Yatzy - Game 3 strategies
Yatzy - Game 3 strategies
Perhaps some of the techniques below can be disproved mathematically, but below are things
I do for high scores:
Don't waste time on a game that is going nowhere. Once I get too many bad rolls which cannot
be entered anywhere satisfactorily, I restart the game. I only open a game with the following
scores:
30 in 6's, 25 in 5's, 20 in 4's, 15 in 3's, or Yatzy (with 1's or 2's)
4-of-a-kind used in 1's-5's (4, 8, 12, 16, or 20)
24 in 4-of-a-kind
27 or 29 in Chance
Small or Large Straight
26-28 in Full House
22 in 2 Pairs
Anything else is restarted after the first turn. It is questionable whether to start with 18
in 3-of-a-kind or 12 in pairs. They are the highest scores which can be obtained, but occur
so often and I almost always need to dump 2 or 3 6's or 5's in there sometime later. Though
it is possible to get a 369 with a 0 in 1's to start, there is a much better chance of a great
game starting it with an excellent turn. It can be argued that the best strategy is to only
start games with 5 of-a-kind, but so often they are followed with bad turns and can occur
later in the game. I am currently placing 15 in 3's sometimes if it is the first 5 of-a-kind
not late in a game. An argument can be made that all 5 of-a-kinds should be placed in the
left section if possible and only reserve Yatzy for those with dice which have already been
used there. Doing so would produce many fewer high scores, but would more likely produce the
ultimately high score. 354 is extremely high, but I had one game in which I needed a 6 with
2 dice the to complete a 24 in 4-of-a-kind and then would need a 22 in 2 Pairs (could've done
that rolling 2 5's the previous roll) and 15 in 3's for either a 361 or 364 (don't recall
which). Even the 24 & 22 with 9 in 3's would be at least 355. Also has a game in which I had
Yatzy & 3's left with a total of 295. This calculation shows (if I made no mistakes - hard to
do probably) that the odds of getting both at that point for the record was about 1 in 131 -
not good, but realistically possible.
So I guess the above paragraph is really saying that you can choose to put up alot of lower
scores or fewer high scores and try using a strategy to get them a little more often. I wonder
how many games it would take a computer to get a 370 or 374 - that sort of calculation is a
little beyond my capabilities¤.
I can play for a game of 340 or better for an hour without finishing a game.
Too many good scores ruin a game. 25 in Chance, 9 in 3's, 15 in 3 of-a-kind, etc. are often
necessary to keep a game going, but better scores are generally needed. If more than one of
these are among the first several turns, I restart.
Always place 5-of-a-kind in 6's & 5's if they are open, and usually 4's. Very high scores occur
when that can be combined with a 50 in Yatzy. I had 97 in the left section once.
In general, place a roll of 4 6's as 24 in 4-of-a-kind first, unless it is 29 - then place it
in Chance (can only do 1 better). Don't take away the possibility of a 30 in 6's early in the
game (as I recall, I started the 354 game on 7/29/2006 with a 3 in 1's & 29 in Chance - later
getting a 20 in 4's, 30 in 6's, and a Yatzy in the next to last turn).
The next 354 game (10/13/2006), I only had 2 Yatzys - 50 in that & 30 in 6's. Here were the
scores (didn't write them down first time):
1's: 3 Pairs: 12
2's: 8 2 Pairs: 22
3's: 12 3ok: 18
4's: 16 4ok: 24
5's: 20 SS: 15
6's: 30 LS: 20
Total: 89 FH: 28
Bonus: 50 Y: 50
Left Total: 139 CH: 26
Right Total: 215
Grand Total: 354
In that game, I had 1's & 3 of-a-kind left with 2 turns to play and rolled as I recall 3 1's
the first roll. Thus I decided to roll for 1's rather than giving myself 2 chances to get
a good score in 3 of-a-kind, but got no more than 3 1's (and followed it with the 18 in
3 of-a-kind to tie the record). Earlier in the game I stopped after 2 rolls with a 6 6 5 5 4
for a 26 in Chance (after having Full House & 2 Pairs). Though the probability of doing better
than that rerolling the 4 is only 1/3 and that of doing worse is 1/2, it might be better to
try that. Doing so in this particular game would not only have changed that roll, but it seems
every subsequent one (at least with that dice - don't know what the program does). So it is
unlikely this game would've been nearly as good, but it is probably the better strategy in
general for a very high score.
