fun.

Cards to discard


  1:  7♥  8♣

  2:  8♣  9♣ 10♠  J♠

  3:  9♣  4♠

  4:  6♥  4♦

  5:  Q♠  8♣  6♦  7♠

  6:  5♦  7♦

  7:  7♥  2♠  5♠  J♣

  8:  8♣  9♠

  9:  3♦  6♠  4♠  

 10:  7♠  Q♥

 11:  4♠  5♠  6♠

 12:  NONE

 13:  2♣  4♣  5♣

 14:  3♣  4♣  5♣

 15:  J♠  2♥  8♦  Q♠

 16:  NONE

 17:  3♦  4♦  6♦  5♠

 18:  2♦  8♥  5♥  9♣  6♦

 19:  8♠  Q♦  K♠  6♠

 20:  3♣  4♦  J♥  2♣ 10♥

Edited by bushindo
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1: 7♥ 8♣

2: K♠

3: 9♣ 4♠ J♣

4: 6♥ 4♦

5: Q♠ A♥

6: 5♦ 7♦

7: 7♥ J♣

8: 10♥

9: 5♦

10: 7♠ Q♥

11: 7♣

12: 2♠

13: 3♠

14: 2♠

15: 2♥ 8♦

16: 6♣

17: 5♠

18: 2♦

19: Q♦

20: J♥ 10♥

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1: 10♦ A♠ 10♥ 7♥ 8♣ - discard 7♥ 8♣
2: K♠ 8♣ 9♣ 10♠ J♠ - K♠
3: 9♣ 8♦ 8♥ 4♠ J♣ - 9♣ 4♠ J♣
4: 6♥ K♦ A♦ K♥ 4♦ - 6♥ 4♦
5: Q♠ 8♣ A♥ 6♦ 7♠ - 8♣ 6♦ 7♠
6: 3♣ K♠ 5♦ 7♦ 3♥ - 3♣ 5♦ 7♦ 3♥ [my thinking is that it's much more likely to get a high pair by discarding 4 than to get 2 pair by keeping the 3s]
7: K♠ 7♥ 2♠ 5♠ J♣ - 7♥ 2♠ 5♠ J♣
8: J♦ 10♣ 10♥ 8♣ 9♠ - 10♥ [not sure if this is smart--expected value might be higher for keeping 10s]
9: 3♦ 6♠ 5♠ 4♠ 5♦ - 5♦
10: 5♠ 7♠ 5♦ A♠ Q♥ - 5♠ 7♠ 5♦ Q♥
11: 4♠ 5♠ 6♠ 7♠ 7♣ - 7♣ [9/52 chance of getting another spade; 8/52 chance of getting 3 or 8]
12: 2♠ 3♣ 4♣ 5♣ 6♣ - 2♠ [risky, but 2/52 chance of straight flush, 9/52 chance of flush makes it worth it, I think]
13: 3♠ 2♣ 3♣ 4♣ 5♣ - 3♠
14: 2♠ 2♣ 3♣ 4♣ 5♣ - 2♠
15: J♠ K♠ 2♥ 8♦ Q♠ - J♠ 2♥ 8♦ Q♠
16: 2♠ 3♠ 4♠ 5♠ 6♣ - 6♣
17: 3♦ 4♦ 6♦ K♦ 5♠ - 5♠
18: 2♦ 8♥ 5♥ 9♣ 6♦ - All?
19: 8♠ Q♦ A♠ K♠ 6♠ - A♠ K♠
20: 3♣ 4♦ J♥ 2♣ 10♥ - J♥

6 correct.

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This seems fun.

Cards to discard


1: 7♥ 8♣
2: 8♣ 9♣ 10♠ J♠
3: 9♣ 4♠
4: 6♥ 4♦
5: Q♠ 8♣ 6♦ 7♠
6: 5♦ 7♦
7: 7♥ 2♠ 5♠ J♣
8: 8♣ 9♠
9: 3♦ 6♠ 4♠
10: 7♠ Q♥
11: 4♠ 5♠ 6♠
12: NONE
13: 2♣ 4♣ 5♣
14: 3♣ 4♣ 5♣
15: J♠ 2♥ 8♦ Q♠
16: NONE
17: 3♦ 4♦ 6♦ 5♠
18: 2♦ 8♥ 5♥ 9♣ 6♦
19: 8♠ Q♦ K♠ 6♠
20: 3♣ 4♦ J♥ 2♣ 10♥

10 correct.

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1 more time

Cards to discard


  1:  7♥  8♣

  2:  K♠

  3:  9♣  4♠ J♣

  4:  6♥  4♦

  5:  Q♠  8♣  6♦  7♠

  6:  5♦  7♦ 

  7:  7♥  2♠  5♠  J♣

  8:  J♦  8♣  9♠

  9:  3♦  6♠  4♠ 

 10:  7♠  Q♥ 

 11:  7♣

 12:  NONE

 13:  3♠

 14:  2♠

 15:  J♠  2♥  8♦  Q♠

 16:  NONE

 17:  5♠

 18:  2♦  8♥  5♥  9♣  6♦

 19:  8♠  Q♦  K♠  6♠

 20:  3♣  4♦  J♥  2♣ 10♥

Edited by bushindo
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1 more time

Cards to discard


1: 7♥ 8♣
2: K♠
3: 9♣ 4♠ J♣
4: 6♥ 4♦
5: Q♠ 8♣ 6♦ 7♠
6: 5♦ 7♦
7: 7♥ 2♠ 5♠ J♣
8: J♦ 8♣ 9♠
9: 3♦ 6♠ 4♠
10: 7♠ Q♥
11: 7♣
12: NONE
13: 3♠
14: 2♠
15: J♠ 2♥ 8♦ Q♠
16: NONE
17: 5♠
18: 2♦ 8♥ 5♥ 9♣ 6♦
19: 8♠ Q♦ K♠ 6♠
20: 3♣ 4♦ J♥ 2♣ 10♥

10 correct, having 7 overlaps with your earlier score of 10.

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