as the name of the game lets expect PENTAKUBUS is to be played with 5 dice.
the aim is to collect as many points as possible by throwing different
dice combinations. so first of all it is a game of chance, but
nevertheless tactical considerations do matter too. by the way playing
it all by oneself is possible just as well as playing with several
game partners. the rules are reminding of games like Yam or Yahtzee
(which is also known by many other names) but also contain elements
of further combinatory gambles like Roulette, Rummy or Poker.
the throws
a complete game claims to fill in 72 fields (by each player). every throw
may lead to points, bonus points and possibly even to "slam points" and
always finishes with an entry in one of those 72 fields.
one throw consists of up to three attempts, and following the first and the
second attempt it is allowed to let stand any number of dice with their
actual figure on top. this of course includes the possibility of letting
stand all of the five dice after attempt #1 for instance and thus renounce
a second or even a third try. furthermore it is also possible to reload and
cast again within the last attempt some of the dice originally let stand.
with every new attempt all of the preceding ones become invalid anyhow!
as already said before each throw has to be concluded by filling in exactly one
empty field. entering zero points in this case amounts canceling the field.
the goal is now to fill in the 72 fields the best way (i.e. with as many
points as possible). how to manage this will (hopefully) follow from the
explanations coming up next.
if more than one single player is taking part in the game it may be useful to take turns
at throwing the dice and entering the fields. note: any throw should be ended by filling in
a field and reloading the dice cup, before it's the next player's turn...
an erroneous entry can only be corrected as long as the dice are still on the table.
once the dice are back in the cup, any changes are no longer permitted! in the case of an
incorrect entry noticed later (e.g. in a field beyond reach in the left or right column)
this must of course be withdrawn. the affected player has to cross out an allowed field instead.
the playing board
those 72 fields spread over three (vertical) columns with 4 sections
containing 6 fields each. consequently every column consists of 24 fields.
these are numbered on the far right, with the numbers in pink showing those fields
in which necessarily all five dice are involved in the scoring!
the six fields in the first section (A) belong to the single combinations.
that means in the first row only cast 1s count,
in the second the 2s and so on up until the sixth row,
where the 6s only count. while filling
in the fields thus the points on the not rated dice each stay disregarded.
this remark also will be valid for all the following fields! so in the best
case these first six fields can accumulate 105 points, what in practice
however seems to be rather unlikely to occur... it is worth yet to watch
the here achieved sum of points, because by reaching 60 points or more
one gets another 60 points as a bonus!
in section B we find the easy combinations as there are the following:
the 7th row counts the cast points of the lower half only
(that is 1, 2 or 3), the 8th row on the other hand those of the
upper half (4, 5 or 6).
in row #9 exclusively odd numbers are valid
(1, 3, 5) and in row #10 vice versa the even numbers
(2, 4, 6). field #12 is called major or high
and should be filled in with points as much as possible (best case with 30
which is corresponding to five 6s naturally). the name of field #11 is
minor or low and could make suppose to fill in
a rather small amount of points. but on the contrary this field too is meant
to pick up as many points as possible, but on condition that the filled in
number is lower than that in the major field or at most the same.
in other words: major has got to contain a number of points that is higher
than minor or at least the same.
and in this section one can get a bonus too, that is another 30 points
by achieving a sum of 110 points within fields #7 to #12. by the way
optimum in section two was 160 +30 points.
the third section (C) contains the middle combinations. field #13 only is to
be filled in, if all of the 5 dice are within the lower third, the middle or
all five are in the upper third.
the valid number of points is then composed of the sum on all the five dice
plus a bonus of 40 points. unlike field #7 and #8 here points can
only be won, if indeed all of the 5 dice match the mentioned requirements.
if all dice only show 2 and 3 or maybe 1 and 6, of course
they lie in a third as well but this is not evaluated here... - and a field "all dice
in upper or lower half" (as within Hexakubus) does not exist here.
