5 dice alternative to Craps

Quote: MattUK

Quote: charliepatrick

- Two pairs, keep one pair (K Q J or T if possible) - If you only keep the highest pair it's 99.68%.
- Pair, play the pair unless there's an easy straight draw (e.g. KKQJT).
- Draw one card to Two Pairs.

Ehm... Charlie... it's a wrong thread. This one is about a dice game. :-)

Possibly you missed my earier comment that I was going to use poker dice as it makes it easier to think about Straights etc. So it was easier for me to see that "KQJT" is an outside straight draw than "5432". Sorry if that caused confusion.

Quote: MattUK

You can't talk about the second decision without resolving the first one (what to do in all cases). Probably that's why you have that funny 23.84% RTP. It's like American Roulette with 29 zeros. Please consider looking on that Japanese website and translate using Google.
https://www.a-taichi.com/tablegame/nicedice.html
It's not easy to go through his tables, but he did it Wizard-style. Ideally, we should also have EV for any case to be able to compare them and see the increase (unless it was a risk worth taking, but the roll decreases the EV, which in principle may happen).
Here is his summary. I am trying to figure out what the oriental names describe.
https://drive.google.com/file/d/1lk10B2ZT5NfidfwgKf-ZDxDXPlTEGZdS/view?usp=sharing

Here's the first roll strategy:
Five Of A Kind - hold all five
Four Of A Kind - hold the four; reroll the other die
Straight - hold all five
Full House - hold all five
Three Of A Kind - hold the three; reroll the other two dice
Two Pair - hold one of the pair; reroll the other three dice
2345 and a Pair - keep 2345, and break up the pair
One Pair (other than with 2345) - hold the pair, and reroll the other three
12346, 12356, 12456, 13456 - reroll the 1 and the 6, and keep the other three

The RTP is 99.5376324%


Here's the complete strategy:

Five Of A Kind - hold all five

Four Of A Kind - hold the four; reroll the other die

Straight - hold all five

Full House - hold all five

Three Of A Kind - hold the three; reroll the other two dice

Two Pair
First roll: hold one of the pair; reroll the other three dice
Second roll: hold both pairs; reroll the other die

One Pair
First roll: if you have 2345, keep the 2345, and break up the pair; otherwise, hold the pair, and reroll the other three
Second roll: if you have a 1 and a 6, hold the pair, and reroll the other three; otherwise, keep the 4 to a straight (even an inside straight), and break up the pair

12346, 12356, 12456, 13456
First roll: reroll the 1 and the 6, and keep the other three
Second roll: reroll the 1 (or the 6), and keep the other four

I still think that to mark the job done we should have return values for every case. Can the EV go backward (when the risk is worth taking but backfires)?

Initial roll	First decision	Second decision
Five of a kind	stand	stand
Four of a kind	reroll unmatched die	reroll unmatched die
Straight	stand	stand
Full House	hold Three of a kind, reroll the other two dice	stand
Three of a kind	reroll the other two dice	reroll the other two dice
Two Pair	hold inside Pair if you can; reroll the other three dice	hold both pairs; reroll the other die
One Pair and 2345	hold the 2345; reroll the paired die	hold the 2345; reroll the paired die
One Pair and one Outsider*	hold the pair, and reroll the other three	hold 4 to a Straight, and reroll the paired die
One Pair and both Outsiders*	hold the pair, and reroll the other three	hold the pair, and reroll the other three
13456, 12456, 12356, 12346	reroll both Outsiders*, and hold the other three	reroll either Outsider*, and hold the other four

* Outsiders are numbers 1 and 6

The above table takes into account subsequent discussion and corrections.

Everything is correct except for the One Pair (and even then, One Pair and 2345 is correct):

Initial roll	First decision	Second decision
One Pair and 2345	hold the 2345; reroll the paired die	hold the 2345; reroll the paired die
One Pair, and either but not both 1 or 6**	hold the pair, and reroll the other three	hold 4 to a straight, and reroll the paired die
One Pair, and both 1 and 6**	hold the pair, and reroll the other three	hold the pair, and reroll the other three

** note the 1 and 6 may or may not be in the pair; for example, both 1 2 4 4 5 and 2 3 5 6 6 are "one pair, and either but not both 1 or 6," and both 1 1 2 3 6 and 1 4 4 5 6 are "one pair, and both 1 and 6"

Apologies for using AKQJT9 rather than 123456 (or 654321). I now agree with the RTP but there's still the need to hold an inside Pair when you choose which Pair to hold from Two Pairs after the first roll. Also note you only keep the Trips after the first roll with a Full House, but you keep the Full House after the second roll.

