Hand Groups Formula, Part 2

[YY] Having finished his homework¹, Yuri the Young Gun has reconvened with Benny and Joey, “Okay, I got it. For convenience, here’s the full-table formula again.

• If your hole cards are paired:
• Aces through Tens are Hand Group 1; Nines through Sixes are 11 minus the denomination (i.e., Group 2 through 5), and Fives and lower are Hand Group 6. [As a list, AA-TT:1, 99:2, 88:3, 77:4, 66:5, 55-22:6.]
• If your hole cards are unpaired:

1. If your higher card is an Ace, start with 1, King is 3, Queen is 4, Jack is 5, and otherwise 11 minus half the denomination rounded up; e.g., a Seven is 7 (11 – 4). [As a list, A:1, K:3, Q:4, J:5, T:6, 9:6, 8:7, 7:7, 6:8, 5:8, 4:9, 3:9.]
2. Add 1 if the gap is 1; add 2 if the gap is 2; add 4 if the gap is 3; and add 5 if the gap is bigger (e.g., for QJ add nothing, for J9 add 1, for 74 add 2, for J7 add 4, and for A2 [gap of 11] add 5).
3. Subtract 1 if the gap is 0 or 1 and the higher card is a Jack or lower.
4. Subtract 2 if your cards are suited. Anything under 1 is Hand Group 1, while anything over 10 is Hand Group 10.”

[YY] “Joey, compared to full table play, how do you think values should change when you’re heads up?”

[JJ] “What do you mean?” queried Joey.

[YY] “Each set of hole cards has three basic features: high card strength, connectedness, and suitedness. Which of these is worth more and which is worth less with fewer opponents?”

[JJ] “Well, my dad says that you want to play your suited connectors against more opponents so that you can get paid off when you hit. That means that being suited and connected (other than pairs) are worth less heads up.”

[YY] “Excellent! You’re really picking up this game quickly. On the flip side then, pairs and high cards are worth more, since they’ll hold up more often against just a single opponent. Those adjustments lead to the following formula for Heads-Up Hand Groups.”

• If your hole cards are paired:
• Aces through Sixes are Hand Group 1; Fives and lower are 7 minus the denomination (i.e., Group 2 through 5). [As a list, AA-66:1, 55:2, 44:3, 33:4, 22:5.]
• If your hole cards are unpaired:

1. If your higher card is an Ace, start with -1, King is 1, Queen is 2, Jack is 4, Ten is 6, and otherwise 12 minus half the denomination rounded up; e.g., a Seven is 8 (12 – 4). [As a list, A:-1, K:1, Q:2, J:4, T:6, 9:7, 8:8, 7:8, 6:9, 5:9, 4:10, 3:10.]
2. Add 1 if the gap is 2; add 2 if the gap is 3; and add 4 if the gap is bigger (e.g., for QJ or J9 add nothing, for 74 add 1, for J7 add 2, and for A2 [gap of 11] add 4).
3. Subtract 1 if your cards are suited. Anything under 1 is Hand Group 1, while anything over 10 is Hand Group 10.”

[BB] “That sounds very close to the full table formula, but with most of the numbers changed.”

[YY] “Yes, I think that might make it hard to remember those two formulas separately. So instead, I present you the combined formula.”

Hand Groups Formula for Any Size Table²

• Paired Hole Cards
• Aces through Tens: Hand Group 1.
• Nines and Lower: Start with half the table size rounded up. Add 6. Subtract the denomination. Anything under 1 is Hand Group 1; anything over 6 is Hand Group 6.
• Unpaired Hole Cards: For the purposes of this formula, table sizes are 8-10 players (big), 5-7 players (medium), and 2-4 players (small).

1. Higher Card Score:

• Aces through Tens: Start with 1 for an Ace and 16 minus the demonimation otherwise. For Aces through Queens, subtract 1 (medium) or 2 (small). For Jacks, subtract 1 (medium and small). [As a list, A:1/0/-1, K:3/2/1, Q:4/3/2, J:5/4/4, T:6.]
• Nines and Lower: Start with 12 (big) or 13 (medium and small). Subtract half the denomination rounded up.

• Gap of 1: Add 1 (big and medium).
• Gap of 2: Add 2 (big and medium) or 1 (small).
• Gap of 3: Add 4 (big and medium) or 2 (small).
• Gap of 4+: Add 5 (big and medium) or 3 (small).

[YY] “Some examples to help you learn this:

• 9♠9♥ with 8 players (big): 8 / 2 = 4 + 6 – 9 = Hand Group 1.
• 7♦7♣ with 5 players (medium): 5 / 2 = 3 + 6 – 7 = Hand Group 2.
• 3♥3♦ with 4 players (small): 4 / 2 = 2 + 6 – 3 = Hand Group 5.
• A♣K♣, 8 players (big): 1 (higher card) + 0 (gap adjustment) – 0 (gap readjustment) – 2 (suited adjustment) = -1 = Hand Group 1.
• A♥Q♠, 6 players (medium): 0 (higher card) + 1 (gap adjustment) – 0 (gap readjustment) – 0 (suited adjustment) = Hand Group 1.
• A♣10♦, 2 players (small): -1 (higher card) + 2 (gap adjustment) – 0 (gap readjustment) – 0 (suited adjustment) = Hand Group 1.
• J♥10♥, 8 players (big): 5 (higher card) + 0 (gap adjustment) – 1 (gap readjustment) – 2 (suited adjustment) = Hand Group 2.
• 10♠8♠, 2 players (small): 6 (higher card) + 0 (gap adjustment) – 0 (gap readjustment) – 1 (suited adjustment) = Hand Group 5.
• 9♦4♥, 9 players (big): 12 – 5 (higher card) + 5 (gap adjustment) – 0 (gap readjustment) – 0 (suited adjustment) = 12 = Hand Group 10.
• 7♥5♠, 5 players (medium): 13 – 4 (higher card) + 1 (gap adjustment) – 1 (gap readjustment) – 0 (suited adjustment) = Hand Group 9.
• 7♣2♦, 2 players (small): 13 – 4 (higher card) + 3 (gap adjustment) – 0 (gap readjustment) – 0 (suited adjustment) = 12 = Hand Group 10.

[YY] “Class dismissed.”³

Concise Hand Groups Formula for Any Size Table

• Paired Hole Cards
• AA-TT: 1
• 99-22: T/2 + 6 – D (max of 6)
• Unpaired Hole Cards

1. Higher Card Score:

• A-T: 16 – D [-1 for A] [-1 for A/K/Q medium] [-2 for A/K/Q small] [-1 for J medium/small]
• 9-3: 12 – D/2 [+1 medium/small]

• 1: +1 big/medium
• 2: +2 big/medium, +1 small
• 3: +4 big/medium, +2 small
• 4+: +5 big/medium, +3 small

T = Table Size (number of players at table); 8-10 (big), 5-7 (medium), and 2-4 (small)

D = Denomination (A = 14, K = 13, Q = 12, J = 11)

Footnotes:

1. Sunday’s blog post was about Pre-Flop Hand Groups, and Tuesday’s was the Hand Groups Formula for full tables.
2. I’d like to call this the Chen-Jen Hand Groups Formula for Any Table Size, but if Bill would like his name taken off, no offense will be taken. I just liked the rhyme.
3. Behind the scenes note: THETA Poker Pro of course has no problems memorizing 169 values, so it uses one static lookup table for 10-player tables and another for heads-up. For all table sizes in between, it just takes the weighted average of the two values (e.g., the 5-player value is 3/8 of the 10-player value plus 5/8 of the heads-up value).