The 352 game the same day included 5 Yatzy's, 4 of which I recall occurred in the first 5 turns
(maybe first 6) - Yatzy, 30 in 6's, 20 in 4's, 30 in Chance, & Large Straight (maybe 18 in
3 of-a-kind also) - thought I might get a perfect game for awhile 
As I recall, these were
the scores (didn't write them down, though am certain of where I placed the 5 of-a-kinds, the
total of 87 in the left section, and the 24 in FH):
1's: 3 Pairs: 12
2's: 10 2 Pairs: 22
3's: 9 3ok: 18
4's: 20 4ok: 24
5's: 15 SS: 15
6's: 30 LS: 20
Total: 87 FH: 24
Bonus: 50 Y: 50
Left Total: 137 CH: 30
Right Total: 215
Grand Total: 352
Scoring well in Chance is quite helpful for a high game. The 15 in 5's came soon after the run
of Yatzy's as I recall, and 18 in 3 of-a-kind was already used. A thing to note is how I used
the 30 in Chance instead of 24 in 4 of-a-kind, and finished the game with the 24 in 4 of-a-kind.
I did not want to give away the other 6 but use it somewhere if possible.
Just as the 15 in 5's above (though I am very happy with the 352), 3 4's is a game ruiner -
nothing really good can be done with it. A perfect score is 374. 354 is only 20 points worse,
and a 12 in 4's is 8 worse. The best which can be done with that roll is place an 8 in Pairs,
but even then a 12 in pairs is among the easiest of things to roll. Pairs is (of course)
typically used like a "chance" category - somewhere to dump 2 6's when rolling for something
better. 3 3's only loses 6 points and thus isn't as bad. Thus I always try to get a 16 or 20
in 4's, and try to avoid rolling for 4's in a game with some high scores already unless I have
3 or more after the first roll. I can tolerate 12 in 4's if I already have a 25 in 5's &/or
30 in 6's. 15 in 3-of-a-kind or 10 in Pairs aint so bad. 24 in 6's is similar to 12 in 4's
in a sense that it loses 6 points from a perfect score and is seldom part of a game with very
high score.
Rolling for a straight after getting an excellent roll like a 30 in 6's during a game that
is going good is quick way to ruin it, though this is exactly what sustains it in many cases.
Yet straights can also pop up in the first or subsequent roll when they are possible - a
convenient way to get them when nothing else good comes. So it is often good to keep a single
6 and roll instead of using 2 3 4 4 6 and trying for a 5 for example early in the game.
Really good games have to sometimes be ruined to try for a great one.
Suppose the first roll of the game is 1 2 3 5 6. It is good to roll for a straight and get one
of them to begin with, because they are so difficult to get - then you can focus more on Yatzy,
4 of-a-kind, and high scores in the left section without having to sneak another straight in
somewhere (most potentially great games are ruined because I cannot get a straight which is
needed - though they often already include a Yatzy). I roll for the Small Straight in this case,
because I may be left with a Straight and Pairs or Chance as the last 2 categories. In that
case, I want to be rolling for the Large Straight instead of the Small, because being able to
keep a 6 can improve the score in Pairs or Chance. Every category must be scored in for a great
game (except perhaps 1's or 2's), and the chances when rolling for them are the same.
If the first roll is 6 6 5 5 X, for which X is 1-4, I try for 27 or 28 in Full House. This is
not an easy thing to get, though in the first 354 game I was fortunate to have rolled it in
the last turn (a 6 & 5 kept after the first roll, then another 6 & 5, then a 6). Same for 22
in 2 Pairs - I put the score there if I don't get the 5 or 6 (if I can). Many potentially high
scores were ruined because I could not get a good score if any in 2 Pairs - can be deceptively
difficult. Yet if 2 Pairs & Chance are open and Full House used, I sometimes place a 6 6 6 5 5
in Chance though a 6 6 5 5 5 in 2 Pairs (depends on the situation). The average score in Chance
when rolling for it is 23 1/3 (23.333...).