Initial Roll Outcome
Nice Dice (AAAAA)	6	10.000
Quads (AAAAK)	150	2.958
Straight (AKQJT)	240	2.000
Trips or FH (AAA??)	1 500	1.539
Easy Str Draw (KKQJT)	240	1.111
2 Pair As & 9s (AA99x)	120	0.742
2 Pairs (AAKKx)	1 680	0.755
Pair As or 9s (AAxxx)	1 200	0.742
Pair Ks Qs Js Ts (KKxxx)	2 160	0.755
Inside Straight (AKQJ9)	480	0.634
First Roll
Nice Dice (AAAAA)	10.000	Stand
Fiver (AAAAA)	4.000	Stand
Quads (AAAAK)	2.958	Draw one
Full House (AAAKK)	1.000	(i) Stand
Full House (AAAKK)	1.539	(ii) Keep Trips
Straight (AKQJT)	2.000	Stand
Trips (AAAxx)	1.539	Keep Trips
Pair / E Str (KKQJ T)	0.755	(i) Keep Pair (K Q J or Ts)
Pair / E Str (KKQJ T)	1.111	(ii) Straight Draw
Pair / I Str (AAKQJ)	0.742	(i) Keep Pair (As or 9s)
Pair / I Str (AAKQJ)	0.611	(ii) Straight Draw
Pair / I Str (KKxxx)	0.755	(i) Keep Pair (K Q J or Ts)
Pair / I Str (KKAQJ)	0.611	(ii) Straight Draw
Two Pair (AA99x)	0.742	(i) Keep Pair (As or 9s)
Two Pair (AA99x)	0.556	(ii) Keep Two Pair
Two Pair (KK99x)	0.755	(i) Keep Pair (K Q J or Ts)
Two Pair (KK99x)	0.556	(ii) Keep Two Pair
Pair (As or 9s) (AAxxx)	0.742	(i) Keep Pair (As or 9s)
Pair (Ks Qs Js Ts) (KKxxx)	0.755	(i) Keep Pair (K Q J or Ts)
Inside Str (AKQJ9)	0.611	(i) Keep 4 to Str
Inside Str (AKQJ9)	0.634	(ii) Keep KQJ
Inside Str (AKQT9)	0.611	(i) Keep 4 to Str
Inside Str (AKQT9)	0.634	(ii) Keep KQT
Inside Str (AKJT9)	0.611	(i) Keep 4 to Str
Inside Str (AKJT9)	0.634	(ii) Keep KJT
Inside Str (AQJT9)	0.611	(i) Keep 4 to Str
Inside Str (AQJT9)	0.634	(ii) Keep QJT
Second Roll
Nice Dice (AAAAA)	10.000	Stand
Fiver (AAAAA)	4.000	Stand
Quads (AAAAK)	2.750	Draw one
Full House (AAAKK)	1.000	(i) Stand
Full House (AAAKK)	0.944	(ii) Keep Trips
Straight (AKQJT)	2.000	Stand
Trips (AAAxx)	0.944	Keep Trips
Pair / E Str (KKQJ T)	0.285	(i) Keep Pair
Pair / E Str (KKQJ T)	0.667	(ii) Straight Draw
Pair / I Str (AAKQJ)	0.285	(i) Keep Pair
Pair / I Str (AAKQJ)	0.333	(ii) Straight Draw
Two Pair (AAKKx)	0.285	(i) Keep Pair
Two Pair (AAKKx)	0.333	(ii) Keep Two Pair
Pair (AAxxx)	0.285	Keep Pair
Inside Str (AKQJ9)	0.333	Keep 4 to Str
Return to player	99.537 632%

Quote: charliepatrick

I now agree with the RTP but there's still the need to hold an inside Pair when you choose which Pair to hold from Two Pairs after the first roll. Also note you only keep the Trips after the first roll with a Full House, but you keep the Full House after the second roll.

Great! Now we're in agreement with the Japanese guy! He had 10 different outcomes, wrote "Hold one pair in the middle" for the first decision having Two Pair and that it's better to reroll two unmatched dice having Full House.
I am beginning to understand what ThatDonGuy had in mind. Somehow Mr. Toyoda did the same, but that was too fragmented piece of the puzzle to understand without background, sorry.
I corrected the table and added "if you can" because it's still possible to roll 1166 and inside number.
PS: Nice Dice is not formed after a decision. They can be removed from the bottom two return tables. Hope that doesn't change anything.