corresponding to this field #14 demands all the five
dice showing an odd number or in reverse all of the five
showing an even number. that again delivers a number of points consisting
of the sum on the five dice plus a bonus of 25. and here the difference to the
rating of fields #9 and #10 should be observed too.
at a first glance the requirement "sum of 17 or 18" for
field #15 seems to be harmless, since the sum of points on five dice is most likely
to be 18 (respectively 17). but the difficulty is often to get this sum just
when you need it... reaching these 17 or 18 points delivers another bonus of
13 points which makes 30 or 31 altogether.
still remaining the three straight sequences: field #16 demands a
short straight, that is a sequence of (at least)
three successive numbers on the
five dice (e.g. 2,3,4 - or 4,5,6 as the best variant). since again
the points on the not rated dice stay disregarded, the possible number
of points to gather here amount from 6 to 15 plus a bonus of 8. the
middle straight (four numbers in a row)
in field #17 delivers corresponding
10 to 18 points plus bonus 15. and eventually the
long straight (successive numbers on
all the five dice = "straight flush") is to be entered in
field #18. the sequence 1 to 5 is worth 15 points whereas 2 to 6 yields
20 points. with the bonus of 34 this makes 49 resp. 54 points altogether.
by the way, the four possible "cross combinations" of street and pair/triplet
(such as within Hexakubus) are not evaluated here...
furthermore this section (as well as the following last one) lacks an
overall bonus, cause each field has already got its own bonus. the
optimum score for this section #3 amounts to 308 points per row
(incl. the slam bonus, look below).
finally section D holds the difficult combinations. the start
at field #19 still turns out easy with a simple twin
(a pair of dice points), which yields consequently only a bonus of 5
in addition to the points on the two rated dice. field #20 follows up
with double twins which of course includes
quadruplets as well. this delivers apart from
the points on the 4 involved dice a bonus of 10. field #21 requires a
triplet that then yields the points on the 3 rated
dice plus a bonus of 12. not necessary to mention that quadruplets or
even quintuplets may be "abused" as triplets too.
the combination triplet plus twin (a so-called
"full house") fits in field #22 and yields besides the points on the 5 dice
a bonus of 31. and a quintuplet may serve as full house just as well.
a quadruplet in field #23 even is worth 38 bonus
points in addition to the points on the 4 rated dice. and eventually
field #24 is kept for the ultimate dice combination, the
quintuplet or "pentayam"! that one yields
the sum of points on the 5 dice plus 55 more bonus points.
thus at best 334 points per row (incl. the slam bonus, look below) can be
gathered within this last section.
at the end the points of the whole column are to be added up and
entered in the last field (sum of column).
the probability
for those who are interested in mathematical details the following chapter
contains the exact probability for each of the dice combinations within
section C and D.
in view of statistics it's about "variations (including repitition)",
and the relative frequency of occurrence with the first attempt only has been
examined. (i.e. i only dealt with the probabilities of slams - see below.)
as already said before the quintuplet in the last field is the rarest
variation - it only comes up in 6 of the 7776 (= 6 to the power of 5) cases,
which equals a probability of lower than 0.001 (lower than one permille thus).
better of course does the (pure) quadruplet which occurs 150 times.
including the special case of quintuplet (quadruplet plus one more) this
amounts to 156 cases. and this way the probability increases with every
field ascending in section D up to the single pair. that one fails to appear
in only 720 of all 7776 cases, and then it necessarily must be a straight -
a short straight at least! and in section C we have different
probabilities too for each field; the exact data are as follows:
field #
13
14
15
16
17
18 |
description
third
odd/even
sum17/18
3straight
4straight
5straight |
frequency
96
486
1 560
3 480
1 200
240 |
p in ‰
012
063
201
448
154
031 |
|
field #
19
20
21
22
23
24 |
description
■■
■■ ■■
■■■
■■■ ■■
■■■■
■■■■■ |
frequency
7 056
2 256
1 656
306
156
6 |
p in ‰
907
290
213
039
020
001 |
|
dealing with the probabilities after three attempts you get a different result of course,
but these data are much more difficult to procure. at least that has been calculated concerning
the quintuplet, whose probabality then increases up to 0.045. that would mean to get 3 quintuplets
in an average complete game (in case of heading for it within each of the 72 throws).