I try to dump bad rolls such as 2 or 1 of one dice in 1's & 2's if I must, though in the first
354 game I made it a point to roll for 2's rather than a straight and got a 6 as I recall.
Alot of 6's are needed in any great game - so when in doubt, I keep the 6's. Same thing for
entering scores - when in doubt, I enter the score which loses the fewest points from the best
possible in that category.
Try to avoid mistakes such as not highlighting a dice, entering a score in 1's or Pairs when
meaning to roll or elsewhere by mistake (especially Full House & Chance which are next to each
other and the same score), double-clicking a roll (especially if such can ruin a game that is
already going great, but even doing this early in the game could ruin what would've been one),
and forgetting to get the code and accidentally entering a previous score or nothing.
Probability of 5-of-a-kind is 1 about in 21.7 (.04603 or 4.6%) when rolling for it.
When rolling for a specific dice such as 6's, the chance of a 5-of-a-kind is about 1 in 75
(.01327 or 1.3%), 4-of-a-kind is about 1 in 10 (.10443 or 10.4%), 3-of-a-kind is about 5 in
14 (.35485 or 35.5%), and 2-of-a-kind is about 7 in 10 (.69884 or 69.9%). That is useful if
for example 3 or 4-of-a-kind and a couple other spaces are left late in a game. Exact numbers
are shown below.
Probabilities are nice to know, but a great game is necessarily an incredibly lucky one. So my
idea is simply to increase the poor odds as much as possible as going along and trying to be in
a position to take advantage of some lucky rolls at the end (would be nice to have 1's, Chance,
or Pairs as the last category, but that rarely can be done with a very high score). For example,
getting to the point where a Yatzy is needed on the last roll to break the record 15 times
should on the average accomplish that. I.e., (1-.046)^15 = .493, which means about 51% of the
time the Yatzy is obtained in 15 rolls. That was my basic strategy to break the 348, though
I had only one game prior to that when a Yatzy on the last roll would've done it (2 when 30
in 6's would've), and a couple others of slightly better probability with 2 turns left - about
1 in 9 & 1 in 15. The one which did it had a much worse probability at that point, and still
only 1 in 3 when rolling for the last 5 or 6.
-----
These probabilities are correct if I made no mistakes:
5-of-a-kind: 347897/7558272 = .04603
For a specific dice (3-of-a-kind or more, etc.):
5-of-a-kind: 6240321451/470184984576 = .013272
4-of-a-kind: 12274918019/117546246144 = .10443
3-of-a-kind: 27807523471/78364164096 = .35485
2-of-a-kind: 82145855519/117546246144 = .69884
1-of-a-kind: 439667406451/470184984576 = .93509
Here are examples of how I calculated these for a Yahtzee of a specific dice and any Yahtzee.
-----
¤ Such a program would show for example where to place a 5-of-a-kind the first roll to obtain
a certain score in the fewest expected games. E.g., to get a 314 or higher quickest, it may be
best to place only 30 in 6's but everything else in Yatzy, to get a 330 or higher, add 5's to
that, etc. It would probably show not to start with a 5 in 1's instead of 50 in Yatzy unless
you are going for 367 or so 
This web site:
Solitaire Yahtzee: Optimal Player and Proficiency Test
includes a program to maximize the expected score of the standard Yahtzee game with bonus chips,
and provides links to other calculations done and programs of the sort. These are all to my
knowledge done using decimal rather than rational math, though there is a rational number
(i.e., numerator divided by denominator as those shown above, though both huge) which is the
average score of Yahtzee played optimally. In the Trivia link:
Optimal Solitaire Yahtzee Player: Trivia
and in this PDF document:
Optimal Solitaire Yahtzee Strategies
Tom Verhoeff shows the decimal value to be 254.59 using bonus chips and 245.87 without them.