It was laborious, but here is the final answer. My thanks to charliepatrick and thatdonguy for doing the calculations.

"First decision" table

Initial roll	First decision	Expected value
Five of a kind	stand	4.000
Four of a kind	reroll unmatched die	2.958
Straight	stand	2.000
Three of a kind	hold three of a kind, reroll the other two dice	1.539
Full House	hold three of a kind, reroll the other two dice	1.539
insider Pair and 2345	hold the 2345, reroll the paired die	1.111
insider Pair and 1 or 6	hold the pair, reroll the other three	0.755
11 or 66 Pair	hold the pair, reroll the other three	0.742
Two Pair 11 and 66	hold either pair, reroll the other three	0.742
any other Two Pair	hold insider pair, reroll the other three	0.755
13456, 12456, 12356, 12346	reroll 1 and 6, hold three insiders	0.634

"Second decision" table

Second roll	Second decision	Expected value
Five of a kind	stand	4.000
Four of a kind	reroll unmatched die	2.750
Straight	stand	2.000
Three of a kind	reroll the other two dice	0.944
Full House	stand	1.000
insider Pair and 2345	hold the 2345, reroll the paired die	0.667
insider Pair and 1 or 6	hold 4 to a Straight, reroll the paired die	0.333
Two Pair 11 and 66	hold both pairs, reroll the other die	0.333
any other Two Pair	hold both pairs, reroll the other die	0.333
13456, 12456, 12356, 12346	reroll 1 or 6, hold the other four	0.333
11 or 66 Pair	hold the pair, reroll the other three	0.285

Strategy for Two Pairs can be merged into "hold insider Pair if you have it, otherwise either pair; reroll the other three". Likewise, some cells can be merged if EV is ditched.
Note that the EV decrease in the second round for every outcome, except instant winners Five of a kind and Straight.
For the avoidance of doubt, Yahtzee aka Nice Dice returning 10x is an automatic winner after the initial roll and hence cannot be formed.
Just maybe some people would like to have all five columns in a landscape and print out horizontally. It can be done, but not exactly using html.
And finally, here's their "handy video tutorial":
https://www.linkedin.com/posts/roxor-gaming_roxorgaming-wemakeplaymorerewarding-igaming-activity-6689475826964033536-aZk9/

I have a feeling this game will have a much greater probability of success in inner city casinos...

Quote: MattUK

It was laborious, but here is the final answer. My thanks to charliepatrick and thatdonguy for doing the calculations.

"First decision" table

Initial roll	First decision	Expected value
Five of a kind	stand	4.000
Four of a kind	reroll unmatched die	2.958
Straight	stand	2.000
Three of a kind	reroll the other two dice	1.539
Full House	hold Three of a kind, reroll the other two dice	1.539
insider Pair and 2345	hold the 2345, reroll the paired die	1.111
insider Pair and 1 or 6	hold the pair, reroll the other three	0.755
11 or 66 Pair	hold the pair, reroll the other three	0.742
Two Pair 11 and 66	hold either pair, reroll the other three	0.742
any other Two Pair	hold insider pair, reroll the other three	0.755
13456, 12456, 12356, 12346	reroll 1 and 6, hold three insiders	0.634

"Second decision" table

Second roll	Second decision	Expected value
Five of a kind	stand	4.000
Four of a kind	reroll unmatched die	2.750
Straight	stand	2.000
Three of a kind	reroll the other two dice	0.944
Full House	stand	1.000
insider Pair and 2345	hold the 2345, reroll the paired die	0.667
insider Pair and 1 or 6	hold 4 to a Straight, reroll the paired die	0.333
Two Pair 11 and 66	hold both pairs, reroll the other die	0.333
Tany other Two Pair	hhold both pairs, reroll the other die	0.333
13456, 12456, 12356, 12346	reroll 1 or 6, hold the other four	0.333
11 or 66 Pair	hold the pair, reroll the other three	0.285

The tables can be slightly simplified by giving recommendation "hold insider Pair if you have it, reroll the other three".
Note that the EV decreas in the second round for every outcome, except Five of a kind and Straight.

Nice table. Only commented is EV on first roll for five of a kind. I thought this got the nice dice bonus?

Quote: unJon

Nice table. Only commented is EV on first roll for five of a kind. I thought this got the nice dice bonus?

I skipped that because it does not require a decision, but it's included in the 99.54% RTP.