so much for the statistics. |
|
by the way there is also existing an mega-version of this
game with even 6 ten-sided dice! within that extended variant called
HEXAKUBUS then 3 x 40 = 120 fields would exist to be filled in, and there
you will have different probabilities of course... |
the peculiarity of "slam"
all of those described combinations for the 24 fields can be put
together within the three attempts per throw. but on the other hand
any desired combination seems to be more precious if thrown in one single
attempt. (for this purpose it doesn't matter which of the three possible
attempts is to be rated. so it can be the third attempt as well, with
only three or four dice in the cup for instance.
main point is that all the dice participating in the rated combination have been
thrown within one single attempt.) this peculiarity is called "slam" and for
certain combinations within section #3 and #4 it may yield some further bonus
points. except for the short straight (field #16) and the single twin
(field #19) such slam points are available; these two combinations do
appear so often that a slam bonus does not seem to be apprporiate.
like bonus points in all fields those slam points as well increase with the
rarity of their possible occurrence. thus the slam in field #13 (all in a third)
with 12 points extra turns out pretty lush. in #14 (all odd or all even) a slam
is worth 8 bonus points. in field #15 (sum of 17 or 18) and #17 also (4 in a row)
a slam gives 6 points extra, and in field #18 (5 in a row) even 10 points.
in field #20 (double twins) and #21 (triplet) a slam yields 5 points extra each,
in #22 (fullhouse) 9 and in #23 (vierling) 11 points. and eventually a "slammed"
quintuplet (which is a so-called "grand slam") gets 15 slam points in addition.
thus the gs6 (grand slam by six = throwing five 6s in one cast) turns out to be
the most valuable combination to get, and it provides a score of 30 + 55 + 15 = 100
points! but pay attention to this: any slam-bonus can only be won by
renouncing further attempts (where still available) during the current throw!
by the way the optimum scores mentioned above (section C and D)
already include these possible slam bonuses. so the maximum amount of
points to be achieved in one column is 997 points.
the three columns
now it's not only about filling in one column with 24 fields but three of them.
the remarkable point is that the order of entering the fields is free to
choose in the central column but compulsory in the two others.
the left column inflicts descending direction (i.e. the fields have
to be filled in from top to bottom) whereas in the right column the
opposite (ascending) direction is obligatory (so these latter fields
must be filled in from bottom to top). but nevertheless all three columns
may be treated independently from each other, which means that one
can start in the right column (below) for example although not yet
having completed or even begun the other two columns.
to some relief of this difficult task may contribute the "two field rule",
valid in both of the outer columns: despite the obligatory entering
direction not only the next empty field has to be filled in, but also
the next but one can be used for this purpose. but this ease (entry
option on the next two fields) is connected with the permanent obligation,
that once an empty field has been skipped, the concerned column is closed
for further entries until this left out field is filled in. in other
words: a this way created gap first has got to be closed before going
on in the concerned column is permitted.
so at the beginning of the game after the first throw one has the
choice betwen exactly 28 fields: the opening entry may be put in one
of the first two fields of the left column (left above), in field #23 or
#24 of the riht column (right below) or in one of the 24 fields of the
central column.
at the end of course all of the three column sums have to be added up
to determine the final score (the entire sum of points over all 72
fields). the optimum score of 3 x 997 = 2991 points is but a
theoretical benchmark, surely not within reach in practice. but
after all the current "world record" already amounts to 2438 points!
however an overall score of 2000 points or more should be called
quite satisfactory... (after more than 700 documented scores the long-term
average seems to be somewhere between 2000 and 2100 points!)
hier you may find a game board
this page in german