Doing the Yahtzee Proficiency Test at the site above, I usually get mean deltas per choice in
the .1 to .3 range, though occasionally get a game where I make every correct choice. A problem
I have is that I don't like playing with bonus chips, so don't generally think in a way to
optimize score with them but rather without them. Below is an example I just decided to do
which was close to being optimal (3rd game done when typing this) - a good game in which most
decisions were obvious. Both times I was wrong are probably because keeping the pair rather
than the non-pair of higher dice made a Yahtzee and thus a subsequent bonus one(s) more likely.
Felix Holderied has a program for the game with no bonus chips or Yahtzees as Jokers here:
Über das perfekte Kniffel-Spiel
Note that maximizing your average greatly differs from trying to get one game with a very high
score.
Game Details and Analysis
==== ======= === ========
# Game State Tot Roll You Expect SD OSYP Expect SD Delta
-- ----------------- --- ------- ----- ------ -- ----- ------ -- -----
1 123456TFHSLYC;63- 0 33566;2 66___ 253.94 59
; 13666;1 666__ 258.13 58
; 16666;0 6 24 268.23 53
2 12345_TFHSLYC;39- 24 12345;2 L 40 275.70 49
3 12345_TFHS_YC;39- 64 22666;2 666__ 283.09 52
; 36666;1 6666_ 292.31 54
; 26666;0 F 26 282.03 44
4 12345_T_HS_YC;39- 90 45566;2 5566_ 279.84 41 55___ 281.81 43 1.97 **
; 55666;1 H 25 283.29 40
5 12345_T__S_YC;39- 115 22446;2 44___ 282.37 39
; 44445;1 4444_ 297.12 45
; 34444;0 4 16 288.12 35
6 123_5_T__S_YC;23- 131 13556;2 55___ 288.12 34
; 14555;1 555__ 290.07 34
; 35556;0 5 15 286.93 31
7 123___T__S_YC; 8- 146 33446;2 33___ 286.17 30
; 11336;1 33___ 283.53 28
; 33346;0 3 44 284.35 27
8 12____T__S_YC; 0- 190 12234;2 S 30 283.95 25
9 12____T____YC; 0- 220 23334;2 333__ 283.92 29
; 33366;1 66___ 281.59 22
; 22666;0 T 22 281.30 22
10 12_________YC; 0- 242 11255;2 11___ 280.54 20
; 11556;1 556__ 278.40 18 11___ 278.80 18 0.40 *
; 15556;0 1 1 277.05 17
11 _2_________YC; 0- 243 22335;2 22___ 277.53 16
; 22346;1 22___ 275.56 14
; 22456;0 2 4 274.34 13
12 ___________YC; 0- 247 22345;2 5____ 272.49 10
; 15566;1 5566_ 274.80 11
; 55566;0 C 27 276.30 10
13 ___________Y_; 0- 274 12236;2 22___ 275.45 8
; 22333;1 333__ 275.39 8
; 33334;0 Y 0 274.00 0
-- ----------------- --- ------- ----- ------ -- ----- ------ -- -----
274 2.37
# Choices (one option = no choice) : 33 total
# Choices identical to OSYP : 31
# Choices with delta < 0.01 : 0 <>
# Choices with 0.01 <= delta < 1.00 : 1 *
# Choices with 1.00 <= delta < 5.00 : 1 **
# Choices with 5.00 <= delta : 0 ***
Mean delta per choice : 0.072
Score Card
===== ====
1 Aces
4 Twos
9 Threes
16 Fours
15 Fives
24 Sixes
35 UPPER SECTION BONUS
22 Three of a Kind
26 Four of a Kind
25 Full House
30 Small Straight
40 Large Straight
0 Yahtzee
27 Chance
0 EXTRA YAHTZEE BONUS
----- -------------------
274 GRAND TOTAL
0 Yahtzees Rolled
0 Jokers Applied
©2006, Joseph Bartlo
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